Why do we move? Part 6

Have you ever watched a time-lapse video of peak hour road traffic at night time viewed from a tall building or stars moving across the night sky? Beautiful aren’t they? It’s quite moving to watch many wheeled-machines operated by humans on a chair behind a wheel or galactic balls of matter moving across your field of vision. 

But you know what also moves? Everything. The planets in our solar system, the Sun, the Moon the stars in and out of our Milky way galaxy, the asteroids, other galaxies, and even you, although you are sitting still in your chair or bed reading this right now with a confused and skeptical look on your face. You are still relative to the Earth’s surface, but relative to anything outside Earth, you are moving quickly across the vast space of the universe. Also the Earth is spinning and surface friction is dragging you along with the spin, so you don’t feel nauseous. When you gaze at the bright stars in the clear night sky, they seem still and silent from your perspective. But in actuality, they are moving faster than you can fathom. Your naked eye doesn’t have the capabilities of measuring blue and red shift emanating off the stars to determine its distance from your eyes and its motion relative to your eyes. 

https://en.wikipedia.org/wiki/Motion_(physics)

But what is motion?

According to physics, motion is defined as the change in the object’s position over time. It can mathematically be described in terms of displacement, distance, velocity, acceleration, speed, and time. The motion of a body is observed according to a frame of reference from the observer and the change in the body’s position relative to that frame is measured.

If the object’s position doesn’t change relative to a given frame of reference, the object is at rest, motionless, immobile, stationary, or has a constant / time-invariant position relative to its surroundings. Wahlin (1997) explained that without an absolute frame of reference, absolute motion cannot be determined. Therefore, everything in our universe can be interpreted to be in motion. 

Motion applies to various physical systems:

— Objects, Bodies

— Matter particles, Matter fields 

— Radiation, Radiation fields, Radiation particles 

— Curvature, Space-time 

— Images, shapes and boundaries 

— Waves, quantum particles 

https://en.wikipedia.org/wiki/Timeline_of_classical_mechanics

https://en.wikipedia.org/wiki/History_of_classical_mechanics

(a) Early Mechanics 

— 4th century BC: Aristotle invented the system of Aristotelian physics, later disproved. Babylonian astronomers calculate Jupiter’s position using the mean speed theorem. 

— 260 BC: Archimedes works out the principle of the lever and connects buoyancy to weight. 

— 60 AD: Hero of Alexandria published Metrica, Mechanics (on means to lift heavy objects), and Pneumatics (on machines working on pressure). 

— 350 AD: Themistius stated, that static friction is larger than kinetic friction. 

— 6th century: John Philoponus observed 2 balls of different weights will fall at nearly the same speed, postulated the equivalence principle. 

— 1020: Ibn Sīnā published his theory of motion in The Book of Healing. 

— 1021: Al-Biruni used 3 orthogonal coordinates to describe points in space. 

— 1000-1030: Alhazen and Avicenna developed the concepts of inertia and momentum. 

— 1100-1138: Avempace developed the concept of reaction force. 

— 1100-1165: Hibat Allah Abu’l-Barakat al-Baghdaadi discovered that force is proportional to acceleration rather than speed, a fundamental law in classical mechanics. 

— 1121: Al-Khazini published The Book of the Balance of Wisdom, in which developed the concepts of gravity-at-a-distance. He suggested that gravity varies depending on its distance from the centre of the universe, namely Earth. 

— 1340-1358: Jean Buridan developed the theory of impetus. 

— 14th century: Oxford Calculators and French collaborators proved the mean speed theorem. Nicole Oresme derived the times-squared law for uniformly accelerated change. However, he regarded this discovery as a purely intellectual exercise that is irrelevant to the description of any natural phenomena. This consequently lead to the failure of recognising any connection with the motion of accelerating bodies. 

— 1500-1528: Al-Birjandi developed the theory of “circular inertia” to explain Earth’s rotation. 

— 16th century: Francesco Beato and Luca Ghini experimentally contradicted aristotelian view on free fall. Domingo de Soto suggested that bodies falling through a homogeneous medium are uniformly accelerating. Soto, however, failed to anticipate many of the qualifications and refinements contained in Galileo's theory of falling bodies. Furthermore, he failed to recognise that a body would fall with a strictly uniform acceleration only in a vacuum, which would otherwise eventually reach a uniform terminal velocity, as Galileo recognised. 

— 1581: Galileo Galilei noticed the timekeeping property of the pendulum. 

— 1589: Galileo Galilei rolled balls on inclined planes to demonstrate that different weights fall with the same acceleration. 

— 1638: Galileo Galilei published Dialogues Concerning Two New Sciences (which were materials science and kinematics) where he developed Galilean transformation, among other concepts. 

— 1645: Ismaël Bullialdus argued that “gravity” is directly proportional to the inverse square of the distance. 

— 1651: Giovanni Battista Riccioli and Francesco Maria Grimaldi discovered the Coriolis effect. 

— 1658: Christiaan Huygens experimentally discovered that balls placed anywhere inside an inverted cycloid reach the lowest point of the cycloid in the same time and thereby experimentally demonstrated that the cycloid is the tautochrome. 

— 1668: John Wallis suggested the law of conservation of momentum. 

— 1676-1689: Gottfried Leibniz developed the concept of vis viva, a limited theory of conservation of energy. 

(b) Formation of classical mechanics 

— 1687: Isaac Newton published his Philosophiae Naturalis Principia Mathematica, in which he formulated Newton’s laws of motion and Newton’s law of universal gravitation. 

— 1690: James Bernoulli showed that the cycloid is the solution to the tautochrone problem. 

— 1691: Johann Bernoulli showed that a chain freely suspended from 2 points formed a catenary. 

— 1696: James Bernoulli showed that the catenary curve had the lowest centre of gravity of any chain hung from 2 fixed points. 

— 1707: Gottfried Leibniz probably developed the principle of least action. 

— 1710: Jakob Hermann demonstrated that Laplace-Runge-Lenz vector is conserved for a case of the inverse-square central force. 

— 1714: Brook Taylor derived the fundamental frequency of a stretched vibrating string in terms of its tension and mass per unit length by solving an ordinary differential equation. 

— 1733: Daniel Bernoulli derived the fundamental frequency and harmonics of a hanging chain by solving an ordinary differential equation. 

— 1734: Daniel Bernoulli solved the ordinary differential equation for the vibrations of an elastic bar clamped at 1 end. 

— 1739: Leonhard Euler solved the ordinary differential equation for a forced harmonic oscillator and noticed the resonance. 

— 1742: Colin Maclaurin discovered his uniformly rotating self-gravitating spheroids. 

— 1743: Jean le Rond d’Alembert published his “Traite de Dynamique”, in which he introduced the concept of generalised forces and D’Alembert’s principle. 

— 1747: D’Alembert and Alexis Clairaut published 1st approximate solutions to the 3-body problem. 

— 1749: Leonhard Euler derived the equation for Coriolis acceleration. 

— 1759: Leonhard Euler solved the partial differential equation for the vibration of a rectangular drum. 

— 1764: Leonhard Euler examined the partial differential equation for the vibration of a circular drum and found 1 of the Bessel function solutions. 

— 1776: John Smeaton published a paper on experiments relating power, work, momentum and kinetic energy, and supported the conversation of energy. 

— 1788: Joseph Louis Lagrange presented Lagrange’s equations of motion in the Méchanique Analitique. 

— 1789: Antoine Lavoisier stated the law of conservation of mass. 

— 1803: Louis Poinsot developed idea of angular momentum conservation (previously known nly in the case of conservation of areal velocity). 

— 1813: Peter Ewart supported the idea of the conservation of energy is his paper “On the measure of moving force”. 

— 1821: William Hamilton began his analysis of Hamilton’s characteristic function and Hamilton-Jacobi equation. 

— 1829: Carl Friedrich Gauss introduced Gauss’s principle of least constraint. 

— 1834: Carl Jacobi discovered his uniformly rotating self-gravitating ellipsoids. 

— 1835: Louis Poinsot noted an instance of the intermediate axis theorem. William Hamilton stated Hamilton’s canonical equations of motion. 

— 1838: Liouville began work on Liouville’s theorem. 

— 1841: Julius Robert von Mayer wrote a paper on the conservation of energy but his lack of academic training leads to its rejection. 

— 1847: Hermann von Helmholtz formally stated the law of conservation of energy. 

— First half of 19th century: Cauchy developed his momentum equation and his stress tensor. 

— 1851: Léon Foucault showed the Earth’s rotation with a large pendulum, also called Foucault pendulum. 

— 1870: Rudolf Clausius deduced virial theorem. 

— 1902: James Jeans found the length scale required for gravitational perturbations to grow in a static nearly homogeneous medium. 

— 1915: Emmy Noether proved Noether’s theorem, from which conservation laws are deduced. 

— 1952: Parker developed a tensor form of the virial theorem. 

— 1978: Vladimir Arnold stated the precise form of Liouville-Arnold theorem. 

— 1983: Mordehai Milgrom proposed Modified Newtonian dynamics. 

— 1992: Udwadia and Kalaba created the Udwada-Kalada equation.

https://en.wikipedia.org/wiki/Mechanics

https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion

https://en.wikipedia.org/wiki/Inertia

What are the laws of motion? 

In physics, the motion of large objects is described through 2 related sets of mechanics. Motions of large-scale and familiar objects in the universe (such as cars, projectiles, planets, cells, animals and humans) are  described by classical mechanics. On the other hand, motions of tiny atomic and sub-atomic objects are described by quantum mechanics. 

This is a painting of Isaac Newton (1643–1727), the physicist who formulated the laws. 

The foundation of classical mechanics was laid by Newtons Three laws of motion. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. They apply to objects idealised as single point masses, whilst the object’s size the shape is negligible for the sake of simplicity. 

The 3 laws of motion were first compiled by Isaac Newton in his 1687 Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). He used them to explain and investigate the motion of many physical objects and systems.

(1) Newton’s First Law 

Khan Academy: 
https://www.youtube.com/watch?v=CQYELiTtUs8&t=116s

CrashCourse: 
https://www.youtube.com/watch?v=kKKM8Y-u7ds&t=152s

Translated from Latin = 

 Law I: Every Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed. 

The first law states that if the net force (the vector sum of all forces acting on an object) is zero (0), then the velocity of the object is constant. Velocity is a vector quantity expressing both the object’s speed and the direction of its motion. So, if an object’s velocity is constant, its speed and direction of its motion are both constant. 

Mathematically, the first law is stated when the mass is a constant that is not 0kg, as: 

ΣF = 0 <=> dv/dt = 0 

Consequently, 

— An object that is at rest will stay at rest unless a force acts upon it. 

— An object that is in motion will not change its velocity unless a force acts upon it. 

This is known as uniform motion. 

In other words, an object will continue to move at a constant speed in a certain direction forever unless a force is exerted upon it. If an object is still (i.e. at rest), it will keep still forever unless a force is exerted upon it to make it move in a certain direction. This law explains why you can magically whip a tablecloth beneath the dishes on a tabletop without dragging the dishes off the table by the sliding tablecloth and the dishes will miraculously remain on the table in their initial stillness. 

https://www.youtube.com/watch?v=nSj_MVUsBzg

If an object is moving in any direction, it will continue to move in that direction without turning, speeding up or slowing down along a straight path indefinitely. For example, space probes are continuously moving in outer space after leaving Earth’s gravitational boundary. In order to change the motion of a moving object, you have apply a force against the object’s tendency to remain in its state of motion. 

In current physics, an observer defines itself as in inertial frame by hooking a spring onto a stone. They then rotate the spring in any direction, and observe the stone as static and the length of the spring remains constant. 

Let’s say there are 2 observers, A and B. If observer B is moving at a constant velocity relative to A, then both A and B will observe the same physics phenomena. If A verified Newton’s first law, then B will verify it as well. This is Einstein’s equivalence principle. Therefore, the definition of an inertial frame doesn’t mention absolute space or faraway stars, but only refers to local objects that can be reached and measured. 

Since a particle not subject to forces will continue to move (related to inertial frame) in a straight line at a constant speed, Newton’s 1st law is often referred to as “Law of Inertia”. Thus, to ensure the  uniform motion of a particle relative to an inertial reference frame, the total net force acting on the particle is zero. With this in mind, the first law can be rephrased as follows: 

Beatty (2006) stated that 

“in every material universe, the motion of a particle in a preferential reference frame Φ(phi) is determined by the action of forces whose total vanished for all times when and only when the velocity of the particle is constant in Φ. That is, a particle initially at rest or in uniform motion in the preferential frame Φ continues in that state unless compelled by forces to change it.”

Note that Newton’s first law is valid only in an inertial reference frame. Thornton (2004) asserted that any reference frame in uniform motion with respect to an inertial frame is also an inertial frame, i.e. Galilean invariance or the principle of Newton relativity.

What is Inertia? 

Inertia is defined as the tendency for a physical object to resist any change to its current velocity (including speed, or direction of motion). This means moving objects tend to maintain a straight trajectory at a constant speed in the absence of any forces. Inertia comes from the Latin word, iners, meaning ‘idle, sluggish’. It is one of the properties that give objects mass, a quantitative property of physical systems. Isaac Newton defined inertia as his first law in his Philosophiæ Naturalis Principia Mathematica as stated: 

The vis insita, or innate force of matter, is a power of resisting by which every body, as much as in it lies, endeavours to preserve its present state, whether it be of rest or of moving uniformly forward in a straight line. 

Another form of inertia is ‘rotational inertia’ (—> moment of inertia), which involves a rotating rigid body maintaining its state of uniform rotational motion. Its angular momentum remains constant, unless an external torque is applied to the body. This is also called ‘conservation of angular momentum’. Rotational inertia depends on the object to remain structurally intact as a rigid body, so it demonstrates practicality. For example, a gyroscope uses the property of rotational inertia to resist any change in the axis of rotation. I will discuss more about angular momentum later in this blog post.

(2) Newton’s Second Law 

Khan's Academy: 
https://www.youtube.com/watch?v=ou9YMWlJgkE

Translated from Latin = Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress’d. 

The second law states that the rate at which the body’s momentum changes is directly proportional to the force applied to the body, and this momentum change occurs in the direction of the applied force. 

F = dp/dt = d(mv)/dt. 

The second law can also be stated in terms of an object’s acceleration. Since Newton's second law is valid only systems in which mass is constant, m can be taken outside the differentiation operator by the constant factor rule in differentiation. Thus,

F = m*(dv/dt) = m*a, 

— F = Net force applied 

— m = Mass of the body 

— a = Body’s acceleration 

This means the net force applied to a body produces a proportional acceleration. In other words, if a body is accelerating (changing velocity), then there is a force on it. 

It suggests that the second law merely defines F, rather than a precious observation of nature. Current physics restate the 2nd law in several measurable steps: 

I. Define the term 'one unit of mass' by a specified stone. 

II. Define the term 'one unit of force' by a specified spring with specified length. 

III. Measure by experiment or prove by theory (with a principle that every direction of space are 

      equivalent), that force can be added as a mathematical vector 

IV. Conclude that F = m*a 

The 2nd law also implies the ‘conservation of momentum’. It states that when the net force on the body is zero, the momentum of the body is constant. Therefore any net force is equal to the rate of change of the momentum. Since the 2nd law is an approximation, it doesn’t demonstrate accuracy at higher speeds due to the effects of relativity. Therefore, in modern terms, Newton’s 2nd law can be interpreted as: 

The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed.

This can be expressed mathematically as: 

F = p’ 

— p’ = Time derivative of the momentum (p). 

Motte’s 1729 translation of Newton’s Latin continued with Newton’s commentary of the second law of motion is as follows: 

If a force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

https://en.wikipedia.org/wiki/Impulse_(physics)

https://en.wikipedia.org/wiki/Momentum

What is Momentum? 

Khan Academy: 
https://www.youtube.com/watch?v=XFhntPxow0U

i. Single particles 

Since momentum has a direction, it can be used to predict the resulting direction and speed of the motion of objects after their collision with each other. 

p = m*v 

— m = mass of the object 

— v = velocity (vector quantity) 

— p = momentum 

The SI unit of momentum is kg*m/s.

For example, if a 1kg model train is initially travelling east at 2 m/s in a straight and level section of track, then the train has a momentum of 2 kg*m/s due north relative to the ground. 

ii. Many particles 

The momentum of a system filled with many particles is the vector sum of their momenta. If 2 particles have respective masses m₁ and m₂, and velocities v₁ and v₂, the total momentum is: 

p = p₁ + p₂ = m₁*v₁ + m₂ *v₂  

If there are more than 2 particles, their momenta can be summed more generally with the following: 

p = Σ_i m_i * v_i 

If a system of particles has a centre of mass, we can determined a point by the weighted sums of their positions:

If 1 or more of the particles is moving, then the system’s centre of mass will be moving as well (unless the system is in pure rotation around it). 

p = m*v_cm 

— m = Total mass of the particles 

— v_cm = Velocity of the centre of mass 

— p = Momentum of the system 

This is known as Euler’s first law.

How does momentum relate to force? 

If the net force (F) applied to a particle is constant, and is applied for a time interval (Δt), the momentum of the particles changes by an amount: 

Δp = F*Δt

In differential form, the rate at which the particle’s momentum changes is equal to the instantaneous force (F) acting on it: 

F = dp/dt

Therefore,  

— Δp = Change in linear momentum from time t₁ and t₂. 

This is often called the impulse-momentum theorem (similar to the work-energy theorem). 

As a result, an impulse is regarded as the object’s momentum change caused by the application of a resultant force. If the net force experienced by a particle changes as a function of time, F(t), the change in momentum (or impulse J) between times t₁ and t₂ is: 

— F = Resultant force applied 

— t₁ = Time when the impulse begins 

— t₂ = Time when the impulse ends 

— m = Mass of the object (assuming it’s constant). 

— v₁ = Object’s initial velocity when the time interval begins 

— v₂ = Object’s final velocity when the time interval ends 

Impulse is measured in the derived units of the Newton second (1 N*s = 1 kg*m/s) or dyne second (1 dyne*s = 1 g*cm/s). Therefore it has same units and dimensions (MLT^-1) as momentum. In English engineering units, they are slug*ft/s = lbf*s. 

Impulse also refers to a fast-acting force or impact, which is often idealised as the change in the body’s momentum caused by a resultant force over no change in time. This type of change is known as a ‘step chance’, which can not be physically possible. However, it can be a useful model for computing the effects of ideal collisions (such as in game physics engines). 

https://en.wikipedia.org/wiki/Specific_impulse

How about variable mass?

Newton’s 2nd law for variable mass allows impulse and momentum to be used as analysis tools for jet- or rocket-propelled vehicles. The impact imparted by rockets can be normalised by unit of propellant expended to produce a performance parameter called “specific impulse”. This concept is used for the derivation of the Tsiolkovsky rocket equation, which relates the vehicle's propulsive change in velocity to the engine's specific impulse (or nozzle exhaust velocity) and the vehicle's propellant-mass ratio. I’ll discuss the physics of rockets in detail later below. 

Under the assumption of that the mass (m) of a body is constant, we can rewrite it as:

F = d(mv)/dt = m*(dv/dt) = m*a 

— Net force (F) is equal to the particle’s mass (m) multiplied by its acceleration (a). 

For example: A model train of mass 1 kg accelerates from rest to a velocity of 10 m/s due east in 5 seconds. 

— The net force required to produce this acceleration is (1*(10/5)) = 1*2 = 2 Newtons due east. 

— The change in momentum is 1*10 = 10 kg*m/s due east. 

— The rate at which the train’s momentum changes is 10/5 = 2 N/s. 

What is conservation of momentum? 

In a closed system (i.e. no matter can be exchanged with its surroundings nor can be acted by external forces), the total momentum is constant. This is known as the “law of conservation of momentum”. For example, let’s label 2 interacting particles F₁ and F₂. The 3rd law states that forces between those particles are equal and opposite. The 2nd law states that: 

F₁ = dp₁/ dt & F₂ = dp₁/ dt.

Therefore:

dp₁/ dt = - dp₁/ dt

— The negative sign indicates the forces are opposing each other. Equivalently: 

d/dt (p₁ + p₂) = 0 

If the velocities of the particles are u₁ & u₂ before their interaction, and are v₁ & v₂, then:

m₁*u₁ + m₂*u₂ = m₁*v₁ + m₂*v₂ 

This law holds no matter how complicated the force is between particles. If there are more than 2 particles, the momentum exchanged between each set of particles adds up to zero, so the total change in momentum is zero. This conservation law applies to all interactions, including collisions and separation caused by deflagrations and explosions. 

Exercise 1: 

— A wheeled, 1000kg cannon is initially at rest. It then fires a 2kg cannonball horizontally across the field. The ball leaves the cannon travelling at 200 m/s. At what speed does the cannon recoil as a result? 

2*200 = - 1000*v 

400 = - 1000*v 

v = - 4/10 = - 0.4 m/s 

Exercise 2a: 

— 2 freight cars are rolling along the tracks hurtling towards each other. A tank car filled with water has a mass of 50 kg and is moving at 3 m/s south and a boxcar filled with gears has a mass of 60 kg at 2 m/s north. What will be the resultant velocity of the 2 freight cars after they couple together during the collision? Friction is negligible, so let’s ignore it for simplicity’s sake. 

p₁ = 50*3 + 60*(-2) = 150 - 120 = 30 kg*m/s 

p₂  = (50+60)*v 

Since p₁ = p₂, therefore, 

30 = 110*v 

v = 30 / 110 = 3/11 = 0.2727… m/s south

Exercise 2b: It’s measured the collision between the 2 freight cars lasted 0.5 seconds. What is the resultant net force of the collision? 

F = 50*3*0.5 + 60*(-2)*0.5 = 75 - 60 = 15 N south.

Dependence on reference frame 

Measuring momentum depends on the motion of the observer.

For example, if an apple is sitting in a descending glass elevator, an observer A outside of the elevator looks into it and notices the apple moving. According to A, the apple has momentum and velocity that is not zero (p > 0, v = v₁ > 0). However if an observer B is inside the elevator, the apple does not appear to moving from their perspective, so B thinks the apple has zero momentum and zero velocity (p = v = 0) .  

Both observers have a frame of reference, in which, they observe motions. If the elevator is descending at a steady velocity (v₁), they will see behaviour that is consistent with those same physical laws. 

Suppose a particle is in position x in a stationary frame of reference. From the point of view of another frame of reference, the particle moving at a uniform speed ‘u’, the position (represented by a primed coordinate, x’) changes with time (t) as: 

x’ = x — u*t 

This is called a Galilean transformation. If the particle is moving at speed dx/dt = v in the first frame of reference, thus in the second frame of reference, the particle is moving at speed: 

v’ = dx’ / dt = v - u 

Since u is constant, the accelerations at both frames of reference remain the same. 

a’ = dv/ / dt = a 

Therefore, momentum is conserved in both reference frames. As long as the force has the same form in both reference frames, Newton’s 2nd law is unchanged. Forces such as Newtonian gravity, which depend only on the scalar distance between objects, satisfy this criterion. Goldstein (1980) labelled this independence of reference frame as Newtonian relativity or Galilean invariance. 

To simplify calculations of motion, a change of reference frame is required. For example, understanding the physics behind the collision of 2 particles requires the choice of a reference frame, with one particle initially at rest. A commonly used reference frame is the centre of mass frame, one that moves with the centre of mass. In this reference frame, the total momentum is zero.

How does momentum apply to collisions? 

CrashCourse: 
https://www.youtube.com/watch?v=Y-QOfc2XqOk

By itself, the law of conservation of momentum inadequately explains the motion of particles after a collision. Another concept of motion is kinetic energy, which has to be known. In real life, kinetic energy is not necessarily conserved. 

A. Elastic collisions 

https://en.wikipedia.org/wiki/Elastic_collision

Elastic collisions are encounters between 2 bodies in which the total kinetic energy of the 2 bodies is the same. In an ideal, perfectly elastic collision, there is no net conversion of kinetic energy into forms of energy such as heat, sound, vibration, or potential energy. 

When small objects collide, kinetic energy first converts to potential energy associating with a repulsive force between the particles. That is, when the particles move against this force, i.e. the angle between the force and the relative velocity is obtuse or larger than 90^o. Then this potential energy converts back to kinetic energy. That is, when the particles move with this repulsive force, i.e. the angle between the force and the relative velocity is acute or less than 90^o. 

Examples include atom collisions e.g. Rutherford backscattering.

 As long as black-body radiation (not shown) doesn’t escape a system, atoms in thermal agitation undergo essentially elastic collisions. On average, two atoms rebound from each other with the same kinetic energy as before a collision. Five atoms are coloured red so their paths of motion are easier to track. This is a screenshot of an animation. Click on the wikipedia page link for elastic collisions to watch the animation.

 A special case of an elastic collision is the 2 bodies having equal mass, in which case they exchange their momenta.

Describe the equations of elastic collisions

i. One-dimensional Newtonian 

Consider particles 1 and 2 with masses m₁, m₂, and velocities u₁, u₂ before collision, and velocities v₁, v₂ after collision. The conservation of the total momentum before and after the collision is expressed by:

m₁*u₁ +  m₂* u₂ = m₁*v₁ +  m₂* v₂

Likewise, the conservation of the total kinetic energy is expressed by: 

0.5* m₁*(u₁)² +  0.5*m₂*(u₂)² = 0.5*m₁*(v₁)² +  0.5*m₂*(v₂)²

If both u₁ & u₂ are known, then these equations can be solved directly to find v₁, v₂.

If m₁ = m₂, we have a trivial solution: 

v₁ = u₂, v₂ = u₁

— This corresponds to the bodies exchanging their initial velocities to each other. 

As expected, the solution is invariant under the addition of a constant to all velocities. This is analogous to using a frame of reference with constant translational velocity. Nevertheless, derivation of the equations requires a change in the reference frame to make one of the known velocities zero. Then the unknown velocities in the new frame of reference can be determined, and then converted back to the original frame of reference. 

For example: 

Imagine 2 billiard balls colliding with each other on a pool table: 

— Ball 1: Mass = 1 kg, initial velocity = 3 m/s 

— Ball 2: Mass = 2 kg, initial velocity = -4 m/s 

Calculate the balls’ velocities after the collision. 

v₁ = [(1 - 2)/ (1 + 2)] * 3 + [(2*2)/ (1 + 2)]*(-4) 

= (-1/3) * 3 + (4/3)*(-4) 

= -1 + (-16/3) 

= -19/3 = -6.33m/s 

v₂ = [(2 - 1)/ (1 + 2)] *(-4) + [(2*1)/ (1 + 2)]*(3) 

= (1/3)*(-4) + (2/3)*3 

= -4/3 + 2 

= 2/3 = 0.67 m/s

Scenario 1: Elastic collision of unequal masses  

Object 1: Mass = 2m, Velocity before collision = v 

Object 2: Mass = m, Velocity before collision = v 

Object 1: Velocity after collision = 1/3 v 

Object 2: Velocity after collision = 5/3 v

Scenario 2: Elastic collision of equal masses  

Object 1: Mass = m, Velocity before collision = v 

Object 2: Mass = m, Velocity before collision = 0 

Object 1: Velocity after collision = 0 

Object 2: Velocity after collision = v

Scenario 3: Elastic collision of masses in a system with a moving frame of reference  

(A)

Object 1: Mass = m, Velocity before collision = v 

Object 2: Mass = m, Velocity before collision = -v/2 

Object 1: Velocity after collision = -v/2 

Object 2: Velocity after collision = v 

(B)

Object 1: Mass = m, Velocity before collision = v 

Object 2: Mass = m, Velocity before collision = v/2 

Object 1: Velocity after collision = v/2 

Object 2: Velocity after collision = v

A limiting case involves m₁ being larger than m₂, such as a ping-pong paddle hitting a table tennis ball. In this case, the object with heavier mass hardly changes velocity, while the object with lighter mass would bounce off and change its velocity significantly. 

In the case of an object with a large u₁, the value of v₁ would be small if the masses are approximately the same. In other words, hitter a lighter particle would not change its velocity much, whereas hitting a heavier particles causes the fast particle to rebound at higher speeds. 

To derive the above equations for v₁ & v₂, the equations for kinetic energy and momentum can be rearranged: 

m₁ * ((v₁)² - (u₁)²) = m₂ *((u₂)² - (v₂)²) m₁ * (v₁ - u₁) = m₂ * (u₂ - v₂) 

Divide each side of the top equation by each side of the bottom equation, and use the formula (a² - b²) / (a - b) = a + b, to give: 

v₁ + u₁ = u₂ + v₂  => v₁ - v₂ = u₂ - u₁ 

That is, the relative velocity of 1 particle with respect to the other is reversed by the collision. 

Now the above formulas follow from solving a system of linear equations for v₁ & v₂ regarding m₁, m₂, u₁ & u₂ as constants: 

Once v₁ is determined, v₂ can be found by symmetry. 

ii. Centre of mass frame 

With respect to the centre of mass, both velocities are reversed by the collision. This is, a heavier particles moves slowly toward the centre of mass, bounces back with the same low speed. Whereas a lighter particles moves quickly toward the centre of mass, and bounces back with the same high speed. 

The collision doesn’t change the velocity of the centre of mass. To understand why this is the case, consider the centre of mass at time before collision (t) and time after collision (t’).

Hence, the velocities of the centre of mass before and after collision are:

The numerators of v_x- and v_x’ are the total momenta before and after collision. Since momentum is conserved, we have v_x- = v_x’-. 

iii. One-dimensional relativistic 

According to Einstein’s theory of special relativity:

— p = Momentum of any particle with mass 

— v = Velocity 

— m = Mass of the particle 

— c = Speed of light 

In the centre of momentum frame, the total momentum equals zero. Therefore:

— m₁ & m₂ = Rest masses of the 2 colliding bodies 

— u₁ & u₂ = Velocities before collision 

— v₁ & v₂ = Velocities after collision 

— p₁ & p₂ = The bodies’ momenta before and after collisions respectively 

— c = Speed of light in vacuum = 299,792 km/s 

— E = Total Energy, the sum of rest masses and kinetic energies of the two bodies. 

Since the total energy and momentum of the system are conserved and their rest masses remain constant, this solution demonstrates the momentum of the colliding body is determined by the rest masses of the colliding bodies, the sum of rest masses and kinetic energies of the two bodies. Relative to the centre of momentum frame, the momentum of each colliding body does not change magnitude after collision, but the body’s direction of movement is reversed. 

In comparison with classical mechanics, the total momentum of the 2 colliding bodies is dependent on the reference frames. Classical mechanics tells us that in the centre of momentum frame,

It seems the solution agrees with the relativistic calculation u₁ = -v₁, despite their differences. 

One of the postulates in the theory of special relativity states that: 

The laws of physics, such as conservation of momentum, should be invariant (i.e. never changing) in all inertial frames of reference. So, in a general inertial frame where the total momentum could be arbitrary.  

— p_T = Total momentum of the 2 moving bodies in the same system 

— E = Total energy 

— v_c = Velocity of the centre of mass 

Relative to the centre of momentum frame, the total momentum equals zero. It can be shown that: 

v_c = (p_T * c²)/E 

The velocities before the collision in the centre of momentum frame, u₁’ & u₂’, are evaluated to be: 

When u₁ << c, and u₂ << c, then:

Therefore, the classical calculation holds true, when the speed of both colliding bodies is much lower than the speed of light (~300 million m/s). 

iv. Relativistic derivation using hyperbolic functions

https://en.wikipedia.org/wiki/Rapidity

When we use the parameter of velocity (s, usually called the rapidity), we get the formula:  

v/c = tanh (s) 

— tanh = Hyperbolic tangent function 

Hence we get:

— sech = Hyperbolic secant function 

Relativistic energy and momentum are expressed as follows: 

— cosh = Hyperbolic cosine function 

— sinh = Hyperbolic sine function 

When we sum the equations of energy and momentum colliding masses m₁ & m₂, (velocities: v₁, v₂, u₁, u₂ correspond to the velocity parameters s₁, s₂, s₃, s₄), after dividing by adequate power c, we get: 

m₁*cosh(s₁) + m₂*cosh(s₂) = m₁*cosh(s₃) + m₂*cosh(s₄) m₁*sinh(s₁) + m₂*sinh(s₂) = m₁*sinh(s₃) + m₂*sinh(s₄)

Then we sum the above equations to yield the dependent equation: 

m₁* e^s₁ + m₂*e^s₂ = m₁*e^s₃ + m₂*e^s₄ 

Next we subtract the squares from both sides of “momentum” equations from “energy” equations, and then use the identity cosh²(s) - sinh²(s) = 1. After these steps, we simplify our equation to yield: 

2*m₁*m₂*[cosh(s₁) * sinh(s₂) - cosh(s₂) * sinh(s₁)] = 2*m₁*m₂ *[cosh(s₃) * sinh(s₄) - cosh(s₄) * sinh(s₃)] 

If mass is non-zero, then we get: 

cosh(s₁ - s₂) = cosh(s₃  - s₄) 

Since functions cosh(s) are even, we can get 2 solutions: 

s₁ - s₂ = s₃ - s₄ s₁ - s₂ = - s₃ + s₄

The last equation leads to a non-trivial solution. Therefore we can solve s₂ and substitute into the dependent equation to obtain e^s₁ and then e^s₂. This leads to: 

This is a solution to the problem, but expressed by the parameters of velocity. Therefore we return the substitution step to yield the solution for velocities, which are:

Next we substitute the previous solutions and replace it with:

After a long transformation and substituting the following:

This leads to: 

Describe momentum in 2 dimensions

To understand the collision between 2 bodies in 2 dimensions, we have to split the overall velocity of each body into 2 perpendicular velocities. 

— 1 tangent to the common normal surfaces of the colliding bodies at the point of contact 

— 1 other along the line of collision. 

Since the collision only imparts force along the line of collision, velocities tangent to the point of collision do not change. Therefore, velocities along the line of collision can then be used in the same equations as a one-dimensional collision. The final velocities can then be calculated from the 2 new component velocities, but this depends on the point of collision.

 This sequence of images illustrates two-dimensional elastic collision between 2 circular bodies. 

When we view the collision from a centre of momentum frame, the velocities of the 2 bodies are in opposite directions all the time, with their magnitudes inversely proportional to the masses. Depending on the shapes of the bodies and point of contact, the directions of the velocities may change. 

e.g. When 2 spherical objects collide, their angle depends on the distance between the (parallel) paths of the centres of the 2 bodies. There are a few ways for any non-zero change of direction to be possible. 

— If the distance between the paths of the centres of the 2 bodies is zero, then the velocities are reversed in the collision. 

— If the distance between the paths of the centres of the 2 bodies is close to the radii of the spheres, then the 2 bodies deflect only slightly. 

Let’s assume that the 2nd particle is at rest before the collision. This means there is a relationship between the angles of deflection of the 2 particles, Θ₁ & Θ₂, and the angle of deflection (θ) in the the system of the centre of mass. This relationships can be expressed mathematically as follows:

The magnitudes of the velocities of the particles after the collision are:

i. 2 moving objects 

The final x and y velocities components of the 1st ball can be calculated as:

— v₁ & v₂ = Scalar sizes of the 2 original speeds of the objects 

— m₁ & m₂ = Masses of the object 

— θ₁ & θ₂ = Movement angles 

— v_1x = v₁* cos(θ₁), v_1y = v₁* sin(θ₁). This means direct movement towards the right is either a -45° angle, or a 315°angle. 

— φ = Contact angle 

— If you want to evaluate the ‘x’ and ‘y’ velocities of the 2nd ball, substitute all the '1' subscripts with '2' subscripts. 

Derivation of this equation is based on the fact that the interaction between the 2 bodies can be calculated along the contact angle. This means the objects’ velocities can be worked out in 1 dimension through rotation of the x- and y-axis to be in parallel with the contact angle of the objects. Then the axes are rotated back to the original orientation to gain the true ‘x’ and ‘y’ components of the velocities. 

To represent the above equations without angles, the changed velocities can be computed using the centres x₁ & x₂ at the time of contact as:

— The angle brackets indicate the inner product (or dot product) of the 2 vectors.

B. Inelastic collisions 

https://en.wikipedia.org/wiki/Inelastic_collision

This collision involves kinetic energy not being conserved due to the presence of the internal friction. In real-life collisions between macroscopic bodies, some of the total kinetic energy is transformed into vibrational energy of the atoms, producing heat, which deforms the bodies. For example, collisions between gaseous or liquid molecules tend to be inelastic because some kinetic energy is exchanged between the molecules' translational motion and their internal degrees of freedom with each collision. 

Beer & Johnston (1996) stated that despite kinetic energy being lost during inelastic collisions, particles still obey the law of conservation of momentum nonetheless. e.g. A ballistic pendulum obey the conservation of kinetic energy only when the block swings to its largest angle. 

In nuclear physics, incoming particles collide with each other to strike each other’s nuclei, causing it to become excited or disintegrate. Deep inelastic scattering helps probe the structure of subatomic particles similar to Rutherford scattering. Such experiments were conducted at the Stanford Linear Accelerator (SLA) in the 1960s, where they used high-energy electrons to collide with protons. As in Rutherford scattering, deep inelastic scattering of electrons by proton targets lead to the discovery of minimal physical interaction between most of the incident electrons. They found the electrons would often pass straight through one another, with only a few bouncing off each other. This suggested that the charge in the proton is concentrated in tiny lumps, which supports Rutherford’s discovery of the positive charge being concentrated inside the atom’s nucleus. 

The formula for the velocities after a one-dimensional inelastic collision is:

— v_a = Final velocity of the 1st object after the collision 

— v_b = Final velocity of the second object after the collision 

— u_a = Initial velocity of the first object before the collision 

— u_b = Initial velocity of the second object before the collision 

— m_a = Mass of the first object 

— m_b = Mass of the second object 

— C_R = Coefficient of restitution 

If C_R = 1, then the collision is elastic. 

If C_R = 0, then the collision is perfectly inelastic. 

In a centre of momentum frame, the formulas simplify to become: 

v_a = -C_R*u_a v_b = -C_R*u_b 

For 2- and 3-dimensional collisions, the velocities in these formulas are the components perpendicular to the tangent line/plane at the point of contact. 

The normal impulse: 

J_n = [(m_a*m_b) / (m_a + m_b)] * (1 + C_R) * (u_b - u_a) 

This gives the velocity updates:

What’s a perfectly inelastic collision? 

A perfectly inelastic collision occurs when the maximum amount of kinetic energy of a system is lost. This means the coefficient of restitution is zero, and the colliding particles stick together. This shows that kinetic energy is lost as the 2 particles bound together. The maximum kinetic energy in the system is lost to this bonding energy. 

— If the surface has zero friction, momentum of the 2 body system is conserved. 

— If the surface has friction, the momentum of the 2 bodies is transferred to the surface that the bound bodies are sliding upon. 

— Similarly, if there is air resistance, the momentum of the bodies can be transferred to the air. 

 This is a bouncing basketball captured with a stroboscopic flash at 25 images per second. Every time the ball bounces off the ground, the impact is inelastic. This means kinetic energy is dissipating at every bounce. Ignoring air resistance, the square root of the ratio of the height of one bounce to that of the preceding bounce gives the coefficient of restitution (C_R) for the ball/surface impact. 

The equation below mathematically explains the two-body (Body A, Body B) system collision in this example. The momentum of this system is conserved due to the lack of friction between the sliding bodies and the surface. 

m_a*u_a + m_b*u_b = (m_a + m_b)*v 

— v = Final velocity. 

Therefore, the above equation can be rearranged: 

Example 1a : A 5kg carriage rolls along a smooth road towards a stationary 5kg carriage at 4 m/s north. The collision between the 2 carriages is perfectly inelastic and friction is negligible. Calculate the final velocity of the 2 carriages bound together. 

v = (5*4 + 5*0) / (5+5) = 20/10 = 2 m/s north 

i.e. Both carriages are moving at 2 m/s north. 

1b: How much kinetic energy is lost during this collision? 

0.5*5*(4^2) + 0.5*5*(0^2) = 0.5*5*(2^2) + 0.5*5*(2^2) + Loss 

0.5*80 + 0 = 0.5*20 + 0.5*20 + Loss 

40 = 10 + 10 + Loss 

40 - 10 -10 = Loss 

Loss = 20 J 

This reduction of total kinetic energy is equal to the total kinetic energy before the collision in a centre of momentum frame with respect to the system of 2 particles. This is because the kinetic energy after the collision in a centre of momentum frame is zero. Therefore, most of the kinetic energy before the collision is initially contained in the particle with the smaller mass. In another frame of reference, some kinetic energy may be transferred from one particle to the other in addition to the reduced kinetic energy. This demonstrates the relativity of collisions from different frames of reference.

What about partially inelastic collisions? 

Collisions in the real world are commonly partially inelastic. This type of collision involves colliding objects not sticking together with some kinetic energy lost. The lost kinetic energy may be converted to friction, sound and heat. 

Momentum in multiple dimensions 

Real motion has both direction and velocity, which is represented by a vector. A 3D coordinate system has x, y, z axes, with velocity having components, v_x, v_y & v_z, in the x-, y- and z-directions respectively. The vector is expressed:

v = (v_x, v_y, v_z)

Similarly, the momentum is also a vector quantity, hence is expressed as, 

p = (p_x, p_y, p_z) 

The above equations work in vector form if the scalars ‘p’ and ‘v’ are replaced by vectors ‘p’ and ‘v’. Therefore each vector equation represents 3 scalar equations. For example, p = m*v represents 3 equations: 

p_x = m*v_x p_y = m*v_y p_z = m*v_z 

The kinetic energy equations are exceptions to the above replacement rule, since the equations are still one-dimensional. However, each scalar represents the magnitude of the vector, for example, 

v² = v_x² + v_y² + v_z² 

Each vector equation represents 3 scalar equations, however only 2 coordinates are often required to work out the 3rd coordinate. Feymann asserted that each component can be obtained separately and the results combined would produce a vector result. Rindler (1986) explained that a simple construction involving a centre of mass frame can be used to demonstrate two colliding objects heading off at right angles when a moving sphere hits a stationary elastic sphere. 

Each vector equation represents 3 scalar equations, however only 2 coordinates are often required to work out the 3rd coordinate. Feymann asserted that each component can be obtained separately and the results combined would produce a vector result. Rindler (1986) explained that a simple construction involving a centre of mass frame can be used to demonstrate two colliding objects heading off at right angles when a moving sphere hits a stationary elastic sphere. 

Objects of variable mass 

The concept of momentum plays a fundamental role in explaining the behaviour of variable-mass objects such as a rocket’s ejection of rocket fuel or a star’s accretion of gas. To analyse such objects, we have to treat the object’s mass as a function that varies with time, m(t). Therefore the momentum of the object at time ’t’ is p(t) = m(t) * v(t). We then try to invoke Newton’s 2nd law of motion by stating the relationship between the external force (F) on the object and its momentum p(t), by expressing it as F = dp/dt. 

F = m(t) * dv/dt - u* (dm/dt) 

— u = Velocity of the ejected/accreted mass as seen in the object's rest frame. 

— v = Velocity of the object itself as seen in an inertial frame. 

P(t+dt) = (m - dm)*(v + dv) + dm*(v - u) = m*v + m*dv - u*dm = P(t) + m*dv - u*dm 

This equation shows the rate at which both the momentum of the object as well as the momentum of the ejected/accreted mass (dm) over time. When we consider these factors together, the object and the mass (dm) constitute a closed system in which total momentum is conserved. 

Describe the relativistic effects of momentum 

i. Lorentz invariance 

Because Newtonian assumes absolute time and space exist outside of any observer, this gave rise to Galilean invariance. It also lead to predictions that the speed of light varies from one reference frame to another, which is contrary to previous observation. In Einstein’s special theory of relativity, he postulated the equations of motion are independent on the reference frame, with the assumption that the speed of light (c) is invariant. This lead to the suggestion that position and time in 2 different reference frames are related by the Lorentz transformation rather than the Galilean transformation. 

Let’s consider one reference frame moving relative to another at velocity v in the x-direction. The Galilean transformation would give the following coordinates of the moving from as: 

t’ = t x’ = x - u*t 

Meanwhile the Lorentz transformation gives: 

t’ = γ*(t - (v*x)/c²)

x’ = γ*(x - v*t) 

— γ = Lorentz factor 

With mass fixed, Newton’s 2nd law is not invariant under a Lorentz transformation. However, it can be made invariant by making the inertial mass (m) of an object a function of velocity:

m = γ*m₀

— m₀ = Object’s invariant mass 

So the modified momentum, p = γ*m₀*v, obeys Newton’s 2nd law, F = dp/dt. 

With the domain of classical mechanics, relativistic momentum is closely approximate to Newtonian momentum. At low velocities,  γ*m₀*v ~ m₀*v (the Newtonian expression for momentum.) 

ii. Four-vector formulation 

In the theory of special relativity, physical quantities are expressed in terms of four-vectors that include time as a 4th coordinate along the 3 space coordinates. These vectors are represented by capital letters such as U and V for velocities. The expression for the four-momentum depends on the expression of the coordinates. If we want all components of the four-vector to dimensions of length, we can give time its normal units or multiply it by the speed of light (c). If the latter scaling is used, an interval of proper time (τ), then it can be defined by: 

c² * dτ² = c² *dt² - dx² - dy² - dz² 

This equation is invariant under Lorentz transformations. This invariance can be ensured mathematically in 1 of 2 ways: 

I) By treating the four-vectors as Euclidean vectors and multiplying time by √-1 

II) Keeping time a real quantity and embedding the vectors in a Minkowski space. In a Minkowski space, the scalar product of 2 four-vectors U = (U₀, U₁, U₂, U₃) and V = (V₀, V₁, V₂, V₃) is defined as,

U*V = U₀*V₀ - U₁*V₁ - U₂*V₂ - U₃*V₃

In all the coordinate systems, the (contravariant) relativistic four-velocity is defined by, 

U = dR/dr = γ*(dR/dr), 

and the (contravariant) four-momentum is 

P = m₀*U, 

— m₀ = Invariant mass. 

If R = (ct, x, y, z) (in Minkowski space), then, 

P = γ*m₀ (c,v) = (m*c, p). 

Using Einstein’s mass-energy equivalence (E = m*c²), the above equation can be rewritten as, 

P = (E/c, p) 

Thus, it shows the conservation of four-momentum is Lorentz-invariant. This implies conservation of both mass and energy. 

The magnitude of the momentum four-vector is equal to m_o*c: 

||P||² = P*P = γ² * (m₀)² * (c² - v²) = (m₀*c)², 

and is invariant across all reference frames. 

The relativistic energy-momentum relationship holds even for massless particles such as photons. If we let m₀ = 0, this means E = p*c. 

Imagine you’re playing a game of relativistic “billiards”, you hit a stationary particle with a moving particle and the collision is elastic. The angle of the paths formed by the 2 particles after the collision is acute. Rindler (1986) stated that this scenario is not like the non-relativistic case where they travel at right angles. 

Rindler (1991) also developed the relationship between the four-momentum of a planar wave and a wave four-vector:

For a particle, the relationship between the temporal components, E = ħ*ω, is the Planck-Einstein relation. Also the relation between spatial components, p = ħ*k, describes a de Broglie matter wave.

What is generalised momentum? 

Constraints limiting motion makes Newton’s laws difficult to apply to numerous kinds of motion. For example, a bead on an abacus is constrained to move along its wire and a pendulum bob is constrained to swing at a fixed distance from the pivot. Goldstein (1980) incorporated many such constraints through changes of the normal Cartesian coordinates to a set of generalised coordinates. Advanced mathematical methods introduced a “generalised momentum” or canonical / conjugate momentum to include both concepts of linear momentum and angular momentum in order to solve mechanics problems in generalised coordinates. Several papers referred the product of mass and velocity as “mechanical, kinetic or kinematic momentum” to distinguish it from generalised momentum. 

i. Lagrangian mechanics 

A Lagrangian is defined as the difference between the kinetic energy (T) and the potential energy (V): 

L = T - V

The generalised coordinates are represented as a vector q = (q₁, q₂, …, q_N) and time differentiation is represented by a dot over the variable. This means the equations of motion (known as the Lagrange or Euler-Lagrange equations) 

If the coordinate ‘q_i’ is not a Cartesian coordinate, the associated generalised momentum component ‘p_i’ doesn’t necessarily have the dimensions of linear momentum. Goldstein (1980) argued that even if q_i is a Cartesian coordinate, p_i won’t be the same as the mechanical momentum if the potential depends on velocity. Lerner and Trigg (2005) suggested that kinematic momentum should be represented by the symbol Π. 

In this mathematical framework, a generalised momentum is associated with the generalised coordinates. Its components are defined as:

Each component ‘p_j’ is a conjugate momentum for the coordinate ‘q_j’. Now if a given coordinate ‘q_j’ doesn’t appear in the Lagrangian, then p_j is a constant. 

Goldstein (1980) stated this is the generalisation of the conservation of momentum. Even if the generalised coordinates are just the ordinary spatial coordinates, the conjugate momenta aren’t necessarily the ordinary momentum coordinates e.g. electromagnetism. 

ii. Hamiltonian mechanics 

The Lagrangian (i.e. a function of generalised coordinates and their derivatives) is substituted by a Hamiltonian that is a function of generalised coordinates and momentum, which is defined as:

— Momentum is obtained by differentiating the Lagrangian as above. Goldstein (1980) derived the Hamiltonian equations of motion as:

As in Lagrangian mechanics, if a generalised coordinate doesn’t appear n the Hamiltonian, its conjugate momentum component is conserved. 

iii. Symmetry and conservation 

Conservation of momentum is a mathematical consequence of the homogeneity (shift symmetry) of space. Note the position in space is the canonical conjugate quantity to momentum. Hand and Finch (1986) suggested conservation of momentum is a consequence of the law of physics’ independence on position, which is a special case of Noether’s theorem. 

Describe the electromagnetic properties of momentum

i. Particle in a field 

Maxwell’s equations tells us that the forces between particles are mediated by electric and magnetic fields. The electromagnetic force known as Lorentz force on a particle with charge (q) due to the presence of an electric field (E) and a magnetic field (B) is: 

F = q*(E + v*B) 

— It has an electrical potential ψ(r,t) and magnetic vector potential A(r,t). In the non-relativistic regime, its generalised momentum is expressed as: 

P = m*v + q*A 

while in relativistic mechanics, this becomes: 

P = γ*m*v + q*A 

ii. Conservation 

In Newtonian mechanics, the law of conservation of momentum is derived from the law of action and reaction, which states that every force has a reciprocating equal and opposite force. Griffiths (2013) indicates certain circumstances where moving charged particles exert forces on each other in non-opposite directions. Nevertheless, the combined momentum of the particles and the electromagnetic field is conserved. 

— Vacuum: 

Jackson (1975) explained that Lorentz force imparts a momentum to a particle, so the particle must impart a momentum to the electromagnetic fields according to Newton’s 2nd law. In a vacuum, the momentum per unit volume is: 

g =  [1 / (μ₀*c²)]*E*B 

— μ₀ = Vacuum permeability 

— c = Speed of light 

The momentum density is proportional to the Poynting vector (S), which derives the directional rate of energy transfer per unit area: 

g = S/c² 

If momentum is conserved over the volume (V) over a region (Q), any change in the momentum of matter through the Lorentz force would need to be balanced by changes in the momentum of the electromagnetic field and outflow of momentum. Therefore, if P_mech is the momentum of all the particles in Q, then the particles can be treated as a continuum. Hence, Newton’s 2nd law gives us:

The electromagnetic momentum is expressed as:

The equation for conservation of each component (i) of the momentum is given as: 

— The term on the right side of the equals sign is an integral over the surface area (Σ) of the surface (σ) representing momentum flow into and out of the volume. 

— n_j = A component of the surface normal of S. 

— The quantity T_ij is called the Maxwell stress tensor, which is expressed as:

— Media 

The above derivations are for the microscopic Maxwell equations, which can be applied to electromagnetic forces in a vacuum (or on a very small scale in media). There is difficulty defining momentum density in media due to the arbitrary division into electromagnetic and mechanical fields. The definition of electromagnetic momentum density is then tweaked to give: 

g = (1/c²)*E*H = S/c² 

— H-field (H) is related to the B-field and the magnetisation (M) by the following equation: 

B = μ₀*(H+M) 

Jackson (1975) stated that the electromagnetic stress tensor is dependent on the properties of the media. 

Describe the quantum mechanics of momentum 

In quantum mechanics, momentum is defined as a self-adjoint operator on the wave operator. The Heisenberg uncertainty principle defines limits on the accuracy of knowing the momentum and position of a single observable system at once. Furthermore, position and momentum are considered conjugate variables in quantum mechanics. 

For a single particle described in the position basis the momentum operator is expressed as: 

— ∇ = Gradient operator 

— ħ = Reduced Planck constant 

— i = Imaginary unit 

This is a common form of the momentum operator. However, in other bases, the momentum operator can be expressed in other forms such as in momentum space. 

p*ψ(p) = p*ψ(p) 

— p = Operator acting on a wave function ψ(p) yields that wave function multiplied by the value p.

For both massive and massless objects, relativistic momentum is related to the phase constant (β): 

p = ħ*β 

Electromagnetic radiation (including visible light, ultraviolet light, and radio waves) is carried by photons. Even though photons (the particle aspect of light) are massless, they still carry momentum. This means applications such as the solar sail can be made. 

Momentum in deformable bodies and fluids

i. Conservation in a continuum 

In fields such as fluid dynamics and solid mechanics, there is not much feasibility in following the motion of individual atoms or molecules. Therefore, physicists would need to use a continuum to estimate materials, in which a particle or fluid parcel at each point is assigned the average of the properties of atoms in a small region nearby. In particular, these materials have a density (ρ) and velocity (v) that depend on time (t) and position (r), thus their momentum per unit volume is ρ*v.

This diagram illustrates the motion of a material body. 

Think of a column of water in hydrostatic equilibrium. All the forces on the water are perfected balanced and the water is motionless. On any given drop of water, 2 types of forces balance out each other. 

I. Gravity acts directly on each atom and molecule inside. The gravitational force per unit volume is ρ*g, where g is the gravitational acceleration. 

II. Sum of all the forces exerted on its surface by the surrounding water. The force from below is greater than the force from above by the amount required to balance gravity. The normal force per unit area is the pressure (p). Feynman evaluated the average force per unit volume inside the droplet as the gradient of the pressure, which lead to the force balance equation. 

-∇p + ρ*g = 0 

If the forces are unbalanced, the droplet increases in acceleration. Tritton (2006) argued this positive acceleration isn’t simply the partial derivative ∂v/∂t due to the fluid in a given volume changing with time. Therefore, he implied the material derivative is necessary.

Since this material derivative includes the rate of change at a point and the changes due to advection caused by fluid carrying past a certain point, this concept can apply to any physical quantity. Per unit volume, the rate of change in momentum is equal to ρ*(Dv/Dt), which is equal to the net force on the droplet. 

Forces that can change the momentum of a droplet include the gradient of the pressure and gravity, as well as surface forces deforming the droplet. A shear stress (τ) exerts a force parallel to the surface of the droplet, which is proportional to the rate of deformation or strain rate. If the fluid is moving faster on one side than another, creating a velocity gradient, this creates a shear stress. If the speed in the x-direction varies with z, the tangential force in the x-direction per unit area normal to the z-direction is:

— μ = Viscosity. It’s also flux, or flow per unit area, of x-momentum through the surface. 

If we include the effect of viscosity, the momentum balance equations for the incompressible flow of a Newtonian fluid are: 

ρ*(Dv/Dt) = -∇*p + μ*∇²*v + ρ*g

— These are the Navier-Stokes equations.

The momentum balance equations can be applied to more general materials, including solids. For each surface with a normal force in the i-direction and a force in the j-direction, there is a stress component σ_ij. The 9 components making up the Cauchy stress tensor (σ) includes both pressure and shear. The Cauchy momentum equation expresses the local conservation of momentum: 

ρ*(Dv/Dt) = ∇*σ + f 

— f = Body force 

This equation broadly applies to deformations of solids and liquids. The relationship between the stresses and the strain rate depends on the properties of the material. I’ll delve into the details of viscosity in another post. 

ii. Acoustic Waves 

A disturbance in a medium gives rise to oscillations, or waves, that propagate away from their source. In a fluid, small changes in pressure (p) are described by the acoustic wave equation:

— c = Speed of sound 

Gubbins (1992) found similar equations can be obtained for propagation of pressure (P-waves) and shear (S-waves) in a solid. 

The flux, or transport per unit area, of a momentum component (ρ*v_j) by a velocity (v_i) is equal to ρ*v_j*v_j. Linear approximate leads to the above acoustic equation, which suggests the time average of this flux is zero. LeBlond & Mysak (1980) claimed non-linear effects can give rise to a non-zero average. McIntyre (1981) thought momentum flux could occur despite the wave itself not having a  mean momentum. 

(3) Newton’s Third Law 

Khan Academy: 
https://www.youtube.com/watch?v=By-ggTfeuJU

Translated from Latin = Law III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

The 3rd law states that all forces between 2 objects exist in equal magnitude and opposite direction. So if object A exerts a force F_A on object B, then object B simultaneously exerts a force F_B on object A. The 2 forces are equal in magnitude and opposite in direction: F_A = -F_B . 

 This diagram illustrates Newton’s 3rd law in which 2 skaters push against each other. Let’s name the left skater Alyssa and the right skater Mitchell. Alyssa exerts a normal force N_1->2 on Mitchell directed towards the right, and Mitchell exerts a normal force N_2->1 on Alyssa directed towards the left. The magnitude of both forces are equal, but they have opposite directions, as dictated by Newton’s 3rd Law. 

This law means all forces are interactions between different bodies, or different regions within one body. Thus there is no such thing as a force that is not accompanied by an equal and opposite force. In some situations, the magnitude and direction of the forces are determined entirely by 1 of the 2 bodies. 

For example, Body A exerts a force (F_A) on Body B, which is called the “action force”. The force exerted by Body B (F_B) on Body A is called the “reaction force”. Some physicists call this the “action-reaction law”. 

Other situations involve both bodies jointly determining the magnitude and directions of the forces, which defeats the aim of separately identifying which one is the “action” force or “reaction” force. A 1992 study stated that there is no purpose in separate labelling of the “action” and “reaction” forces in interactions where both forces occur simultaneously. They argued both forces are part of a single interaction, and neither force exists without the other. 

Examples of Newton’s 3rd law in action include: 

• A bipedal human walking by pushing their feet against the floor, and the floor pushes against their feet. 
• Tires of a car pushing against the road surface, while the road surface pushes pack on the tires. Therefore the tires and road simultaneously push against each other. 
• A swimmer pushes the water backwards, while the water simultaneously pushes the swimmer forwards, meaning the both the person and the water push against each other. 

Hewitt (2006) described a situation where friction is minimal, minimising the action and reaction forces. A person or car on ice experiences difficulty exerting the action force to produce the needed reaction force. 

How important and valid are Newton’s 3 laws? 

For over 200 years, many experiments and observations verified Newton’s 3 laws, and there were accurate approximations at the scales and speeds of everyday life. Combined with Newton’s law of universal gravitation and the mathematical techniques of calculus, Newton’s laws of motion provided a unified quantitative explanation for a wide range of physical phenomena. 

Although Newton’s laws helped generate decent approximations for macroscopic objects under everyday conditions, they don’t seem to work at minuscule scales, at extremely high speeds, or in extremely strong gravitational fields. They fail to explain phenomena such as conduction of electricity in a semiconductor, optical properties of substances, errors in non-relativistically corrected GPS systems and superconductivity. To explain these phenomena, it requires more sophisticated physical theories, including general relativity and quantum field theory. 

In quantum mechanics, concepts such as force, momentum, and position are defined by linear operators that operate on the quantum state. At speeds significantly slower than the speed of light, Newton’s laws function just as precisely as these operators as they are for classical objects. However, at speeds close to the speed of light, only the 2nd law holds in the original form (F = dp/dt), where F and p are four-vectors. 

How does Newton’s laws relate to the conservation laws? 

In modern physics, the laws of conservation of momentum, energy, and angular momentum are generally more valid than Newton's laws, since they apply to both light and matter, and to both classical and non-classical physics. They all simply state that “momentum, energy and angular momentum cannot be created or destroyed.” 

Because force is the time derivative of momentum, the concept of force becomes redundant and subordinate to the conservation of momentum, thus is left out of fundamental theories (e.g. quantum mechanics, quantum electrodynamics, general relativity, etc.). The standard model explains in detail how the 3 fundamental forces, known as ‘gauge forces’, originate out of exchange by virtual particles. Other forces, such as gravity and fermionic degeneracy pressure, also arise from the conservation of momentum. Note that the conservation of 4-momentum in inertial motion via curved space-time results in gravitational force in the theory of general relativity. Applying the space derivative (a momentum operator in quantum mechanics) to the overlapping wave functions of a pair of fermions (particles with half-integer spin) results in shifts in the maxima of compound wave-function away from each other, which is observable as the "repulsion" of the fermions. 

Newton expounded his 3rd law of motion within a world-view that presumed instantaneous action at a distance between material particles. However, modern physics omitted this ‘action at a distance’ concept except for subtle effects involving quantum entanglement. In reference to Bell’s theorem, no local model can reproduce the predictions of quantum theory. Despite only being an approximation, modern engineering and all other practical applications involved the motion of vehicles and satellites, demonstrating the extensive use of the concept of action at a distance. 

In the 19th century, the discovery of the 2nd law of thermodynamics by Carnot demonstrated that not every physical quantity is conserved over time. This disproved the validity of inducing the opposite metaphysical view from Newton's laws. Therefore, a "steady-state" worldview based solely on Newton's laws and the conservation laws does not take entropy into account. 

https://en.wikipedia.org/wiki/Euler%27s_laws_of_motion
http://emweb.unl.edu/NEGAHBAN/EM373/note19/note19.htm

What are Euler’s laws of motion?

In classical mechanics, these equations of motion are an extension of Newton’s laws of motion for point particle to rigid body motion. They were formulated by Leonhard Euler about 50 years after Isaac Newton formulated his laws in 1687. 

(1) Euler’s First Law 

This law states that the linear momentum of a body (p or G) is equal to the product of the mass of the body (m) and the velocity of its centre of mass (v_cm). 

p = m*v_cm 

Because there is an equal and opposite force, internal forces between the particles making up a body don’t contribute to the change in the body’s total momentum, which results in no net effect. The law is also stated as: 

F = m*a_cm 

— a_cm = dv_cm/ dt : Acceleration of the centre of mass

— F = dp/dt : Total applied force on the body, which is the time derivative of the previous equation (m is a constant) 

How were these laws derived?

The internal forces in a deformable body don’t necessarily distribute equally throughout i.e. the stresses vary from one point to the next. This variation of internal forces throughout the body is governed by Newton’s 2nd law of motion and of conservation of both linear and angular momenta, which are applied to a mass particle that are extended in continuum mechanics to a body of continuously distributed mass for simplicity. Continuous bodies, however, are dictated by the Euler’s laws of motion. If a body is represented as a collection of discrete particles, each governed by Newton's laws of motion, then we can derive Euler's equations from Newton’s laws. Furthermore, Jacob Lubliner (2008) stated that Euler’s equations can be interpreted as axioms that describe the the laws of motion for extended bodies, independently of any particle distribution. 

The total body force applied to a continuous body with mass (m), mass density (ρ), and volume (V), is the volume integral integrated over the volume of the body:

— b = Force acting on the body per unit mass (dimensions of acceleration, which can be misinterpreted as the “body force”) 

— dm = ρ dV : An infinitesimal mass element of the body 

Body forces and contact forces acting on the body lead to corresponding moments (torques) of those forces relative to a given point. Thus, the total applied torque (M) about the origin is given by: 

M = M_B + M_C 

— M_B & M_C respectively = This indicates the moments caused by the body and contact forces. 

Thus, the sum of all applied forces and torques (with respect to the origin of the coordinate system) acting on the body can be given as the sum of a volume and surface integral:

— t = t(n) : Surface traction, integrated over the surface of the body, in turn ’n’ denotes a unit vector normal and directed outwards to the surface (S). 

Let the coordinate system (x₁, x₂, x₃) be an inertial frame of reference,

— r: Position vector of a point particle in the continuous body with respect to the origin of the coordinate system 

— v = dr/dt : Velocity vector of that point

Euler’s First Axiom or Law concerns the law of balance of linear momentum or balance of forces. It states that, in an inertial frame, the rate at which the change of linear momentum (p) of an arbitrary portion of a continuous body with respect to time is equal to the total applied force (F) acting on that portion. This statement can be expressed mathematically as,

— v = Velocity 

— V = Volume 

— dp & dL = Material derivatives of linear momentum and angular momentum respectively 

https://en.wikipedia.org/wiki/History_of_classical_mechanics

https://en.wikipedia.org/wiki/Classical_mechanics

What are mechanics? 

Mechanics is a branch of physics concerned with the motions of macroscopic objects. 

There are many different topics on classical mechanics: 

— Newtonian mechanics = Original theory of motion (kinematics) and forces (dynamics) 

— Analytical mechanics = Reformulation of Newtonian mechanics with an emphasis on system energy, rather than on forces. The 2 main branches include: 

- Hamiltonian mechanics = A theoretical formalism based on the principle of conservation of energy. 

- Lagrangian mechanics = Another theoretical formalism based on the principle of least action. 

— Classical statistical mechanics = This generalises ordinary classical mechanics to understand systems in an unknown state, which often derives thermodynamic properties. 

— Celestial mechanics = The motion of bodies in space: planets, comets, stars, galaxies etc 

— Astrodynamics = Spacecraft navigation etc. 

— Solid mechanics = This investigates elasticity, plasticity, viscoelasticity exhibited by deformable solids.

— Fracture mechanics = This studies the propagation of cracks in solids. 

— Acoustics, sound = This studies density variation propagation in solids, fluids and gases. 

— Statics = This looks into semi-rigid bodies in mechanical equilibrium. 

— Fluid Mechanics = This studies the motion of fluids. 

— Soil Mechanics = This investigates the mechanical behaviour of soils. 

— Continuum mechanics = The mechanics of continua (both solid and fluid) 

— Hydraulics = The mechanical properties of liquids 

— Fluid statics = Liquids in equilibrium 

— Applied mechanics = Engineering mechanics 

— Biomechanics = This looks into the mechanical aspects of biological solids, fluids etc. 

— Biophysics = Physical processes in living organisms 

— Relativistic or Einsteinian mechanics = Universal gravitation, relating to special relativity and general relativity. 

In terms of detail, I’ll discuss the bolded topics in the next few blogs post and unbolded topics in other blog posts. 

There are also many branches of quantum mechanics: 

— Schrödinger wave mechanics = Describes the movements of the wave-function of a single particle.

— Matrix mechanics = An alternative formulation of mechanics that considers systems with a finite-dimensional state space. 

— Quantum statistical mechanics = This topic generalises ordinary quantum mechanics to consider systems in an unknown state, which often leads to the derivation of thermodynamic properties. 

— Particle physics = Motion, structure, and reactions of particles 

— Nuclear physics = Motion, structure, and reactions of nuclei 

— Condensed matter physics = Condensed matter physics = Quantum gases, solids, liquids, etc. 

I’ll discuss quantum mechanics in another post.

What is classical mechanics? 

They describe the motion of macroscopic objects, ranging from projectiles to machinery, astronomical objects, such as spacecraft, planets, stars and galaxies. If we know the present state of object, we can predict by the laws of classical mechanics how it will move in the future (determinism) and how it has moved in the past (reversibility). The earliest development of classical mechanics are often referred to as Newtonian mechanics, which consists of the concepts proposed and the mathematical methods made by Isaac Newton, Gottfried Wilhelm Leibniz and others in the 17th century to describe the motion of bodies under the influence of a system of forces. 

Later, more abstract methods were developed, which lead to  the reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. Throughout the the 18th and 19th centuries, these theories extended beyond Newton's work, particularly through their use of analytical mechanics. 

To model the motion of real-world objects as point particles (objects with negligible size), we have to characterise a small number of parameters including its position, mass, and the forces applied to it. In reality, the kind of objects that classical mechanics can describe always have a non-zero size. That means the physics of very small particles, such as the electron, is more accurately described by quantum mechanics. Objects with non-zero size have demonstrated to have more  complicated behaviour than hypothetical point particles, because of the additional degrees of freedom e.g. a baseball spinning during its motion. However, if these point particles are treated as composite objects i.e. consisting of a large number of collectively acting point particles, then the results of such particles can be used to study such objects. That means the centre of mass of a composite object behaves like a point particle. 

Classical mechanics uses common-sense notions of the existence and interaction between matter and forces. This assumes matter and energy have definite, knowable attributes such as location in space and speed, as well as instantaneously acting forces. 

https://en.wikipedia.org/wiki/Kinematics
https://en.wikipedia.org/wiki/Position_(geometry)

What are kinematics?

Kinematics describes the motion of points, bodies (objects) and systems of bodies (groups of objects) without considering the forces that cause them to move. Physicists often refer kinematics as the “geometry of motion” and view it as a branch of mathematics. Kinematics problems outline the geometry of the system and declare the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then geometric arguments are used to determine the position, velocity and acceleration of any unknown parts of the system. 

The term kinematic is the English version of A.M. Ampère’s cinématique, which he derived from the Greek word κίνημα (kinema) meaning "movement, motion”, which itself is derived from κινεῖν (kinein) meaning “to move”. Although kinematic and cinématique are related to the French word cinéma, they don’t share any roots from it. Douglas Harper believed both words share a common root word, which is the Greek word for movement and from the Greek γρᾰ́φω (grapho) meaning “to write”. Since cinéma is an abbreviation of cinématographe, it means "motion picture projector and camera”. 

Describe a particle’s trajectory in a non-rotating frame of reference 

People studying particle kinematics study the trajectory of particles. They define the position of a particle as the coordinate vector from the origin of a coordinate frame to the particle. 

Example: Imagine a light tower 100m west from your home, where the coordinate frame is centred on your home, such that east is in the +x-direction and north is in the +y-direction. That means the coordinate vector to the base of the tower is r = (-100, 0, 0). If the light tower is 20 m high, then the height measured along the z-axis makes the coordinate vector at the top of the tower as r = (-100, 20, 0). 

A 3D coordinate system is often used to define the position of a particle. If a particle is constrained to move within a plane, then a 2D coordinate system is sufficient. All observations wouldn’t be complete without being described with respect to a reference frame. 

This diagram shows the kinematic quantities of a classical particle: 

— mass (m), position (r), velocity (v), acceleration (a).

The position vector of a particle is drawn from the origin of the reference frame to the particle. It indicates both the distance between the point and the origin and the point’s direction from the origin. In 3 dimensions, the position of point P can be written as:

— x_P, y_P, z_p = Cartesian coordinates 

— î, j^, k^ = Unit vectors along the x,y and z coordinate axes respectively. 

The magnitude of the position vector |P| indicates the distance between the point P and the origin.

The direction cosines of the position vector give out a quantitative measure of direction. Note that the position vector of a particle is not unique. The position vector of a given particle is different relative to different frames of reference. 

The trajectory of a particle is a vector function of time, P(t), which defines the curve traced by the moving particle. This is expressed as: 

— x_P, y_P, z_P = Each coordinate have a function of time. 

What is position and its derivatives? 

In geometry, a position or position vector, also known as a location vector or radius vector, is a Euclidean vector that represents the position of a point P in space in relation to an arbitrary reference origin (O). The straight line segment from O to P is the displacement or translation that maps the origin to P, often denoted x, r or s.

i. 3 Dimensions 

In 3 dimensions, any set of 3D coordinates and their corresponding basis vectors can be used to define the location of a point in space. If we use the Cartesian coordinate system, or spherical polar coordinates or cylindrical coordinates: 

— t = A parameter, owing to their rectangular or circular symmetry. 

These different coordinates and corresponding basis vectors represent the same position vector. More general curvilinear coordinates may be used instead and in other contexts such as continuum mechanics and general relativity.

This diagram illustrates a space curve in 3D. The position vector (r) is parameterised by a scalar (t). At r = a (red) line, it is tangential to the curve and the blue plane is normal to the curve.

ii. n dimensions 

To abstract an n-dimensional position vector, we use linear algebra to express it as a linear combination of basis vectors: 

The set of all position vectors forms position space (i.e. a vector space whose elements are the position vectors). Note positions can be added (via vector addition) and scaled in length (via scalar multiplication) to obtain another position vector in the same space. The notion of space is “intuitive” since each x_i (i = 1,2,3,…,n) can have any value. Therefore, the collection of values defines a point in space. 

The dimension of the position space is denoted n (also dim(R) = n). The coordinates of the vector (r) with respect to the basis vectors e_i are denoted x_i. The vector of coordinates forms the coordinate vector or n-tuple (x₁, x₂, …, x_n). 

Each coordinate x₁ may be parameterised with a number of parameters (t), with one parameter x₁(t) describing a curved 1D path, 2 parameters x₁(t₁, t₂) describing a curved 2D surface, 3 x₁(t₁, t₂, t₃) describing a curved 3D volume of space, and so on. 

The linear span of a basis set B = {e₁, e₂, …, e_n) equals the position space R, denoted span(B) = R. 

What are the applications of position vectors? 

i. Differential geometry 

Mathematicians use position vector fields describe continuous and differentiable space curves, in which case the independent parameter needs not be time, but can be (e.g.) arc length of the curve. 

ii. Mechanics 

In any equation of motion, the position vector r(t) function defines the motion of a particle (i.e. a point mass), which is its location relative to a given coordinate system at some time (t). Defining motion in terms of position requires parametrisation of each coordinate by time. Since each successive value of time corresponds to a sequence of successive spatial locations given by the coordinates, the continuum limit of many successive locations is a path the particle traces. For 1-dimensional cases, the position has only 1 component, so it effectively is a scalar coordinate. It can also be a vector in the x-direction, or the radial r-direction. Equivalent notations include:  

What are the derivatives of position? 

For a position vector (r) that is a function of time (t), the time derivatives are computed with respect to time. These derivatives are commonly utilised in the study of kinematics, control theory, engineering and other sciences. 

Velocity: 

v = dr/dt 

— dr = An infinitesimally small displacement (vector) 

Acceleration: 

a = dv/dt = d²r / dt² 

Jerk: 

j = da/dt = d²v / dt² = d³r / dt³

James Stewart (2001) explained the above equations are the 1st, 2nd and 3rd derivatives of position, respectively, that are commonly used in basic kinematics. Higher-order derivatives lead to improvements of approximations of the original displacement function, meaning they accurately represent the displacement function as a sum of an infinite sequence. 

https://en.wikipedia.org/wiki/Distance

What is distance? 

It is a numerical measurement of how far apart objects or points are. Used in physics or everyday life, distance can refer to a physical length or an estimation based on certain criteria (e.g. 2 blocks down the road). In most cases, "distance from A to B" is interchangeable with "distance from B to A”. 

A physical distance has several interpretations: 

• Distance travelled = The length of a specific path travelled between two points, such as the distance walked while navigating a maze. 
• Straight-Line (Euclidean) Distance = The length of the shortest possible path through space, between 2 points, that could be taken if there were no obstacles (usually formalised as Euclidean distance). 
• Geodesic Distance = The length of the shortest path between 2 points while remaining on some surface, such as the great-circle distance along the curve of the Earth. 
• The length of a specific path that returns to the starting point, such as a ball thrown straight up, or the Earth when it completes 1 orbit. 
• Circular Distance = The distance travelled by a wheel, used for vehicular design or mechanical gear design. Since the  circumference of the wheel is 2π multiplied by the radius (assuming to be 1), then each revolution of the wheel is equivalent of the distance 2π radians. In engineering ω = 2π*f is often used, where f = frequency. 
• “Manhattan distance” = Rectilinear distance, named after the number of blocks north, south, east, or west a taxicab must travel on to reach its destination on the grid of streets in parts of New York City. 
• “Chessboard distance” = Known as Chebyshev distance, it is the minimum number of moves a king must make on a chessboard to travel between 2 squares.

This diagram illustrates manhattan distance on a grid. 

“Distance” is also used as an analogy for measuring non-physical entities in certain ways. Computer scientists uses the notion of the “edit distance” between 2 strings. For example, the words “dog” and “dot” vary by only one letter. These are more alike than "dog" and “cat”. This concept of edit distance is used in spell checkers and in coding theory. It can be mathematically evaluated in different ways, such as: 

• Levenshtein distance = In information theory, linguistics and computer science, it is a string metric for measuring the difference between 2 sequences. 
• Hamming distance = In information theory, it is the number of positions at which the corresponding symbols are different between 2 strings of equal length. 
• Lee distance = In coding theory, it is the distance between 2 strings x₁, x₂, … x_n and y₁, y₂,…y_n of equal length n over the q-ary alphabet {0,1,2,…q - 1} of size q > 2. 
• Jaro-Winker distance = In computer science and statistics, it is a string metric that measures an edit distance between 2 sequences.
• Metric space = In mathematics, it is a set for which distances between all members of the set are defined. This means many different types of “distances” can be calculated such as for traversal of graphs, comparison of distributions and curves, and using unusual definitions of “space" e.g. using a manifold or reflections. 
• In graph theory, distance is used to describe social networks, such as with the Erdös number or the Bacon number, which is the number of collaborative relationships away a person is from prolific mathematician Paul Erdös or actor Kevin Bacon, respectively. 

What is directed distance? 

Directed distance can be determined along straight lines and along curved lines. Along straight lines, they are vectors that indicate the distance and direction between a starting point and an ending point. Imagine 3 points in a Euclidean vector space (A, B and C). Draw out the directed distance from point A to point C in the direction of point B along a line AB. This is the same distance from point A to point C if point C falls on the ray AB. However, if point C falls on the ray BA (i.e. if point C  is not on the same side of point A as point B is), then it forms the negative of the previous directed distance. 

For example, to measure the directed distance from the New York City Main Library flag pole to the Statue of Liberty flag pole, we need to mark a starting point (library flag pole), an ending point (statue flag pole), a specified direction (-38^o) and a distance (8.72 km). 

Another type of direct distance is between 2 different particles or point masses at a given time. For example, the distance from the centre of gravity of Earth A and the centre of gravity of Moon B. However, this does not simply motion from Earth to the Moon. 

A direct distance along a curved line is not a vector, thus it is represented by a segment of that curved line defined by endpoints A and B. Information about this distance indicates the direction of an ideal or real motion from one endpoint of the segment to the other. For example, labelling the 2 endpoints as A and B can indicate the direction, assuming the sequence from A to B is ordered, which implies that A is the starting point. 

https://en.wikipedia.org/wiki/Displacement_(geometry)

What is displacement?

In geometry, displacement is a vector that defines the shortest length (distance) from the initial to the final position of a point P that undergone motion. It quantifies both the distance and direction of the net or total motion along a straight line from the initial position to the final position of the point trajectory. 

A displacement (s) can be described as a relative position resulting from the motion of a point particle i.e. the final position x_f of a point relatively to its initial position x_i. Therefore, the corresponding displacement vector can be defined as the difference between the final and initial positions (Δx): 

s = x_f - x_i = Δx

To deal with the motion of a rigid body, we have to consider the rotations of the body when evaluating its displacement. 

Discuss the mathematics of distance

i. Geometry 

In analytic geometry, the distance between 2 points of the 2D plane with x and y axes can be worked out using the distance formula. The distance between (x₁, y₁) and (x₂, y₂) is given by:

Similarly, given points (x₁, y₁, z₁) and (x₂, y₂, z₂) in 3D-space, the distance between them is:

These formulas are derived through the construction of a right-angled triangle with a leg on the hypotenuse of another (with the other leg orthogonal to the plane containing the 1st triangle) and application of Pythagorean theorem. In complicated geometry, this is also called Euclidean distance, derived from the Pythagorean theorem, which doesn’t hold in non-Euclidean geometries. This distance formula can also be extended into the arc-length formula.

ii. Distance in Euclidean space 

In the Euclidean space (Rⁿ), the distance between 2 points is given by the Euclidean distance (2-norm distance). For a point (x₁, x₂, …, x_n) and a point (y₁, y₂, …, y_n), the Minkowski distance of order p (p-norm distance) is defined as: 

• p can be any real number larger than or equal to 1. The triangle inequality doesn’t hold for values of p less than 1. 
• 1-norm distance = Colourfully called the ‘taxicab norm’ or ‘Manhattan distance’, it is the distance a car would drive in a city laid out in square blocks (if there are no one-way streets). 
• 2-norm distance = Euclidean distance, a generalisation of the Pythagorean theorem to more than 2 coordinates. This distance can be calculated by using a ruler to measure the length between 2 points. 
• p-norm distance = Rarely used for values of p other than 1,2 and infinity. 
• Infinity norm distance = Chebyshev distance. In 2D, it is the minimum number of moves kings require to travel between 2 squares on a chessboard. 

— r^—>(0) = Trajectory (path) between the 2 points 

— D = Length of the trajectory 

When r = r*, where r* is the optimal trajectory, the distance is the minimal value of this integral. In a Euclidean case, this optimal trajectory (shortest path between 2 points) is simply a straight line. Straight lines can formally be obtained by solving the Euler-Lagrange equations for the above integral. In non-Euclidean manifolds (curved spaces), where the nature of the space is represented by a metric tensor (g_ab), the integrand has to be modified to include Einstein summation convention. 

iv. Generalisation to higher-dimensional objects 

The Euclidean distance between 2 objects may also be generalised to the case where the objects are no longer considered points but rather higher-dimensional manifolds, such as space curves. Therefore, we can discuss concepts of distance between 2 strings. Since these objects deal with additional concepts such as non-extensibility, curvature constraints, and non-local interactions that enforce non-crossing, they are integral to the notion of distance. The distance between the 2 manifolds is the scalar quantity that results from minimising the generalised distance functional, which represents a transformation between the 2 manifolds:

The above double integral expresses the generalised distance functional between 2 polymer conformation. 

— s = Spatial parameter 

— t = Pseudo-timer 

— This means that r^—>(s,t = t_i) is the polymer / string conformation at time t, and is parameterised along a string length by s. 

— Similarly r^—>(s = S,t) defines the trajectory of an infinitesimal segment of the string during transformation of the entire string from conformation r^—>(s, 0) to conformation r^—>(s, T). 

— λ = This cofactor is a Lagrange multiplier that ensures that the length of the polymer remains the same during the transformation. 

If 2 discrete polymers are inextensible, then the minimal-distance transformation between them no longer involves purely straight-line motion, even on a Euclidean metric. Studies by Plotkin suggested such generalised distances can be applied to the problem of protein folding. This generalised distance is analogous to the Nambu-Goto action in string theory, however there doesn’t seem to be any exact correspondence. This is due to the Euclidean distance in 3D space not being equivalent to the space-time distance minimised for the classical relativistic string. 

v. Algebraic distance 

This metric is often used in computer vision minimised by least squares estimation. For curves or surfaces given by the equation x^TC_x = 0 (such as a conic in homogeneous coordinates), the algebraic distance from the point x’ to the curve is x’^TCx’. This may be an “initial guess” for geometric distance in the refinement of estimations of the curve by more accurate methods, such as non-linear least squares. 

vi. General metric 

In geometry, a distance function on a given set M is a function d: M x M —> R, where R denotes the set of real numbers, that satisfies the following conditions: 

— d(x,y) > 0, and d(x,y) = 0 if and only if x = y. : Distance is positive between 2 different points, and is zero precisely from a point to itself. 

— Symmetry: d(x,y) = d(y,x) : Distance between x and y is the same in either direction

— Triangle inequality: d(x,z) < d(x,y) + d(y,z). : The distance between 2 points is the shortest distance along any path. Such a distance function is known as a metric. Together with the set, it makes up a metric space. 

For example, the usual definition of distance between 2 real numbers x and y: d(x,y) = |x-y|, which satisfies the 3 conditions above, and corresponds to the standard topology of the real line. However distance on a given set depends on the definition being chosen such as d(x,y) = 0 if x = y, and 1. This also defines a metric, but lead to a different topology called the “discrete topology”, which means numbers can’t be arbitrarily close. 

vii. Distances between sets and between a point and a set 

Before we measure the distance between celestial bodies, we have to distinguish the surface-to-surface distance from the centre-to-centre distance. If the former is less than the latter, as for low earth orbits, the first tends to be quoted (altitude), otherwise, e.g. for the Earth–Moon distance, the latter. 

The 2 common definitions for the distance between 2 non-empty subsets of a given metric space: 

(I) 1 version of distance between 2 non-empty sets is the infimum of the distances between any 2 of their respective points.

This is a symmetric premetric. If a collection of sets involve touching or overlapping sets, they are not “separating”, because the distance between 2 different but touching or overlapping sets is zero. Also it is not hemimetric, i.e. the triangle inequality doesn’t hold, except in special cases. Therefore, in special cases, this distance makes a collection of sets a metric space. 

d(A,B) > d(A,C) + d(C,B) 

(II) The Hausdorff distance is the larger of 2 values: 

— The supremum = A point ranging over one set, of the infimum, for a second point ranging over the other set, of the distance between the points. 

— The other value being likewise defined but with the roles of the 2 sets swapped. 

This distance makes the set of non-empty compact subsets of a metric space itself a metric space. 

The distance between a point and a set is the infimum of the distances between the point and those in the set. According to the first-mentioned definition above of the distance between sets, this corresponds to the distance from the set containing only this point to the other set. Therefore, we can simplify the definition of the Hausdorff distance to “the larger of 2 values, one being the supremum, for a point ranging over one set, of the distance between the point and the set, and the other value being likewise defined but with the roles of the two sets swapped”. 

— Infimum = The largest quantity that is less than or equal to each of a given set or subset of quantities.

— Supremum = The smallest quantity that is greater than or equal to each of a given set or subset of quantities. 

viii. Graph theory 

In graph theory, the distance between 2 vertices is the length of the shortest path between those vertices. 

ix. Statistical distances 

In statistics and information geometry, there are many kinds of statistical distances, such as divergences e.g. Bregman divergences and f-divergences. These include and generalise many of the notions of “difference between 2 probability distributions”, which are being studied geometrically, as statistical manifolds. The squared Euclidean distance forms the basis of least squares i.e. the most basic Bregman divergence. In information theory, the relative entropy (Kullback-Leibler divergence) allows the analogous study of the maximum likelihood estimation geometrically, which is the most basic f-divergence, as well as a Bregman divergence. Statistical manifolds corresponding to Bregman divergences are flat manifolds in the corresponding geometry. This allows an analog of the Pythagorean theorem to be used for linear inverse problems in inference by optimisation theory. Other statistical distances include the Mahalanobis distance and the energy distance.

https://en.wikipedia.org/wiki/Speed

https://en.wikipedia.org/wiki/Velocity

What is speed? 

An object’s speed is defined as the magnitude of the change of its position, thus a scalar quantity. Hewitt (2006) believed Italian physicist Galileo Galilei was credited for being the first to measure speed by working out the distance travelled and the time it takes to cover that distance. Galileo defined speed as the distance covered per unit of time.

v = d / t 

— v = Speed 

— d = Distance 

— t = time 

For example, a sprinter who ran 100 metres in a time of 10 seconds, has a speed of 10 metres per second. 

An object’s instantaneous speed is defined as the limit of the average speed as the duration of the time interval approaches zero. It also means speed, at some instant, is assumed constant during a brief period of time. Next time you drive your car, look at the speedometer and try to read your car’s instantaneous speed at any instant. For example, a car travelling at 50 km/h generally travels at that speed constantly for less than an hour due to friction and aerodynamic drag. If we eliminated all variables and factors that decreases the car’s speed, it would travel 50 km after exactly one hour driving at 50 km/h. 

In mathematical terms, the instantaneous speed (v) is defined as the magnitude of the instantaneous velocity (v) (i.e. the derivative of the position (r) with respect to time:

If s is the length of the path (also known as the distance) travelled until time (t), the speed equals the time derivative of s: 

v = ds/dt 

In the case where the velocity is constant (i.e. constant speed in a straight line), this can be simplified to v = s/t. The average speed over a finite time interval is the total distance travelled divided by the time duration. 

An object’s average speed in an interval of time is defined as the distance travelled by the object divided by the duration of the interval. For example, if car A travels 60 kilometres after 1 hour, the car’s average speed is 60 km/h. If car B travels 360 kilometres after 6 hours, that car’s average speed is also 60 km/h. 

However, average speed doesn’t describe the speed variations that took place during shorter time intervals (i.e. the entire distance covered divided by the total time of travel), hence average speed is often quite different from a value of instantaneous speed. So, if the average speed and the time of travel are known, the distance travelled can be calculated through rearrangement of the definition. 

If we use the above equation for an average speed of 60 km/h on a 6-hour trip, the distance covered is found to be 360 km. In graphical language, the slope of a tangent line at any point of a distance-time graph is the instantaneous speed at this point. Moreover, the slope of a chord line of the same graph is the average speed during the time interval covered by the chord. Therefore, 

v_av = s/t 

What’s tangential speed?

Tangential speed (or tangential velocity) is the linear speed of an object moving along a circular path. For example, a point on the outside edge of a turntable or carousal travels a greater distance in one complete rotation than a point closer to the centre. Points on the outer edge of a circle travelling a greater distance means they travel at a greater (linear) speed compared to points closer to the axis. Because the direction motion is tangent to the circumference of a circle, it is known as tangential speed. For circular motion problems, the terms ‘linear speed’ and ‘tangential speed’ are used interchangeably. 

Rotational speed (or angular speed) is defined as the number of revolutions per unit of time. For example, all parts of a rigid carousal or turntable turn about the axis rotation in the same amount of time. Hence, all parts share the same rate of rotation, or the same number of rotations or revolutions per unit of time. Rotational speed is often expressed in revolutions per minute (RPM) or in terms of number of "radians" turned in a unit of time. A full rotation is exactly 2π radians or a little more than 6 radians. When rotational speed is assigned direction, this becomes rotational velocity or angular velocity, which is a vector whose magnitude is the rotational speed. 

There is a direct proportional relationship between tangential speed and rotational speed at any fixed distance from the axis of rotation. The higher the RPMs, the faster the rotational speed. However, tangential speed depends on radial distance (the distance from the axis). For a platform rotating with a fixed rotational speed, the tangential speed in the centre is zero. Meanwhile, towards the edge of the platform, the tangential speed increases proportionally to the distance from the axis. It’s shown by this equation: 

v = Tangential speed 

ω = Rotational speed 

r = Radial distance 

Increases to the rate of rotation increases (ω increases) and the radial distance from the axis (r increases) would lead to increases in tangential speed. When proper units are used for tangential speed, rotational speed and radial distance, the direct proportion of ‘v’ to both ‘r’ and ‘ω’ becomes the exact equation: v = r*ω 

Thus ‘v’ is directly proportional to ‘r’ when all parts of a system simultaneously have the same ω, as for a wheel, disk, or rigid wand. 

What are the units of speed? 

— Metres per second (ms^-1, m/s) = SI derived unit 

— Kilometres per hour (km/h) 

— Miles per hour (mph, mi/h) 

— Knots (nautical miles per hour, kn or kt) 

— Feet per second (fps, ft/s) 

— Mach number (dimensionless) = Speed divided by the speed of sound 

— In natural units (dimensionless) = Speed divided by the speed of light in vacuum (symbol c = 299792458 m/s). 

How to convert between the common units of speed? 

* = Approximate values 

What are examples of different speeds? 

— Approximate rate of continental drift = 0.00000001 m/s 

— Speed of a common snail = 0.001 m/s 

— A brisk walk = 1.7 m/s 

— Average jog = 2.2352 m/s 

— A typical road cyclist = 4.4 m/s 

— A fast martial arts kick = 7.7 m/s 

— Average cruise ship speed = 10.2889 m/s 

— Sprint runners = 12.2 m/s 

— Typical suburban average speed limit in most countries = 13.8 m/s 

— Taipei 101 Observatory Elevator = 16.7 m/s 

— Approximate speed of a cough = 22.352 m/s 

— Typical rural speed limit = 24.6 m/s 

— Category 1 hurricane = 33 m/s 

— Highest recorded human-powered speed = 37.02 m/s 

— Approximate speed of a sneeze = 44.704 m/s 

— Speed of fastest cricket ball delivered by Shoaib Akhtar in 2003 = 44.805556 m/s 

— Muzzle velocity of a paintball marker = 90 m/s 

— Average speed of action potential across myelinated axons of nerve cells = 113.995 m/s 

— Maximum speed of F4 Tornado = 116.23 m/s

— Speed of Maglev Train = 167.5 m/s 

— Cruising speed of a Boeing 747-8 passenger jet = 255 m/s 

— The official land speed record = 341.1 m/s 

— The speed of sound in dry air at sea-level pressure and 20 ^oC = 343 m/s 

— Approximate speed of earth’s spin at the equator = 460 m/s 

— Muzzle velocity of a 7.62x39 mm cartridge = 710 m/s 

— Official flight airspeed record for jet engined aircraft = 980 m/s 

— Space shuttle on re-entry = 7800 m/s 

— Escape velocity on Earth = 11,200 m/s 

— Voyager 1 relative velocity to the Sun in 2013 = 17,000 m/s 

— Average orbital speed of Earth around the Sun = 29,783 m/s 

— Fastest recorded speed of the Helios probes = 70,220 m/s 

— Speed of electricity in a 12-gauge copper wire = 285,102,627 m/s 

— Speed of light in vacuum (c) = 299,792,458 m/s 

What is velocity? 

The velocity of an object is the rate of change of its position with respect to a frame of reference, and in a function of time. It specifies both the object’s speed and direction of motion (e.g. 50 km/h to the south). 

When an object is travelling at a constant speed in a constant direction, it has a constant velocity.  An object with constant velocity is constrained to travel in a straight path. For example, a car travelling at a constant 60 km/hr in a circular path is demonstrated to have constant speed. However, it doesn’t have a constant velocity as its direction is constantly changing. Therefore the car is always accelerating. 

What’s the difference between speed and velocity?

Speed denotes only how fast an object is moving, whereas velocity denotes both how fast and in which direction the object is moving. For example, if a car travels at 60 km/h, we have specified its speed. 

Imagine a toy train rolling along a circular track. When it moves in a circular path and returns to its starting point, its average velocity is zero. Whereas its average speed can be calculated by dividing the circle’s circumference by the time taken to move around the circle. This is because calculating the average velocity requires the displacement between the starting and end points, whereas the average speed considers only the total distance travelled. 

https://en.wikipedia.org/wiki/Acceleration

What is acceleration? 

In mechanics, it is defined as the rate of change of the velocity of an object with respect to time. Because it has both magnitude and direction, acceleration is a vector quantity. The orientation of an object’s acceleration is determined by the orientation of the net force acting on that object. Newton’s 2nd law described the magnitude of an object's acceleration as a combination of 2 causes: 

— Net balance of all external forces acting onto the object i.e. Magnitude is directly proportional to this net resulting force. 

— Object’s mass, depending on the materials it is made out of i.e. Magnitude is inversely proportional to the object’s mass

For example, envisage a train starting from a stationary position (zero velocity, in an inertial frame of reference) and travels along a straight piece of track to the right at increasing speeds. This train is accelerating in the right direction. If the train turns after passing a switch, it accelerates towards a new direction. 

• The forward acceleration of the train is labelled the linear (or tangential) acceleration), causing the passengers’ reaction force they experience, pushing those facing forwards backwards into their seats. 
• During the change in direction, the passengers experience radial (as orthogonal to tangential) acceleration, causing the passengers’ reaction force they experience, pushing them sideways. 
• If the speed of the train decreases, the acceleration is oriented in the opposite direction of the train’s forward velocity, also known as deceleration or retrograde burning (in spacecraft). The reaction force passengers experience during deceleration pushes them forwards. 

Since both acceleration and deceleration involve changes in velocity, they are treated the same mathematically. 

 An object’s average acceleration over a period of time is defined as the change in velocity (Δv) divided by the duration of the period (Δt). 

ā = Δv / Δt

An object’s instantaneous acceleration is the limit of the average acceleration over an infinitesimal interval of time. In calculus, instantaneous acceleration is the derivative of the velocity vector with respect to time: 

The integral of the acceleration function, a(t), is the velocity function, v(t). i.e. The area under the curve of an acceleration vs. time (a vs. t) graph corresponds to velocity. 

Acceleration can also be the 2nd derivative of position (x) with respect to time (t):

Acceleration has the dimensions of velocity (L/T) divided by time, i.e. L*T^-2.. The SI unit for acceleration is metre per second squared (m*s^-2). 

Acceleration has the dimensions of velocity (L/T) divided by time, i.e. L*T^-2.. The SI unit for acceleration is metre per second squared (m*s^-2).  

Acceleration is rate of change of velocity. At any point on a trajectory, the magnitude of the acceleration is given by the rate of change of velocity in both magnitude and direction at that point. The true acceleration at time (t) is found in the limit as time interval (Δt —> 0 of Δv/Δt). 

When an object moves in a circular path, such as a satellite orbiting the Earth, they accelerate due to the constant change in direction, despite its speed remaining constant. In this case, this object undergoes centripetal acceleration (directed towards the centre). 

From top to bottom: 

- Green curve: The integral of the velocity is the distance function s(t).

- Blue curve: The derivative of the acceleration is the velocity function v(t). 

- Red curve: Acceleration function (a) 

The velocity of a particle moving on a curved path as a function of time can be written as: 

v(t) = v(t)* [v(t) / v(t)] = v(t)* u_t(t) 

— v(t) = Speed of travel of along the path 

— u_t = v(t) / v(t) 

— u_t is a unit vector tangent to the path pointing in the direction of motion at the chosen moment in time. 

If we take into account both the changing speed, v(t), and the changing direction of u_t, then the acceleration of a particle moving on a curved path can be calculated using the chain rule of differentiation for the product of 2 functions of time as:

— u_n = Unit (inward) normal vector to the particle’s trajectory (also called the principal normal). 

— r = Instantaneous radius of curvature based upon the osculating circle at time (t). 

These components are called the tangential acceleration and the normal or radial acceleration (or centripetal acceleration in circular motion).

This sequence of diagrams illustrates an oscillating pendulum, with velocity and acceleration marked. It experiences both tangential and centripetal acceleration.

This graph shows components of acceleration for a curved motion. The tangential component (a_t) illustrates the change in speed of traversal, and points along the curve in the direction of the velocity vector (or in the opposite direction). The normal component or centripetal component for circular motion (a_c) illustrates the change in direction of the velocity vector, which is normal to the trajectory. It is pointing toward the centre of curvature of the path. 

Conversions between common units of acceleration 

Position vector (r) always points radially from the origin.

Velocity vector (v) is always tangent to the path of motion.

Acceleration vector (a) is not parallel to radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations. 

These 3 kinematic vectors are displayed in plane polar coordinates. Notice the setup is not restricted to 2D space, but it may represent the osculating plane in a point of an arbitrary curve in any higher dimension. 

https://en.wikipedia.org/wiki/Equations_of_motion

What are the equations of motion? 

In physics, the equations of motion describe the behaviour of a physical system in terms of its motion as a function of time. Generally, they describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables. The variables are usually spatial coordinates and time, but may include momentum components. Hand & Finch (2008) stated the most general choice are generalised coordinates, which can be any convenient variables characteristic of the physical system. In classical mechanics, the functions are defined in a Euclidean space, whereas in relativity, they are defined in curved spaces.  If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics. 

Uniform (or constant) acceleration is defined as the change in the object’s velocity by an equal amount in every equal time period. Whelan and Hodgson (1978) explained the equations of motion apply to a particle moving linearly, in 3D, in a straight line with constant acceleration. Since the position, velocity, and acceleration are collinear (i.e. parallel, lying on the same line), only the magnitudes of these vectors are important. Because the motion occurs along a straight line, this effectively simplifies the problem from 3D to 1D. 

Equations of motion: 

1. v = a*t + v₀ 
2. r = r₀ + v₀*t + 0.5*a*t² 
3. r = r₀ + 0.5*(v + v₀)*t  
4. v² = v₀² + 2*a*(r - r₀)  
5. r = r₀ + v*t - 0.5*a*t²

— r₀ = Initial position 

— r = Final position 

— v₀ = Initial velocity 

— v = Final velocity 

— a = Acceleration 

— t = Time interval

How were these equations derived? 

When we integrate the definitions of velocity and acceleration, which is subject to the initial conditions r(t₀) = r₀, and v(t₀) = v₀, we yield equations [1] and [2]. 

In magnitudes, 

[1] v = a*t + v₀ [2] r = 0.5*a*t² + v₀*t + r₀ 

Equation [3] involves the average velocity 0.5*(v + v₀). Intuitively, the velocity increases linearly, so the multiplication of average velocity and time gives us the distance travelled while velocity increases from v₀ and v. This can be illustrated graphically by plotting velocity against time as a straight line graph. Algebraically, we can follow from the solution of equation [1] for: 

a = (v - v₀)/t  

and then substituting into equation [2]: r = r₀ + v₀*t + 0.5*t*(v - v₀) 

Then we simplify it to get: 
r = r₀ + 0.5*t*(v + v₀)  

or in magnitudes: 
r = r₀ + (0.5*(v + v₀))*t  [3] 

From equation [3], we get: 
t = (r - r₀)*(2/(v + v₀)) 

When we substitute for ’t’ in equation [1]:
v = a*(r - r₀)*(2/(v + v₀)) + v₀ 
v*(v + v₀) = 2*a*(r - r₀) + v₀*(v + v₀) 
v² + v*v₀ = 2*a*(r - r₀) + v₀*v + v₀² 
v² = v₀² + 2*a*(r - r₀)  [4] 

From equation [3], 
2*(r - r₀) - v*t = v₀*t  

Then we substitute it into equation [2]: 
r = 0.5*a*t² + 2*r - 2*r₀ - v*t + r₀ 
0 = 0.5*a*t² + r - r₀ - v*t 
r = r₀ + v*t - 0.5*a*t²  [5] 

In elementary physics, the same formulae are often written in different but familiar notation as: 

[1] v = u + a*t [2] s = u*t + 0.5*a*t² [3] s = 0.5*(u + v)*t [4] v² = u² + 2*a*s [5] s = v*t - 0.5*a*t² 

— v₀ is replaced by u. 

— (r - r₀) is replaced by s. 

These are the SUVAT equations, an acronym short for 

— s = displacement 

— u = initial velocity 

— v = final velocity 

— a = acceleration 

— t = time 

If we want to understand the motion of an object in any direction under constant linear acceleration, the initial position, initial velocity and acceleration vectors may not be collinear and take an almost identical form. Therefore we require the dot product to solve the square magnitudes of the velocities. However the derivations are essentially the same as in the collinear case, although the Torricelli equation [4] can be derived using the distributive property of the dot product as follows: 

v² = v*v = (v₀ + a*t)*(v₀ + a*t) = v₀² + 2*t*(a*v₀) + (a²)*(t²)(2*a)*(r - r₀) = (2*a)*(v₀*t + 0.5*a*t²) = 2*t*(a*v₀) + (a²)*(t²) = v² - v₀²

Hence: 

v² = v₀² + 2*(a*(r - r₀))

This graph shows the trajectory of a particle with initial position vector (r₀) and velocity (v₀), subject to constant acceleration (a), as well as all 3 quantities in any direction, and the position (r(t)) and velocity (v(t)) after time ’t'. 

Kinematics is often applied in projectiles, such as a ball thrown upwards into the air. If we know the initial speed (u), then we can work out the maximum height of the ball’s upwards trajectory before it begins to fall. In that case, the acceleration is local acceleration of gravity (g), or 9.8 m/s². Although these quantities appear to be scalars, the direction of displacement, speed and acceleration should be considered as unidirectional vectors. If we choose ’s’ to measure up from the ground, then the acceleration (a) must be -g (negative gravitational acceleration). Since the force of gravity acts downwards, the acceleration on the ball has to be negative. 

At the highest point, the ball will be at rest: therefore v = 0. Using equation [4], we get: 

s = -0.5*(v² - u²)/g 

After substituting and cancelling the minus signs, we then get: 

s = 0.5*u²/g

Constant circular acceleration

The analogues of the equations of motion can also be written for rotation. These  axial vectors must all be parallel to the axis of rotation, so only the magnitudes of the vectors are necessary. 

1. ω = ω₀ + α*t  
2. θ = θ₀ + ω₀*t + 0.5*α*t² 
3. θ = θ₀ + 0.5*(ω₀ + ω)*t 
4. ω² = ω₀² + 2*α*(θ - θ₀)  
5. θ = θ₀ + ω*t - 0.5*α*t²

— α = Constant angular acceleration 

— ω = Angular velocity 

— ω₀ = Initial angular velocity

— θ = Angle turned through (Angular displacement) 

— θ₀ = Initial angle 

— t = Time taken to rotate from the initial state to the final state.

What is general planar motion? 

The kinematic equations for a particle traversing a path in a plane describe the position r = r(t). They are the time derivatives of the position vector in plane polar coordinates, using the definitions of physical quantities above for angular velocity (ω) and angular acceleration (α). These are instantaneous quantities that change with time. 

The position of the particle is: 

r = r (r(t), θ(t)) = r*ê_r 

— ê_r & ê_θ = Polar unit vectors 

When we differentiate the above equation with respect to time, it gives the equation for velocity: 

v = ê_r*(dr/dt) + r*ω*ê_θ 

— dr/dt = Radial component 

— r*ω = Rotation component 

When we differentiate the above equation with respect to time again, it gives the equation for acceleration: 

a = (d²r/dt² - r*ω²)*ê_r + (r*α + 2*ω*(dr/dt))*ê_θ 

— d²r/dt² = Radial acceleration 

— -r*ω² = Centripetal acceleration 

— 2*ω*(dr/dt) = Coriolis acceleration 

— r*α = Angular acceleration

Special cases of motion can described by these equations are summarised in the table below. 

General 3D motions

In 3D space, the equations in spherical coordinates (r, θ, φ) with corresponding unit vectors (ê_r, ê_θ, ê_φ), the position, velocity, and acceleration generalise respectively to: 

r = r(t) = r*ê_rv = v*ê_r + r*(dθ/dt)*ê_θ + r*(dφ/dt)*sin(θ*ê_φ) a = (a - r*(dθ/dt)² - r*(dφ/dt)² * sin²(θ))*ê_r + (r*(d²θ/dt²) + 2*v*(dθ/dt) - r*(dφ/dt)² * sin(θ) * cos(θ))*ê_θ + (r*(d²φ/dt²)* sin(θ) + 2*v*(dφ/dt)* sin(θ) + 2*r*(dθ/dt)*(dφ/dt)*cos(θ))*ê_φ

In the case of a constant φ, this reduces to the planar equations above. 

What are the dynamic equations of motion? 

See Newton’s 2nd law near the top. 

For a number of particles, the equation of motion for one particle (i) influenced by other particle is:

— p_i = Momentum of particle i 

— F_ij = Force on particle i by particle j 

— F_E = Resultant xternal force due to any agent not part of system. 

Note particle i doesn’t exert a force on itself. 

Euler’s laws of motion are similar to Newton’s laws, but they are applied specifically to the motion of rigid bodies. The Newton-Euler equations combine the forces and torques acting on a rigid body into a single equation. 

Newton’s 2nd law of rotation is expressed similarly in the case of translational bodies: 

τ = dL/dt 

— τ = Torque 

— L = Angular momentum 

This equation means the torque acting on the body is dependent on the rate of change of the object’s angular momentum. Analogous to Newton’s 2nd law, the moment of inertia tensor (I) depends on the distribution of mass about the axis of rotation, and the angular acceleration (α) is defined as the rate of change of angular velocity. 

τ = I*α 

Likewise, for cases involving a number of particles, the equation of motion for one particle i is:

— L_i = Angular momentum of particle i 

— τ_ij = Torque on particle i by particle j 

— τ_E = Resultant external torque (due to any agent not part of system)

— Particle i does not exert a torque on itself. 

A 1983 textbook by Pain showed examples of Newton’s law to describe the motion of a simple pendulum: 

-m*g*sin(θ) = m*[d²(l*θ)/dt²] => d²θ/dt² = -(g/l)*sin(θ) 

as well as a damped, sinusoidally driven harmonic oscillator, 

F₀ * sin(ω*t) = m*(d²x/dt² + 2*ζ*ω₀*(dx/dt) + x*ω₀²)

To describe the motion of masses due to gravity, we combine Newton’s law of gravity with Newton’s 2nd law. Examples include a ball of mass (m) thrown in the air, in air current (such as wind) described by a vector field of resistive forces R = R(r,t) 

- [(G*m*M) / |r|²]*ê_r + R = m*(d²r/dt²) + 0 => d²r/dt² = - [(G*M) / |r|²]*ê_r + A 

— G = Gravitational constant 

— M = Mass of the Earth 

— A = R/m : Acceleration of the projectile due to the air currents at position r and time t. 

The classical N-body problem for N particles each interacting with each other due to gravity is a set of N non-linear coupled second order ODEs: 

— i = 1,2,…, N : Quantities (mass, position, etc.) associated with each particle

I’ll discuss the analytical mechanics section of this post. 

Describe the electrodynamics of motion 

In electrodynamics, the force on a charged particle of charge q is the Lorentz force: 

F = q*(E + v*B) 

If we combine it with Newton’s 2nd law, this gives a 1st order differential equation of motion, in terms of position of the particle: 

m*(d²r/dt²) = q*[E + (dr/dt)*B]

or its momentum: 

dp/dt = q*[E + (p*B)/m]

A 1973 textbook showed the same equation can be obtained using the Lagrangian (and applying Lagrange's equations above) for a charged particle of mass m and charge q:

— A & φ = Electromagnetic scalar and vector potential fields respectively 

 The Lagrangian indicates the canonical momentum in Lagrangian mechanics is given by:

This implies the motion of a charged particle is fundamentally determined by the mass and charge of the particle. This Lagrangian expression was first used to derive the force equation. 

Alternatively, the Hamiltonian expression can be substituted into the equations to obtain the Lorentz force equation: 

 This diagram shows the Lorentz force (F) on a charged particle (of charge q) in motion (instantaneous velocity, v). The E (electric) field and B (magnetic) field vary in space and time. 

https://en.wikipedia.org/wiki/Geodesic

https://en.wikipedia.org/wiki/Geodesics_in_general_relativity

Describe the general relativity of motion 

What is the geodesic equation of motion? 

In curved spacetime, physics becomes mathematically complicated due to the lack of straight lines.  To generalise this complex concept, we replace it with a geodesic of the curved spacetime, which is the shortest length of curve between 2 points. For curved manifolds with a metric tensor (g), the metric  provides the notion of arc length. The differential arc length is given by:

The geodesic equation is a 2nd-order differential equation in the coordinates. Parker (1994) derived the general solution as a family of geodesics:

— Γ^μ_αβ = Christoffel symbol of the second kind, which contains the metric (with respect to the coordinate system).

Given the mass-energy distribution provided by the stress-energy tensor T^αβ, the Einstein field equations are a set of non-linear 2nd-order partial differential equations in the metric. Furthermore, they imply the curvature of spacetime is equivalent to a gravitational field i.e. equivalence principle. Mass falling in curved spacetime is found to be equivalent to a mass falling in a gravitational field because “gravity is regarded as a fictitious force”. The relative acceleration of 1 geodesic to another in curved spacetime can be worked out using the geodesic deviation equation:

— ξ^α = x₂^α - x₁^α : Separation vector between 2 geodesics 

— D/ds = Covariant derivative 

— R^α_βγσ = Riemann curvature tensor, which contains the Christoffel symbols

Generally speaking, the geodesic deviation equation is the equation of motion for masses in curved spacetime, similar to the Lorentz force equation for charges in an electromagnetic field. For flat spacetime, the metric is a constant tensor so the Christoffel symbols are not included in the equation, thus the geodesic equation yields solutions of straight lines. This becomes a limiting case when masses moves according to Newton’s law of gravity. 

In general relativity, rotational motion is described by the relativistic angular momentum tensor, including the spin tensor, which are included in the equations of motion under covariant derivatives with respect to proper time. The Mathisson-Papapetrou-Dixon equations describe the motion of spinning objects moving in a gravitational field. I’ll delve into the details in another post. 

What are the analogues for waves and fields? 

The equations governing the dynamics of waves and fields are partial differential equations, since the waves or fields are functions of space and time. For a particular solution, we need specify both boundary conditions and initial conditions. In different contexts, the wave or field equations are also called “equations of motion”. I’ll delve into the field equations, wave equations and quantum theory in another post. 

https://en.wikipedia.org/wiki/Displacement_(geometry)
https://en.wikipedia.org/wiki/Displacement_(vector)

What is a relative position vector? 

It is a vector that defines the position of 1 point relative to another, meaning the difference in position of the 2 points. Let’s say the position of point A relative to point B is simply the difference between their positions: P_A/B = P_A - P_B 

— The difference between the components of their position vectors. 

If point A has position components P_A = (X_A, Y_A, Z_A) and point B has position components P_B = (X_B, Y_B, Z_B), then the position of point A relative to point B is the difference between their components: 

P_A/B = P_A - P_B = (X_A - X_B, Y_A - Y_B, Z_A - Z_B)

https://en.wikipedia.org/wiki/Relative_velocity

What is relative velocity? 

The velocity of 1 point relative to another is the difference between their velocities V_A/B = V_A - V_B 

— The difference between the components of their velocities. 

If point A has velocity components V_A = (V_Ax, V_Ay, V_Az) and point B has velocity components V_B = (V_Bx, V_By, V_Bz), then the velocity of point A relative to point B is the difference between their components: V_A/B = V_A - V_B = (V_Ax - V_Bx, V_Ay - V_By, V_Az - V_Bz). 

In the case where the velocity is close to the speed of light c (generally within 95%), another scheme of relative velocity called ‘rapidity’, used in special relativity, is based on the ratio of V to c. 

Classical mechanics

i. 1 dimension (non-relativistic) 

Relative motion in the classical, (or non-relativistic, or the Newtonian approximation) involves all speeds being slower than the speed of light, meaning the limit associates with the Galilean transformation.

Imagine a man walking on top of a train from the back edge. At 1:00pm he begins to walk forward at 10 km/h, while the train is moving at 40 km/h. How far has the man walked after 1 hour relative to the ground? 

The figure above depicts the man and train at 2 different times: (1) When the train started moving, and (2) 1 hour later. If we calculate the relative velocity of the man relative to the ground, we add the 2 velocity vectors together. This suggests the man is walking at 50 km/h relative to the ground from the starting point after travelling (via walking and on the train) for 1 hour. Therefore the man has travelled 50 km after 1 hour. Notice the figure includes clocks and rulers to illustrate the question. Although the the logic behind this calculation seem flawless, it makes false assumptions about the perceived behaviour of clocks and rulers. To recognise that this classical model of relative motion violates special relativity, we need to generalise the example into an equation: 

V_M/E = V_M/T + V_T/E = 10 + 40 = 50 km/h

— V_M/E = Velocity of the man relative to the Earth 

— V_M/T = Velocity of the man relative to the train 

— V_T/E = Velocity of the train relative to the Earth 

To legitimise the expressions for “the velocity of A relative to B”, we need to include “the velocity of A with respect to B” and “the velocity of A in the coordinate system where B is always at rest”. Because this equation for relative velocity falsely predicts that different observers will measure different speeds when observing the motion of light, this violates the concept of special relativity. 

ii. 2 dimensions (non-relativistic)

 This vector diagram shows the relative velocities between two particles in classical mechanics.

The figure above shows 2 objects A and B moving at constant velocity. The equations of motion include:

— i = Initial displacement (at time t equal to zero). 

The difference between the 2 displacement vectors (r_B^—> - r_A^—>) represents the location of B as seen from A.

 Hence: 

v_B/A^—> = v_B^—>- v_A^—> 

After substituting v_A/C^—>= v_A^—> & v_B/C^—> = v_B^—>. we get: 

v_B/A^—> = v_B/C^—> - v_A/C^—> => v_B/C^—> = v_B/A^—> + v_A/C^—>

iii. Galilean transformation (non-relativistic) 

Constructing a theory of relative motion consistent with the theory of special relativity requires the adoption of a different convention. If we want to continue working in the (non-relativistic) Newtonian limit, we start with a Galilean transformation in 1 dimension: 

x’ = x - v*t t’ = t 

— x’ : Position as seen by a reference frame that is moving at speed v in the "unprimed" (x) reference frame

If we take the differential of the 1st of the 2 equations above, we get, dx’ = dx - v dt. Since dt’ = dt, we get: 

dx’/dt’ = dx/dt - v 

If we want to recover the previous expressions for relative velocity, we have to assume that particle A follows the path defined by dx/dt in the unprimed reference (and hence dx’/dt’ in the primed frame). Thus:

dx/dt = v_A/Odx’ / dt = v_A/O’ 

— O & O’ : Motion of A as seen by an observer in the unprimed and primed frame, respectively. 

— v = Motion of a stationary object in the primed frame, as seen from the unprimed frame.

Therefore we get: 

v = v_O’/O v_A/O’ = v_A/O - v_O’/O => v_A/O = v_A/O’ + v_O’/O

— The latter form has the desired symmetry. 

Special relativity 

In special relativity, the relative velocity v_B/A^—> is defined as velocity of an object or observer B in the rest frame of another object or observer A. However, v_B/A^—>≠ -v_A/B^—> 

This peculiar lack of symmetry relates to Thomas precession and the fact that 2 successive Lorentz transformations rotate the coordinate system. This rotation doesn’t affect the magnitude of a vector, and thus relative speed is symmetrical: 

||v_B/A^—>|| = ||v_A/B^—>|| = v_B/A = v_A/B 

i. Parallel velocities 

If 2 objects are travelling in parallel directions, the relativistic formula for relative velocity is analogous to the formula for addition of relativistic velocities.

The relative speed is given by the formulas:

ii. Perpendicular velocities 
If 2 objects are travelling in perpendicular directions, the relativistic relative velocity (v_B/A^—>) is expressed as: 

where: 

The relative speed is calculated using the formula:

iii. General case 

The general formula for the relative velocity (v_B/A^—>) of an object or observer B in the rest frame of another object or observer A is: 

— γ_A is the same formula as above. 

The relative speed is calculated using the formula:

What is relative acceleration? 

The acceleration of a point A relative to point B is simply the difference between their accelerations: 

A_A/B = A_A - A_B 

— The difference between components of their accelerations. 

If point A has acceleration components A_A = (A_Ax, A_Ay, A_Az) and point B has acceleration components A_B = (A_Bx, A_By, A_Bz), then the acceleration of point A relative to point B is the difference between their components: 

A_A/B = A_A - A_B = (A_Ax - A_Bx, A_Ay - A_By, A_Az - A_Bz) 

Describe the particle trajectories under constant acceleration 

For a case of constant acceleration, the differential equation can be integrated as the acceleration vector A of a point P is constant in magnitude and direction. This means the velocity, V(t), and then the trajectory P(t) of the particle can be evaluated through integration of the acceleration equation A with respect to time. 

Let’s assume that the initial conditions of the position (P₀) and the velocity (V₀) at time t = 0 are known, the first integration yields the velocity of the particle as a function of time.

A 2nd integration yields its trajectory:

Additional relations between displacement, velocity, acceleration, and time can be derived. Since the acceleration is constant, then 

A = ΔV/Δt = (V - V₀)/t 

can be substituted into the above equation to give: 

P(t) = P₀ + ((V+V₀)/2)*t 

A relationship between velocity, position and acceleration without explicit time dependence can be found by solving the average acceleration for time and substituting and simplifying: 

t = (V - V₀)/A (P - P₀) o A = (V - V₀) o (V + V₀)/2 

— o = Dot product, which is appropriate as the products are scalars rather than vectors. 

2*(P - P₀) o A = |V|² - |V₀|²

We can replace the dot by the cosine of the angle α between the vectors and the and the vectors by their magnitudes. 

2*|P - P₀| * |A| * cos(α) = |V|² - |V₀|² 

Since acceleration is in the same direction of motion, the angle between the vectors (α) is 0, so cos(0) = 1, and: 

|V|² = |V₀|² + 2*|A|*|P - P₀|

 v² = (v₀)² + 2*a*Δx 

This equation reduces the parametric equations of motion of the particle to a cartesian relationship of speed versus position, which is useful when time is unknown. Δx =  ∫(v) dt or Δx is the area under a velocity-time graph. Therefore, we can take Δx by adding the top area and the bottom area. The bottom area is a rectangle (W*H, W = width, H = height), and the top area is a triangle (0.5*B*H, B = base, H = Height). In this case, W = t, H = v₀, meaning A_r = t*v₀, and B = t & H = a*t or A_t = 0.5*B*H = 0.5*a*t². 

Therefore

t*v₀ + 0.5*a*t² = Δx 

Describe particle trajectories in cylindrical-polar coordinates 

The polar coordinates of a particle’s trajectory in the X-Y plane are expressed as P(t) = ((X(t), Y(t) and Z(t)). The coordinate vector P measured in a fixed reference frame F defines the trajectory of a particle P. As the particle moves, its coordinate vector P(t) traces its trajectory, which is a curve in space:

— i,j and k = Unit vectors along the X, Y and Z axes of the reference frame, F, respectively. 

Consider a particle P moving only on the surface of a circular cylinder where R(t) = constant. You can align the Z axis of the fixed frame F with the axis of the cylinder. That means the angle θ around this axis in the X–Y plane can be used to define the trajectory as: 

Introduction of the radial and tangential unit vectors, and their time derivatives from elementary calculus can simplify the cylindrical coordinates for P(t): 

If we use this notation, then P(t) becomes:

— R = A constant in the case of the particle moving only on the surface of a cylinder of radius R. 

Since the trajectory P(t) is not confined to be on a circular cylinder, the radius R varies with time and the trajectory of the particle in cylindrical-polar coordinates is expressed as:

— R, θ, Z might be continuously differentiable functions of time. 

— Note the function notation is dropped for simplicity. 

Therefore the velocity vector (V_P) is the time derivative of the trajectory P(t):

Similarly, the acceleration (A_P), the time time derivative of the velocity (V_P), is expressed as: 

• -R*θ²e_r = Commonly called the centripetal acceleration, this term acts toward the centre of the path’s curvature at the path’s point, 
• 2*R*θe_θ = Coriolis acceleration 

— Constant radius 

If the particle’s trajectory is confined to the surface on a cylinder, then the radius R is constant and the velocity and acceleration vectors simplify. The velocity of V_P, the time derivative of the trajectory (P(t)), is expressed as:

The acceleration vector is expressed as:

— Planar circular trajectories

There’s a special case of a particle trajectory on a circular cylinder occurring when there is no movement along the Z axis:

— R & Z₀ = constants 

In this case, the velocity (V_P) is expressed as:

ω = θ : The angular velocity of the unit vector (e_θ) around the z axis of the cylinder. 

Therefore, the acceleration (A_P) of the particle P can be expressed as:

Hence the radial and tangential acceleration components for circular trajectories are also written as: 

α_r = -R*ω² , α_θ = R*α 

Describe the point trajectories in a body moving in the geometric plane 

To analyse the movement of components of a mechanical system, we attach a reference frame to each component and determine how the various reference frames move relative to each other. If we determine that the components have sufficient structural stiffness, then we can neglect their deformation and use their rigid transformations to define this relative movement. This helps simplify the description of each component’s motion in a complex mechanical system. It becomes a geometric description of each component and geometric association of each part relative to other parts. Since kinematics is described as applied geometry, the movement of a mechanical system can be described using the rigid transformations of Euclidean geometry. 

The coordinates of points in a plane are 2-dimensional vectors in R² (2 dimensional space). To preserve the distance between any 2 points, those points undergo rigid transformations. This set of rigid transformations in an n-dimensional space is called the “special Euclidean group on Rⁿ”, often denoted SE(n). 

— Displacements and motion 

To define a position of 1 component of a mechanical system relative to another, it requires the introduction of a reference frame (M) on one that moves relative to a fixed frame (F) on the other. The rigid transformation (or displacement) of M relative to F defines the relative position of the components. Note that a displacement consists of both rotation and translation. 

The set of all displacements of M relative to F is called the configuration space. A smooth curve from 1 position to another in this configuration space is a continuous set of displacements, known as the motion of M relative to F. The body’s motion consists of a continuous set of rotations and translations. 

— Matrix representation 

Combining the rotation and translation components in the plane R² is expressed as a certain type of 3x3 matrix known as a homogeneous transform. The 3x3 homogeneous transform is constructed from a 2x2 rotation matrix A(φ) and the 2x1 translation vector d = (d_x, d_y): 

These homogeneous transforms perform rigid transformations on the points in the plane, z=1 (i.e. on points with coordinates p = (x,y,1). 

Let p define the coordinates of points in a reference frame (M) that coincides with a fixed frame (F).  If the origin of M is displaced by the translation vector d relative to the origin of F, and then rotated by the angle φ relative to the x-axis of F, then the new coordinates in F of points in M are:

— Homogeneous transforms represent ‘affine transformations’. 

This formulation shows that a translation is not a linear transformation of R². However, Richard Paul (1981) explained that if we use projective geometry so that R² is considered a subset of R³, then the translations become affine linear transformations.  

 This diagram shows the movement of each of the components of the Boulton & Watt Steam Engine (1784), modelled by a continuous set of rigid displacements. 

I’ll discuss the rotation of a body around a fixed axis in the circular motion blog post. 

What is pure translation? 

If a rigid body moves so that its reference frame M does not rotate (∅=0) relative to the fixed frame F, the motion is described as “pure translation”. In this case, the trajectory of every point in the body represents an offset of the trajectory d(t) of the origin of M: 

P(t) = [T(0, d(t))]*p = d(t) + p

Thus, for bodies in pure translation, the velocity and acceleration of every point P in the body are expressed as: 

— A dot (.) denotes the derivative with respect to time 

— V_O = Velocity of the origin of the moving frame M.

— A_O = Acceleration of the origin of the moving frame M 

Recall the coordinate vector p in M is constant, so its derivative is zero. 

Describe the point trajectories in a body moving in 3 dimensions 

— Position 

The movement of a component B of a mechanical system is defined by the set of rotations [A(t) and translations d(t) assembled into the homogeneous transformation [T(t)]=[A(t), d(t)]. If p represents the coordinates of a point P in B measured in the moving reference frame M, then the trajectory of this point traced in F is given by: 

This notation doesn’t distinguish between P = (X,Y, Z,1), and P = (X,Y, Z), 

The equation for the trajectory of P can be inverted to compute the coordinate vector p in M as:

This expression depicts the transpose of a rotation matrix being its inverse: 

[A(t)]^T * [A(t)] = 1 

— Velocity  

The velocity of the point P along its trajectory P(t) can be worked out as the time derivative of this position vector:

— Dot (.) denotes the derivative with respect to time

— Because p is constant, its derivative is zero. 

We can modify this formula to work out the velocity of P by operating on its trajectory P(t) measured in the fixed frame F. If we substitute the inverse transform for p into the velocity equation, then we get: 

The matrix [S] is expressed as: 

[Ω] = ÅA^T : Angular velocity matrix 

If we multiply the above matrix by the operator [S], then formula for the velocity V_P becomes: 

• ω = Angular velocity vector can be evaluated from the components of the matrix [Ω].
• R_P/O = P - d : A vector that represents the position of P relative to the origin O of the moving frame M. 
• V_O = d^* : Velocity of the origin O

— Acceleration 

The acceleration of a point P in a moving body B can be worked out from the time derivative of its velocity vector:

This equation can be expanded firstly by computing:

The formula for the acceleration (A_P) can then be evaluated as:

— α = Angular acceleration vector obtained from the derivative the angular velocity matrix

— R_P/O = P - d : The relative position vector (the position of P relative to the origin O of the moving frame M) 

— A_O = d^** : Acceleration of the origin of the moving frame M.

What are the kinematic constraints? 

They are constraints on the movement of components of a mechanical system, which have 2 basic forms: 

(i) Holonomic constraints = Arising from hinges, sliders and cam joints that define the construction of the system

(ii) Non-holonomic constraints = Imposing on the velocity of the system such as the knife-edge constraint of ice-skates on a flat plane, or rolling without slipping of a disc or sphere in contact with a plane

— Kinematic coupling 

= This exactly constrains all 6 degrees of freedom. 

— Rolling without slipping 

= An object rolling against a surface without slipping demonstrates the velocity of its centre of mass  equalling the cross product of its angular velocity with a vector from the point of contact to the centre of mass:

v_G(t) = Ω*r_G/O 

For objects that don’t tip or turn, this simplifies to v = r*ω. 

— Inextensible cord 

= This occurs when bodies are connected by an idealised cord that remains in tension and cannot change length. A few studies identified the constraint as the sum of lengths of all segments of the cord being the total length, meaning the time derivative of this sum is zero. e.g. A pendulum. Church (1908) gave another example of a drum being pulled by the pull of gravity upon a falling weight attached to the rim by the inextensible cord. Kline (1990) stated the equilibrium problem (i.e. not kinematic) of this type of scenario is the catenary. 

— Kinematic pairs: 

They are the ideal connections between components that form a machine. Reuleaux distinguished between higher pairs and lower pairs.

(i) Lower pair 

It is an ideal joint, or holonomic constraint that have area contact between the links. It helps maintain contact between a point, line or plane in a moving solid (3D0) body to a corresponding point line or plane in the fixed solid body. There are a few cases of lower pairs. 

• Revolute pair, or hinged joint = This requires a line, or axis, in the moving body to remain co-linear with a line in the fixed body. In addition, it required a plane perpendicular to this line in the moving body in order to maintain contact with a similar perpendicular plane in the fixed body. This establishes 5 constraints on the relative movement of the links, creating 1 degree of freedom, which is pure rotation about the axis of the hinge.
• Prismatic joint, or slider = This requires a line, or axis, in the moving body to remain co-linear with a line in the fixed body. In addition, it requires a plane parallel to this line in the moving body in order to maintain contact with a similar parallel plane in the fixed body. This also establishes 5 constraints on the relative movement of the links, hence having 1 degree of freedom, which is identified as the distance of the slide along the line. 
• Cylindrical joint = This requires a line, or axis, in the moving body to remain co-linear with a line in the fixed body. It combines a revolute joint with a sliding joint, meaning it has 2 degrees of freedom. Both the rotation about and slide along the axis defines the position of the moving body. 
• Spherical joint, or ball joint = This requires a point in the moving body to maintain contact with a point in the fixed body. It has 3 degrees of freedom. 
• Planar joint = This requires a plane in the moving body to maintain contact with a plane in fixed body. It also has 3 degrees of freedom. 

ii. Higher pairs 

They are a constraint that requires a curve or surface in the moving body to maintain contact with a curve or surface in the fixed body. e.g. the contact between a cam and its follower is a higher pair called a cam joint. Examples of cam joints are the contact between the involute curves that form the meshing teeth of 2 gears. 

— Kinematic chains 

They are defined as rigid bodies (or links) connected by kinematic pairs (or joints). e.g. mechanisms and robots. The degree of freedom of a kinematic chain is evaluated from the number of links and the amount and type of joints using mobility formula. The mobility formula is used to enumerate the topologies of kinematic chains that have a given degree of freedom. This is known as type synthesis in machine design. 

Consider the planar one degree-of-freedom linkages is assembled from N links and j hinged or sliding joints: 

• N = 1, j = 1 : A 2-bar linkage that is the lever 
• N = 4, j = 4 : The 4-bar linkage 
• N = 6, j = 7 : A 6-bar linkage, which consists of 2 links (or ternary links) supporting 3 joints. There are 3 distinct topologies that depend on how the 2 ternary linkages are connected. Lung-Wen (2001) found the 2 ternary links have a common joint in Watt topology, but lack a common joint and instead connected by binary links in Stephenson topology. 
• N = 8, j = 10 : 8-bar linkage with 16 different topologies 
• N = 10, j = 13 : 10-bar linkage with 230 different topologies 
• N = 12, j = 16 : 12-bar linkage with 6856 topologies

This diagram illustrates a four-bar linkage in the 1876 Kinematics of Machinery

https://en.wikipedia.org/wiki/Jerk_(physics)

What is jerk/jolt? 

In physics, jerk or jolt is defined the rate at which an object’s acceleration changes with respect to time. It is a vector quantity (having both magnitude and direction). It is commonly denoted by the symbol j and expressed in m/s³ (SI units) or standard gravities per second (g/s). 

Jerk can be expressed as the 1st time derivative of acceleration, and so on.

— a^—>= Acceleration 

— v^—>= Velocity 

— r^—>= Position 

— t = time 

3rd-order differential equations of the form:

They may be called ‘jerk equations’. When they are converted to an equivalent system of 3 ordinary 1st-order non-linear differential equations, jerk equations are the minimal setting for solutions that govern chaotic behaviour, which generates mathematical interest in ‘jerk systems’. Chlouverakis & Sprott (2006) stated that systems involving 4th-order derivatives or higher are accordingly called ‘hyperjerk systems’. 

What are the physiological effects and human perception of jerk? 

Human body position is balanced by the forces of antagonistic muscles. In the case of holding up a weight, the post-central gyrus establishes a control loop to achieve an equilibrium. If the force changes abruptly, the muscles are unable to relax or tense quickly enough and overshoot in either direction, ultimately causing a temporary loss of control. The reaction time required in response to changes in force depends on the physiological limitations and the attentive capabilities of the brain.

To avoid vehicle passengers’ losing control over body motion and becoming injured, exposure to both the maximum force (acceleration) and maximum jerk needs to be limited. This is because time is required for the brain to adjust muscle tension and adapt to even limited stress changes. A 2007 report described sudden changes in acceleration causes injuries such as whiplash. Excessive jerks would make rides uncomfortable, but not be injury-inducing. Therefore, engineers spend considerable efforts in designing vehicles that minimise jerks such as elevators, trams and other conveyances. 

For example, imagine you are riding inside a car and it is accelerating and jerking. 

• Skilled and experienced drivers can accelerate smoothly, but learner or beginner drivers often drive with jerky motion. When experienced manual drivers change car gears with a foot-operated clutch, the accelerating force is limited by engine power, whereas inexperienced drivers can cause excessive jerks due to intermittent force closure over the clutch. 
• When a high-powered sports car accelerates, it generates a downwards force that pushes passengers and the driver into their seats. As the car moves from rest, there is a positive jerk as its acceleration rapidly increases. Then there is a small, sustained negative jerk as the force of air resistance increases with the car's velocity, which gradually decreases acceleration and lessens the force that presses the passenger into the seat. When the car reaches maximum speed, the acceleration becomes zero and remains constant. At that point, there is no jerk until the driver decelerates or changes direction. 
• When the driver brakes suddenly or collides with another object, the car’s passengers whip forward with an initial acceleration that is larger during the rest of the braking process. This is due to muscle tension regaining control of the body rapidly after the onset of braking or impact. Unfortunately, these effects aren’t modelled in vehicle testing as cadavers and crash test dummies lack active muscle control. 

How does jerk relate to force and acceleration? 

In classical mechanics of rigid bodies, there are no forces associated with with the derivatives of acceleration. However, there are physical systems that experience oscillations and deformations as a result of jerk. During the construction of the Hubble Telescope, NASA reportedly set limits on both jerk and jounce. 

The Abraham-Lorentz force is defined as the recoil force on an accelerating charged particle emitting radiation, which is proportional to the particle’s jerk and to the square of its charge. 

— F_rad = Force 

— å = Jerk, derivative of acceleration 

— μ₀ = Magnetic constant 

— ε₀ = Electric constant 

— c = Speed of light in free space 

— q = Electric charge of the particle 

A more advanced theory called the "Wheeler-Feynman absorber theory" is applied in a relativistic and quantum environment, and accounting for self-energy. I’ll discuss it in detail in another post. 

Describe jerk in an idealised setting 

Discontinuities in acceleration don’t occur in the real world because of deformation, quantum mechanics effects, and other causes. However, in an idealised setting, it is feasible for a jump-discontinuity in acceleration and, thus, unbounded jerk to occur. For example, an idealised point mass moving along a piecewise smooth, whole continuous path. This jump-discontinuity occurs at points where the path is rough. When we extrapolate from these idealised settings, we can qualitatively describe, explain and predict the effects of jerk in real situations. This can modelled using a Dirac delta function in jerk, scaled to the height of the jump. If we integrate jerk over time across the Dirac delta, it generates the jump-discontinuity. 

Imagine a path along an arc of radius r that tangentially connects to a straight line. The entire path is continuous, and its components are smooth. Now picture a point particle moving at constant speed along this path, meaning its tangential acceleration is zero. The centripetal acceleration can be worked out using the formula v²/r, which is normal to the arc and inward. If the particle passes the connected pieces, it experiences a jump-discontinuity in acceleration dictated by v²/r. Hence, it undergoes a jerk modelled by a Dirac delta, scaled to the jump-discontinuity. 

Think of an ideal spring-mass system with the mass oscillating on an idealised surface with friction. The force on the mass is equal to the vector sum of the spring force and the kinetic frictional force. i.e. F_m = F^—>_s + F^—>_kf .When the mass’s displacement is at maximum or minimum, its velocity sign changes, meaning the magnitude of the force on the mass changes by two times the magnitude of the frictional force. This is due to the continuity of the spring force and the frictional force reversing direction as velocity changes direction. The increase in acceleration equalises the force on the mass divided by the mass. In other words, each time the mass passes through a minimum or maximum displacement, the mass experiences a discontinuous acceleration, and the jerk contains a Dirac delta until the mass stops. Meanwhile, the static friction force adapts to the residual spring force, which leads to the establishment of equilibrium i.e. zero net force and zero velocity. 

Imagine a car that brakes and decelerates. The brake pads generate kinetic frictional forces and constant braking torques on the discs of the wheels. Rotational velocity decreases linearly with constant angular deceleration until it becomes zero. Next, the frictional force, torque, and car deceleration suddenly decreases to zero, indicating a Dirac delta in physical jerk. The real environment smooths down the Dirac delta, which accumulates the effect of damping of the physiologically perceived jerk. However, this scenario ignores the effects of tire sliding, suspension dipping, real deflection of all ideally rigid mechanisms, etc. 

Consider a rope with a particle on its end that is oscillating in a circular path with non-zero centripetal acceleration. When the rope is cut, the particle’s path changes abruptly to a straight path, the force directed inwards suddenly decreases to zero. 

Consider a monomolecular fibre being severed by a laser beam, the particle would experience significant rates of jerk because of the rapidly quick cutting time.

Describe jerk in rotation

Imagine a rigid body rotating about a fixed axis in an inertial reference frame. If its angular position is θ, the formulas of angular velocity, acceleration, and jerk are as follows: 

— Angular Velocity:

Time derivative of θ(t)

— Angular acceleration:

Time derivative of ω(t) 

— Angular Jerk:

Time derivative of α(t) 

Angular acceleration is equal to the torque acting on the body, which is divided by the body’s moment of inertia with respect to the momentary axis of rotation. Therefore, a change in torque results in angular jerk. 

In the case of a rotating rigid body, a kinematic screw theory can model its motion, which includes 1 axial vector, angular velocity (Ω^—>(t)), and 1 polar vector, linear velocity (v^—>(t)). This allows us to define angular acceleration as:

and the angular jerk as: 

For example, imagine a Geneva drive (above diagram), a device used for creating intermittent rotation of a driven wheel (red wheel) by continuous rotation of a driving wheel (blue wheel). During 1 cycle of the driving wheel, the driven wheel’s angular position (θ) rotates by 90 degrees and then remains constant afterwards. Because the driving wheel’s fork (i.e. slow for the driving pin) has a finite thickness, there is discontinuity in the device’s angular acceleration (α) and unbounded angular jerk (ζ) in the driven wheel. 

In applications such as movie projectors and cams, jerk doesn’t preclude the use of the Geneva drive. Normally movie projectors involve the frame-by-frame scrolling of film, but the projector operation is relatively quiet and highly reliable due to its low film load, moderate speed (2.4 m/s, 8.6 km/h) and low friction.

Dual Cam Drives

1/6 per revolution 

1/3 per revolution 

Cam drive systems are mostly dual cam rather than single cam to avoid jerk. However the dual cam has more bulk and is more expensive to design. The dual cam system has 2 cams on 1 axle that shifts a 2nd axle by a fraction of a revolution. The graphic above illustrates step drives of 1/6 and 1/3 rotation for every revolution of the driving axle. Since both arms of the stepped wheel are always in contact with the double cam, there is no radial clearance. Contacts may be combined to avoid jerk (as well as wear and noise) associated with a single follower. For example, a single follower gliding along a slot and changing its contact point from one side of the slot to the other can be avoided by using 2 followers sliding along the same slot, one side each.

  This is a timing diagram over 1 revolution for angle, angular velocity, angular acceleration, and angular jerk. 

Discuss jerk in elastically deformable matter 

When a force (or acceleration) is applied to an elastically deformable mass, it deforms (or changes shape). This deformation is a function of the mass’s stiffness and the magnitude of force. If force changes slowly, then it minimises jerk, and the deformation propagates instantaneously compared to the change in acceleration. The distorted body acts in a quasi-static manner, which means only a changing force (non-zero jerk) would propagate mechanical waves (or electromagnetic waves for a charged particle). Therefore, for any non-zero jerk, we should take into account a shock wave and its propagation through the body. 

Plane wave 

Cylindrical symmetry 

The propagation of deformation is illustrated in the diagrams above, as a compressional plane wave through an elastically deformable material. Angular jerk generates deformation waves that propagate in a circular pattern, which causes shear stress and other modes of vibration. When waves reflect along the boundaries, it creates constructive interference patterns (not shown). This generates stresses that may exceed the material’s limits. The deformation waves generates vibrations that causes noise, wear, and failure, particularly in cases of resonance.

 This diagram illustrates a pole with a massive top.

A block is connected to an elastic pole and a large top. When the block accelerates, the pole bends. When the block stops accelerating, the top with oscillate (damped) under the regime of pole stiffness. Some argue that a greater (periodic) jerk could potentially generate a larger amplitude of oscillation due to small oscillations being damped prior to a shock wave reinforcing them. Furthermore, a larger jerk theoretically increases the  probability of exciting a resonant mode because the larger wave components of the shock wave have higher frequencies and larger Fourier coefficients. 

These graphs showcase a sinusoidal acceleration profile. 

If we want to decrease the amplitude of excited stress waves and vibrations, we have to limit jerk by shaping motion and creating continuous acceleration with flatter slopes. The limitations of abstract models lead to suggestions that the algorithms for reducing vibrations need to include higher derivatives, such as jounce, as well as continuous regimes for both acceleration and jerk. If we want to limit jerk, we need to shape acceleration and deceleration sinusoidally with zero acceleration in between. This makes the curve for speed with respect to time appear sinusoidal with constant maximum speed. Nevertheless, the jerk remains discontinuous at the points where acceleration decreases towards and increases from zero.

How is jerk involved in the geometric design of roads and tracks? 

Roads and railroad tracks are designed to limit the jerk caused by changes in their curvature. Railway designers use 0.35 m/s³ jerk as a goal and 0.5 m/s³ as a maximum. They create track transition curves to limit the jerk during the transition from a straight line to a curve, or vice versa. Note that when objects move at constant speed along an arc, jerk is zero in the tangential direction and non-zero in the inward normal direction. That means transition curves gradually increase the curvature and, ultimately, the centripetal acceleration.

The diagram is a track transition curve, which theoretically limits jerk. The transition is shown in red between the blue straight line and green arc. 

A theoretically optimum transition curve, known as a Euler spiral, linearly increases centripetal acceleration, resulting in constant jerk. In the real-world, the track’s plane is inclined along the curve sections, which causes vertical acceleration. This design is considered for wear on the track and embankment. Other studies explained the Wiener Kurve (Viennese Curve) is considered for reducing wear. A 2007 report stated roller coasters are constructed with track transitions to limit jerk. When a roller coaster car enters a loop, acceleration values approach 4g. Riding in this high acceleration scenario is possible with track transitions. To ensure smooth rides along these track transitions, the track is designed with S-shaped curves, such as figure-8s.

Describe jerk in motion control

In motion control, designers focus on moving a system from one steady position to another (point-to-point motion) in a straight and linear path. The design from the perspective of jerk concerns vertical jerk, that is, the jerk from tangential acceleration is effectively zero since linear motion is non-rotational. 

Example applications of motion control include passenger elevators and machining tools. To ensure convenient elevator rides, vertical jerk needs to be limited. The International Organisation for Standardisation (ISO 18738) specifies measurement methods for elevator ride quality with respect to jerk, acceleration, vibration, and noise. However it needs to satisfy the specified levels for acceptable or unacceptable ride quality. Howkins (2014) reported that most passengers rate a vertical jerk of 2.0 m/s³ as acceptable ride quality and 6.0 m/s³ as intolerable, whereas 0.7 m/s³ is the recommended limit for hospitals. 

To achieve the primary goal for motion control, designers aim to minimise the transition time without exceeding speed, acceleration, or jerk limits. The figure illustrates a 3rd-order motion-control profile with quadratic ramping and de-ramping phases in velocity.

 This motion profile contains the following 7 segments: 

During the 4th segment, the time period (constant velocity) varies with distance between the 2 positions. If this distance is minimised, then the 2nd & 6th segments (constant acceleration) are equally reduced, meaning the constant velocity limit cannot be reached. If this modification fails to sufficiently decrease the crossed distance, then the 1st, 3rd, 5th and 7th segments need to be shortened by an equal amount, meaning the constant acceleration limits cannot be reached. 

Hogan (1984) discussed other motion profile strategies that involve minimising the square of jerk for a given transition time, as well as sinusoidal-shaped acceleration profiles. Such motion profiles apply to machines, people movers, chain hoists, automobiles, and robotics. 

In manufacturing processes, jerk is prevalent in cutting tools in the form of rapid changes in acceleration. This leads to premature tool wear, hence uneven cuts. Consequently, this lead to modern motion controllers including jerk limitation features in its design. In mechanical engineering, jerk, velocity and acceleration are all taken into account during the development of cam profiles. Blair (2005) explained the reason behind these considerations is tribological implications and the actuated body able to follow the cam profile without chatter. 

https://en.wikipedia.org/wiki/Fourth,_fifth,_and_sixth_derivatives_of_position

What is jounce, crackle and pop? 

In physics, the 4th, 5th and 6th derivatives of position are defined as derivatives of the position vector with respect to time. 

(i) 4th derivative = Snap / Jounce 

This is defined as the rate of change of the jerk with respect to time.

The following equations are used for constant snap:

— s^—> = Constant snap 

— j₀^—>= Initial jerk 

— j^—>= Final jerk 

— a₀^—>= Initial acceleration 

— a^—> = Final acceleration 

— v₀^—>= Initial velocity 

— v^—>= Final velocity 

— r₀^—>= Initial position 

— r^—>= Final position 

— t = Time between initial and final states 

The notation s^—>(used by Visser, 2004) should not be confused with the displacement vector. The dimensions of snap are distance per 4th power of time. In SI units, this is "metres per second to the fourth”, m/s⁴, m*s^-4, or 100 gal per second squared in CGS units.

(ii) 5th derivative = Crackle / Flounce 

It is defined as the rate of change of the snap with respect to time. 

The following equations are used for constant crackle:

— c^—> = Constant crackle 

— s₀^—>= Initial snap 

— s^—>= Final snap 

The dimensions of crackle are LT^-5. In SI units, his is m/s5, and in CGS Units, 100 gal per cubed second. 

(iii) 6th derivative = Pop / Pounce

It is defined as the rate of change of the crackle with respect to time. Pop can be defined by any of the following equivalent expressions:

The following equations are used for constant pop:

— p^—>= Constant pop 

— c₀^—>= Initial crackle 

— c^—>= Final crackle 

The dimensions of pop are LT^-6. In SI units, this m/s6, and in CGS units, 100 gal per quartic second. 

https://en.wikipedia.org/wiki/Absement

What is absement?

Absement (or absition) is a measure of sustained displacement of an object from its initial position, i.e. a measure of how far away and for how long. As the object remains displaced and stays constant as the object resides at the initial position, its absement changes. It is the first time-integral of the displacement (i.e. absement is the area under a displacement vs. time graph), so the  displacement is the rate of change (1st time-derivative) of the absement). The dimension of absement is length multiplied by time. Its SI units is metre second (m*s), which corresponds to an object having been displaced by 1 metre for 1 second. 

For example, when the gate of a gate valve (of rectangular cross section) is opened by 6 mm for 1 minute, it yields the absement of 10 mm*s. Maya Burhanpurkar (2014) stated the amount of water flowing through it is linearly proportional to the absement of the gate. Studies claim the word ‘absement’ is a portmanteau of the words ‘absence’ and ‘displacement’. Similarly, ‘absition' is a portmanteau of the words ‘absence’ and ‘position’. 

How does absement occur in nature? 

Whenever the rate of change (f’) of a quantity (f) is proportional to the displacement of an object, f is a linear function of the object’s absement. For example, when the fuel flow rate is proportional to the position of the throttle lever, then the total amount of fuel consumed is proportional to the lever's absement. 

A 2006 paper introduced a flow-based musical instrument called a hydraulophone to study absement. Empirical observations of some hydraulophones found that obstruction of a water jet for a longer period of time resulted in a buildup in sound level. Water accumulates in a reservoir up to a certain maximum filling point beyond which the sound level reached a maximum, or decreased (along with a slow decay upon a water jet being unblocked. Studies use absement to model artificial muscles, as well as real muscle interaction in a physical fitness context, and human posture. 

Jeltsema (2012) stated the displacement can be seen as a mechanical analogue of electric charge, the absement is viewed as a mechanical analogue of the time-integrated charge, a quantity useful for modelling some types of memory elements. 

What are the applications of absement?

Studies use absement to model fluid flow, lagrangian mechanics of electric charge, as well as in physical fitness and kinesiology to model muscle bandwidth and develop a new form a physical fitness training. This gave rise to a new quantity called actergy, which is defined as ‘energy to energy as is to power’. The units of actergy are Joule-seconds, the same as action, which is the time-integral of total energy i.e. time-integral of the Hamiltonian rather than time-integral of the Lagrangian. 

e.g. A vehicle’s distance travelled results from its throttle’s absement. The further the throttle has been opened, and the longer it's been open, the more the vehicle's travelled. 

How does absement relate to PID controllers? 

PID controllers work on a signal that is proportional to a physical quantity (e.g. displacement, proportional to position) and its integral(s) and derivative(s). Therefore, PID are defined in the  context of Integrals and derivatives of a position of a control element. Bratland et al. (2014) stated that, depending on the type of sensor inputs, PID controllers  contain gains proportional to position, velocity, acceleration or the time integral of position (absement). 

e.g. PID = Position, Absement, Velocity 

Below is the ordered list of n-th derivatives of displacement: 

-7: Absop (m*s⁷)

-6. Absackle (m*s⁶)

-5. Absnap (m*s⁵)

-4. Abserk (m*s⁴)

-3. Abseleration (m*s³)

-2. Absity (m*s²)

-1. Absement (m*s) 

0. Displacement (m*s⁰ = m) 

1. Velocity (m*s^-1)

2. Acceleration (m*s^-2)

3. Jerk/ Jolt (m*s^-3)

4. Snap/Jounce (m*s^-4)

5. Crackle/Flounce (m*s^-5)

6. Pop/Pounce (m*s^-6) 

A study by Pei et al. (2015) on memristors and memcapacitors built on the concept of absement, and assigned it as ‘a’, which was represented in plots such as  the graph of absement as a function of displacement. 

It stated the “amplitude of the sinusoidal displacement with period T = 2π/ω and a₀ = A/ω is the value about which the analytic absement a(t) oscillates.” 

Strain absement is the time-integral of strain, which is used in mechanical systems and memsprings. This allows mem-spring models to display hysteretic response in great abundance. 

Absement was originally used in scenarios involving valves and fluid flow, such as the opening of a valve by a long "T"-shaped handle, which varied in angle rather than position. The time-integral of angle is known as “anglement”, which is approximately equal or proportional to absement for small angles, i.e. the sine of an angle is approximately equal to the angle for small angles.

https://en.wikipedia.org/wiki/Non-inertial_reference_frame

https://en.wikipedia.org/wiki/Frame_of_reference

https://en.wikipedia.org/wiki/Inertial_frame_of_reference

https://en.wikipedia.org/wiki/Galilean_transformation

What are frames of reference?

In physics, a frame of reference (or reference frame) consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardise measurements within that frame. 

We need to distinguish between the different definitions of “frame of reference”. 

• Cartesian frame of reference = The type of coordinate system is attached as a modifier. 
• Rotating frame of reference = Emphasis on the state of motion. 
• Galilean frame of reference = Emphasis on the way it transforms to frames considered as related. 
• Macroscopic & Microscopic frames of reference = Distinguished by the scale of their observations. 
• Observational frame of reference = Emphasis upon the state of motion rather than upon the coordinate choice or the character of the observations or observational apparatus. This allows the  study of the effect of motion upon an entire family of coordinate systems that could be attached to this frame. 
• Coordinate system = This definition is employed where the state of motion is not the primary concern. For instance, it is adopted to take advantage of the symmetry of a system. Broadly speaking, the formulation of many physics problems employs generalised coordinates, normal modes or eigenvectors, which are only indirectly related to space and time. 
• An observational frame (such as an inertial frame or non-inertial frame of reference) is a physical concept related to state of motion. 
• A coordinate system is a mathematical concept that amounts to the choice of language used to describe observations. That means an observer in an observational frame of reference has the choice of employing any coordinate system (Cartesian, polar, curvilinear, generalized, …) to describe observations made from that frame of reference. Any change in the choice of this coordinate system would not change an observer's state of motion, thus it would not require a change in the observer's observational frame of reference. 
• Choosing what to measure and with what observational apparatus is independent of the  observer's state of motion and choice of coordinate system. 

https://en.wikipedia.org/wiki/Coordinate_system

What are coordinate systems? 

A coordinate system in mathematics is a a feature of geometry or of algebra, or a property of manifolds (e.g. in physics, configuration spaces or phase spaces). The coordinates of a point r in an n-dimensional space are simply an ordered set of n numbers:

r = [x¹, x², …, xⁿ] 

• In a general Banach space, these numbers could be coefficients in a functional expansion such as the Fourier series. 
• In a physical problem, those numbers could be spacetime coordinates or normal mode amplitudes. 
• In a robot problem, those numbers could be angles of relative rotations, linear displacements, or deformations of joints.

 This diagram shows an observer O, situated at the origin of a local set of coordinates - a frame of reference F. The observer in this frame uses the coordinates (x, y, z, t) to describe a space-time event, shown as a star. 

Let’s suppose these coordinates can be related to a Cartesian coordinate system by a set of functions:

x^j = x^j (x, y, z,…), j = 1,…, n, 

— x, y, z, etc, = n Cartesian coordinates of the point. 

Given these functions, coordinate surfaces are defined by the relations: 

x^j (x, y, z,…) = constant, j = 1,…,n. 

The intersection of these surfaces define coordinate lines. At any selected point, tangents to the intersecting coordinate lines at that point define a set of basis vectors {e₁, e₂, …, e_n} at that point. Papapetrou (1974) expressed it as:

This equation can be normalised to be of unit length.

Zdunkowski & Bott (2003) stated that coordinate surfaces, coordinate lines, and basis vectors are components of a coordinate system.  If the basis vectors are orthogonal at every point, that means the coordinate system is an orthogonal coordinate system. 

A 1979 paper argued an important aspect of a coordinate system is its metric tensor (g_ik), which determines the arc length ds in the coordinate system in terms of its coordinates: 

(ds)² = (g_ik)*(dx^i+k) 

Since it’s part of an axiomatic system, a coordinate system is a mathematical construct. However, there is no necessary connection between coordinate systems and physical motion. Nevertheless, coordinate systems can include time as a coordinate, which describing motion. Thus, Lorentz transformations and Galilean transformations may be viewed as coordinate transformations. 

What are observational frames of reference?

An observational frame of reference, also known as a physical frame of reference, a frame of reference, or simply a frame, is defined as a physical concept related to an observer and the observer's state of motion. However, there is debate regarding the characterisation of an observational frame of reference. Kumar and Barve (2003) argued it was characterised only by its state of motion. The theory of special relativity argued that a frame is an observer as well as a coordinate lattice. This lattice is constructed to be an orthonormal (orthogonal and normalised) right-handed set of space-like vectors perpendicular to a time-like vector. Faber (1983) asserted that the use of general coordinate systems is common in general relativity e.g. the Schwarzschild solution for the gravitational field outside an isolated sphere.

 This diagram shows 3 frames of reference in special relativity. The black frame is at rest. The primed frame moves at 40% of light speed, while the double primed frame moves at 80% of light speed. Note the scissors-like change as speed increases. 

There are 2 types of observational reference frame: inertial and non-inertial. 

What are inertial frames of reference?

In classical physics and special relativity, an inertial frame of reference views a body with zero net force acting upon it meaning it’s not accelerating, i.e. at rest or moving at a constant velocity. Landau and Lifshitz (1960) defined it in analytical terms as a frame of reference that describes time and space homogeneously, isotropically, and in a time-independent manner. Ferraro (2007) explained the physics of a system in an inertial frame lacks causes that are external to the system. Puebe (2009) stated that an inertial frame of reference can be used interchangeably with “inertial reference frame”, “inertial frame”, “Galilean reference frame”, or “inertial space”. 

Introduction

One can only describe the body’s motion relative to something such as other bodies, observers, or a set of space-time coordinates, known as frames of reference. A poor choice of coordinates would lead to more complex laws of motion. e.g. A free body with no external forces acting on it may have been at rest at some instant. In many coordinate systems, the body would begin to move at the next instant, despite no forces acting on it. However, it would seem to remain stationary from a frame of reference. Likewise, if space is not described uniformly or time independently, then a coordinate system would describe the simple flight of a free body in space as a complicated zig-zag trajectory. Landau & Lifshitz (1960) stated that an intuitive summary of inertial frames in which the laws of mechanics take their simplest form in an inertial frames. 

The force (F) is the vector sum of all “real” forces on a particle, such as electromagnetic, gravitational, nuclear and so forth. In contrast, Newton’s 2nd law in a rotating frame of reference, rotating at angular rate Ω about an axis, takes the form: F’ = m*a. 

This equation is the same as in an inertial frame, but F’ is the resultant force of F as well additional terms: 

F’ = F - (2mΩ)*v_B - (m*Ω)*(Ω*x_B) - m*(dΩ/dt)*x_B 

— Ω = This vector expresses the angular rotation of the frame pointing in the direction of the axis of rotation, and with magnitude equal to the angular rate of rotation Ω. 

— * = Vector cross product 

— x_B = This vector locates the body

— v_B = Velocity of the body according to a rotating observer (different from the velocity seen by the inertial observer). 

• The extra terms in the equation are the “fictitious” forces for this frame. They include the Coriolis force, centrifugal force and the Euler force. 
• When Ω = 0, then these terms disappear, i.e. they are zero for a non-rotating inertial frame. 
• Depending on the value of Ω, the terms vary in magnitude and direction in every rotating frame. 
• The terms are ubiquitous in the rotating frame, meaning it affect every particle, regardless of circumstance. 
• The terms don’t have a known apparent source in identifiable physical sources, particularly matter. 
• Fictitious forces don’t reduce with distance, unlike nuclear forces or electrical forces. e.g. The centrifugal force that appears to emanate from the axis of rotation in a rotating frame increases with distance from the axis. 

The stars seemed fixed from our frame of reference but, in actuality, they are not fixed. The stars in our Milky Way Galaxy are rotating with our galaxy, demonstrating proper motions. On the other hand, the stars outside the galaxy move independently, partly due to the continuing expansion of the universe, and partly due to peculiar velocities. Loeb et al. (2005) calculated the Andromeda Galaxy is on a collision course with the Milky Way galaxy at a speed of 117 km/s, with impact predicted to occur in about 3.75 billion years time. The modern concepts of inertial frames of reference has switched from the fixed stars or absolute space to the simplicity of the laws of physics in the frame. 

One question I’ve asked myself “Does the Universe rotate?”. Answering this question requires an understanding of the shape of the Milky Way galaxy using the laws of physics. Studies have noted that definitive observations should be based on larger discrepancies or less measurement uncertainty, such as the anisotropy of the microwave background radiation or Big Bang nucleosynthesis. The flatness of the Milky Way depends on its rate of rotation in an inertial frame of reference. If we attribute its apparent rate of rotation entirely to rotation in an inertial frame reference, then we predict a different type of “flatness”. If we assume part of this rotation is due to rotation of the universe, then we should not include it in the rotation of the galaxy itself. If the laws of physics agree with observations in a model with rotation compared to without it, then physicists tend to prefer the best-fit value for rotation, as long as it’s subject to all other pertinent experiment observations. If we can’t value the rotation parameter and the theory is vulnerable to observational errors, then the laws of physics need to be modified, e.g. dark matter is invoked to explain the galactic rotation curve. Birch (1982) observed that any rotation of the universe is very slow, no faster than once every 60·10¹² years (10⁻¹³ rad/yr), however the presence of such rotation is still debated. If any rotation is detected, interpreting the observations in a frame tied to the universe requires correction for the fictitious forces inherent in such rotations in classical physics and special relativity, or the curvature of spacetime and the motion of matter along the geodesics in general relativity. 

Where the laws of physics are simple for a set of frames

The first postulate of special relativity states “all physical laws take their simplest form in an inertial frame, meaning multiple inertial frames interrelated by uniform translation energy”.

In Section A, §1 of Einstein’s “The foundation of the general theory of relativity”: Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.

Ferraro (2007) explained that this simplicity manifests due to inertial frames demonstrating self-contained physics independent of external causes, while physics in non-inertial frames have external causes. Nagel (1979) and Blagojević (2002) asserted that this principle of simplicity can be applied within Newtonian physics as well as in special relativity. In Gravitation and Gauge Symmetries, p. 4, Blagojević stated that: 

“The laws of Newtonian mechanics do not always hold in their simplest form...If, for instance, an observer is placed on a disc rotating relative to the earth, he/she will sense a 'force' pushing him/her toward the periphery of the disc, which is not caused by any interaction with other bodies. Here, the acceleration is not the consequence of the usual force, but of the so-called inertial force. Newton's laws hold in their simplest form only in a family of reference frames, called inertial frames. This fact represents the essence of the Galilean principle of relativity:The laws of mechanics have the same form in all inertial frames.”

Practically speaking, the equivalence of inertial reference frames asserts that scientists within a box moving uniformly are unable to determine their absolute velocity by any experiment. Einstein (1920) and Feynman (1998) explained that the differences would set up an absolute standard reference frame. Wachter & Hoeber (2006) explained that supplementing this definition with the constancy of the speed of light would lead to the transformation of inertial frames of reference among themselves according to the Poincaré group of symmetry transformations. In Newtonian mechanics, inertial frames of reference are linked by the Galilean group of symmetries. 

What is absolute space?

Newton postulated an absolute space that can be approximated by a frame of reference stationary relative to the fixed stars. An inertial frame was then in uniform translation relative to absolute space. However, a group of scientists called “relativists" believed that absolute space was a defect of the formulation, requiring replacement. In 1885, Ludwig Lange coined the expression “inertial frame of reference” to replace Newton's definitions of "absolute space and time” with a more operational definition: 

A reference frame in which a mass point thrown from the same point in three different (non co-planar) directions follows rectilinear paths each time it is thrown, is called an inertial frame.

Blagojević noted the following inadequacies of the notion of "absolute space" in Newtonian mechanics: 

• The existence of absolute space contradicts the internal logic of classical mechanics since none of the inertial frames can be singled out, according to the Galilean principle of relativity.
• Absolute space fails to explain inertial forces since they are related to acceleration with respect to any one of the inertial frames. 
• Although absolute space acts on physical objects by inducing their resistance to acceleration, it can’t be acted upon: Milutin Blagojević: Gravitation and Gauge Symmetries, p. 5. 

Woodhouse (2003) claimed the special theory of relativity carried the utility of operational definitions further than earlier theories. Robert DiSalle argued the original question asked by Lange “"relative to what frame of reference do the laws of motion hold?” was posed incorrectly. A procedure to construct the laws of motion that can essentially determine a class of reference frames. 

What is Newton’s inertial frame of reference? 

An inertial frame of reference in the realm of Newtonian mechanics validates the Newton’s first law of motion. However, Einstein’s principle of special relativity generalises the notion of inertial frame better that includes all physical laws, not just Newton's first law.

Woodhouse (2003) and Takwale (1980) defined an inertial frame in the field of classical mechanics as: 

“An inertial frame of reference is one in which the motion of a particle not subject to forces is in a straight line at constant speed.”

Therefore, with respect to an inertial frame, this validates Newton’s first law of motion. If we interpret this rule as straight-line motion indicating zero-net force, then the rule fails to identify inertial reference frames as straight-line motion is observed in a variety of frames. Nevertheless, if we interpret the rule through a defined inertial frame, then we can determine the moment of application of zero net force. Einstein (1950) summarised this problem by highlighting the weakness of the principle of inertia involving an argument in a circle. This argument states that a mass moves without acceleration if it is adequately separate from other bodies. Since the mass moves without acceleration, this is evident of the mass’s adequate distance from other bodies.

 This diagram illustrates 2 frames of reference moving with relative velocity (v^->). Frame S’ has an arbitrary but fixed rotation with respect to frame S. They are both inertial frames provided a body not subject to forces appears to move in a straight line. If that motion is seen in one frame, it will also appear that way in the other.

There are a few proposed solutions to this problem: 

1. Rosser (1991) argued that all real forces weaken with distance from their sources in a known manner, meaning no force can be present when a body is sufficiently far from all sources. The view that the distant universe might affect matters (known as Mach’s Principle) challenges this argument. 
2. Identification of all real forces and accounting for them. However, this approach could lead to some things being overlooked, or inappropriate justification for their influence due to Mach’s principle and an incomplete understanding of the universe. 
3. Observing how the forces transform when we shift reference frames. It is thought fictitious forces arise from a reference frame’s acceleration, disappear in inertial frames, and follow complicated rules regarding transformation in general cases.

According to Feynman (1998), Newton proclaimed a principle of relativity in one of his corollaries to the laws of motion. In his Principia, Corollary V, Newton postulated that:

“The motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forward in a straight line”. This principle contrasts with the special principle in 2 ways: 

(1) Being restricted to mechanics, and (2) not mentioning simplicity. However, like the special principle, it shares the invariance of the form of the description among mutually translating reference frames.

How do we separate non-inertial from inertial reference frames?

The absence or presence of fictitious forces can help distinguish inertial and non-inertial reference frames. Borowitz & Bornstein (1968) stated that the observer needs to introduce a fictitious force into his calculations of the motions of objects in the non-inertial frame. Arnol'd (1989) argued the presence of fictitious forces complicated the physical laws available, hence a frame where fictitious forces are present is not an inertial frame according to the special principle of relativity. He explained that “the equations of motion in a non-inertial system differ from the equations in an inertial system by inertial forces. This would allow for the experimental detection of the non-inertial nature of a system”. 

Bodies in non-inertial reference frames are subject to fictitious forces (pseudo-forces), which result from the acceleration of the reference frame itself rather than any physical force acting on the body. Examples of fictitious forces include the centrifugal force and the Coriolis force in rotating reference frames. 

How do we separate “fictitious forces” from “real forces”? Applying the Newtonian definition of an inertial frame would be difficult without this separation. Imagine a stationary object in an inertial frame that is at rest, meaning no net force is applied. However, in a frame rotating about a fixed axis, the object appears to move in a circle, and is subject to centripetal force (composed of the Coriolis force and the centrifugal force). 

So, how do we decide that the rotating frame is a non-inertial frame? 

1. Find the origin of the fictitious forces. In this case, there are no sources for these forces, no associated force carriers, and no originating bodies. 
2. Study a variety of frames of reference. For any inertial frame, the Coriolis and centrifugal forces vanish. Therefore we need to apply the principle of special relativity to identify these frames where the forces disappear. Hence, we can determine that the rotating frame is not an inertial frame. 

 This diagram shows 2 spheres tied with a string and rotating at an angular rate ω. Because of the rotation, the string tying the spheres together is under tension.

 This diagram illustrates an exploded view of rotating spheres in an inertial frame of reference. The black arrows indicate the centripetal forces on the spheres provided by the tension in the tying string.

Newton examined the problem using rotating spheres, shown in the diagrams above. He indicated that non-rotating spheres don't pose any tension in the tying string in every reference frame. If the spheres appear to rotate, the tension in the string measured as zero is accounted for through the observation of the centripetal force supplied by both the Coriolis and centrifugal forces. Furthermore, if the spheres are actually rotating, the tension observed would be measured as the centripetal force required by the circular motion. Therefore, identification of the inertial frame is required when measuring the tension in the string. This means the tension in the strong exerts the centripetal force required by the motion observed by that reference frame. We can conclude that the fictitious forces vanish in this inertial frame of reference. 

Nevertheless, Newton proposed the idea of straight-line accelerations being undetectable in his Principia Corollary VI. He stated that: 

“If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will continue to move among themselves, after the same manner as if they had been urged by no such forces”.

Physicists generalised this principle as the idea of an inertial frame. Consider an observer confined in a free-falling lift. They argue that they are a valid inertial frame, even if they are accelerating under gravity, as along as they are unaware of the ongoings outside the lift. This means the inertial frame is a relative concept. Therefore, inertial frames can be collectively defined as a set of frames that are stationary or moving at constant velocity with respect to each other, so a single inertial frame is defined as an element of this set. 

To apply these ideas, everything we observe in a particular frame of reference needs to be subject to a base-line, common acceleration shared by the frame itself. For example, in an elevator, all objects inside are subject to the same gravitational acceleration, and the elevator itself accelerates at the same rate. 

What are their applications?

Inertial navigation systems use a cluster of gyroscopes and accelerometers to determine their acceleration relative to inertial space. When a gyroscope spins in a particular orientation in inertial space, the law of conservation of angular momentum states that it retains that orientation if no other external forces are applied to it. 3 orthogonal gyroscopes establish an inertial reference frame, while the accelerators measure acceleration relative to that frame. Information from the accelerators and a clock are combined to calculate the change in position. Kennie & Petrie (1993) described inertial navigation as a form of dead reckoning that doesn’t require any external input, meaning it can’t be pushed by any external or internal signal source. 

A device used for navigation of seagoing vessels that detects geometric north is called a ‘gyrocompass’. It doesn’t sense the Earth's magnetic field, but rather uses inertial space as its reference. 

I’ll delve into the details of special relativity and general relativity in another post.

What are some examples of inertial frames of reference?

Example 1: 

Imagine 2 cars travelling along road, both moving at constant velocities. At a particular moment, they are separated by 100 metres. The car in front is travelling at 10 m/s and the car behind is travelled at 15 m/s. To determine the time elapsed for the 2nd car to catch up with the 1st car, we need to understand the 3 “frames of reference” to select from.

 This figure illustrates 2  cars moving at different but constant velocities observed from stationary inertial frame (S) attached to the road and moving inertial frame (S′ ) attached to the first car.

Consider you’re observing both cars from the footpath on the side of the road. You define your “frame of reference" as S. Then we start a stop-clock at the exact moment that the second car passes by you, which is when both cars are 100 metres apart. Since both cars are moving at constant velocity, you can determine their positions  by the following formulas. Let x₁(t) be the position in metres of the 1st car after time t in seconds and x₂(t) as the position of the 2nd car after time t. 

x₁(t) = d + v₁*t = 100 + 10*tx₂(t) = v₂*t = 15*t 

Notice that these formulates predict at t = 0s, the 1st car is 100 metres ahead of the 2nd car, while the 2nd car is positioned beside the observer, i.e. you. To calculate the time at which x₁ = x₂. 

x₁(t) = x₂(t) 100 + 10*t = 15*t 100 = 5*t t = 100/5 = 20 seconds 

Another way to solve this problem is selecting a different frame of reference (S’) situated in the 1st car. In this case, the 1st car is observed as stationary and the 2nd car is approaching from behind at a speed of v₂ - v₁ = 5 m/s. For the 2nd car to catch up to the 1st car, it will take a time of d/(v₂-v₁) = 100/5 = 20 seconds. Note how simpler the problem became after choosing a more suitable frame of reference. 

The 3rd possible frame of reference can be attached to the 2nd car. The working out is identical, except the 2nd is observed as stationary and the 1st car moves backwards towards it at 5 m/s.

Example 2: 

Imagine 2 people standing, facing each other on either side of a east-west street. A car drives past them heading west. For the person facing north, the car was moving towards their left. For the person facing south, the car was moving towards their right. This discrepancy is due to both people using 2 different frames of reference from which to observe the direction of the car’s motion.

This diagram illustrates the simple-minded frame-of-reference example.

Example 3: 

1. Imagine you are standing on the side of a road watching a car drive past you from right to left. In your frame of reference, you define the position where you are standing as the origin, the road as the x-axis and the direction in front of you is the y-axis. To you, the car moves along the x-axis with a certain velocity (v) in the negative x-direction. So your frame of reference is considered an inertial frame of reference because you are not accelerating (ignoring effects such as Earth's rotation and gravity). 
2. Now imagine you are driving the car, and you choose this location as your origin, which becomes your frame of reference. That means the direction to your right as the positive x-axis, and the direction in front of you as the positive y-axis. In this frame of reference, you consider yourself as stationary and the world around you that is moving. For instance, as you drive past a neighbour on the side of the road, you observed them moving with velocity (v) in the negative y-direction. If you are driving east, then east in the positive y-direction. If you turn south, then south becomes your new positive y-direction. 
3. Consider you accelerate the car you’re driving in. As you drive past a bystander on the sidewalk, they evaluate their acceleration as a in the negative x-direction. Assume your acceleration is constant, a second car behind you is moving at constant velocity, what is the second car’s acceleration from your frame of reference? If the 2nd car’s velocity (v) is constant, the driver in that car is in an inertial frame of reference. That means the second car’s driver’s  acceleration is the same as the stationary bystander on the sidewalk in their frame of reference, which is a in the negative y-direction. However, if the second car is accelerating at rate A in the negative y-direction (i.e. decelerating), that means the first car’s acceleration can be evaluated as a’ = a - A in the negative y-direction, which is smaller than the acceleration measured by the bystander. Similarly, if the 2nd car is accelerating at rate A in the positive y-direction (i.e. speeding up), the driver in that car will observe the first car’s acceleration as a’ = a + A in the negative y-direction, which is larger than the acceleration measured by the same bystander. 

When a frame of reference moves close to the speed of light, then the flow of time in that frame of reference doesn’t necessarily apply in another frame. In special relativity, the speed of light is  considered to be the only true constant between moving frames of reference. 

Discuss the remarks

Consider that you own 2 clocks that tick at identical rates. You synchronise them to display the same time. Now put 1 of those clocks on a fast moving train traveling at constant velocity towards the other clock at a certain distance. According to Newton, the rate of time as measured in one frame of reference should be the same as the rate of time in another. He believed in the existence of “universal” time, meaning all other times in all other frames of reference should run at the same rate as this universal time irrespective of their position and velocity. This concept of time and simultaneity was later generalised by Einstein in his 1905 special theory of relativity. It lead to the development of transformations between inertial frames of reference based upon the universal nature of physical laws and their economy of expression, known as Lorentz transformations. 

The definition of inertial reference frame can be extended beyond 3D Euclidean space. Newton assumed a Euclidean space, but special relativity used more general geometric space. Imagine the geometry of an ellipsoid, a “free” particle is defined as 1 at rest or travelling at constant speed on a geodesic path. 2 free particles begin at the same point on the surface, travelling at the same constant speed but in different directions. Both particles eventually collide at the opposite side of the ellipsoid. Since no external forces were acting on those “free” particles, their speed remained constant, meaning no acceleration occurred, confirming Newton's first law. This leads to the suggestion that both particles were in inertial frames of reference. With no external acting forces, only the geometry of the ellipsoid could cause the collision of the 2 particles. Mathematicians commonly described this phenomenon as if we exist in a 4D geometry known as spacetime. To imagine spacetime, the curvature of this 4D space may be responsible for the attraction of the 2 masses in the absence of acting forces, which replaces the force known as gravity in Newtonian mechanics and special relativity. 

What are non-inertial frames of reference? 

Tocaci & Kilmister (1984) defined it as a frame of reference that undergoes acceleration with respect to an inertial frame. For example, an accelerometer at rest in a non-inertial frame will detect a non-zero acceleration. Rindler (1977) and Celnikier (1993) asserted that although the laws of motion are consistent in all inertial frames, they can vary from frame to frame depending on the acceleration in non-inertial frames. 

In classical mechanics, the introduction of additional fictitious forces (also called inertial forces, pseudo-forces, and d’Alembert forces) to Newton’s 2nd law could explain the motion of bodies in non-inertial reference frames. For example, the Coriolis force and the centrifugal force. Albert Shadowitz (1988) thought the the expression for any fictitious force can be derived from the acceleration of the non-inertial frame. Goodman and Warner (2001) argued that the term ‘force’ has to be redefined to include the so-called 'reversed effective forces' or 'inertia forces’ in order for Newton’s 2nd law to hold in the coordinate system. 

In the theory of general relativity, the curvature of space-time causes frames to be locally inertial, but globally non-inertial. The non-Euclidean geometry of curved space-time doesn’t allow the existence global inertial reference frames in general relativity. Specifically, the fictitious force appearing in general relativity is the force of gravity.

How do we avoid fictitious forces in calculations?

We can avoid using non-inertial frames in flat space-time if necessary. Alonso and Finn (1992) stated that measurements with respect to non-inertial reference frames can transform to an inertial frame, which directly incorporates the acceleration of the non-inertial frame as that acceleration as viewed from the inertial frame. Price (2006) evaluated this approach avoids use of fictitious forces, however it is less convenient from an intuitive, observational and a mathematical point of view. For the case of rotating frames in meteorology, Ryder (2007) asserted that transforming all coordinates to an inertial system can simplify the problem. 

How do we detect a non-inertial frame?

Several studies concluded a reference frame’s need for fictitious forces can detect a given frame being non-inertial when explaining observed motions. An example given by Francia (1981) is using a Foucault pendulum to observe the Earth’s rotation. Earth’s rotation seemingly causes the pendulum to change its plane of oscillation as the pendulum’s surroundings move with the Earth. If we view it from an Earth-bound (non-inertial) frame of reference, this apparent change in orientation can be explained by the fictitious Coriolis force. 

A few studies noted the tension in the string between 2 spheres rotating about each other is another famous example. As viewed the rotating reference frame, predicting the measured tension in in the string based upon the motion of the spheres requires the introduction of a fictitious centrifugal force. 

Note that implementing a change in coordinate system, e.g. from Cartesian to polar, without any change in relative motion doesn’t cause the appearance of fictitious forces. This is spite of the fact that the form of the laws of motion varies from one type of curvilinear coordinate system to another.

What are fictitious forces in curvilinear coordinates?

The term “fictitious force” is often used in curvilinear coordinates, particularly polar coordinates. They are non-zero in all frames of reference, inertial or non-inertial, and do not transform as vectors under rotations and translations of the coordinates. 

Using this definition makes it incompatible with non-inertial frames contexts. It is defined by determining the particle’s acceleration within the curvilinear coordinate system, and then separating the double-time derivatives of coordinates from the remaining terms i.e. fictitious forces. Other terms such as “generalised fictitious forces” are used to indicate their connection to the generalised coordinates of Lagrangian mechanics. 

Examples

An accelerated frame of reference often delineates as being the “primed” frame, and primes denote the variables dependent on that frame. e.g. x’, y’, a’. 

The vector from the origin of an inertial reference frame to the origin of an accelerated reference frame is commonly notated as R. Given a point of interest that exists in both frames: 

r = R + r’ 

— r = Vector from the inertial origin to the point  

— r’ = Vector from the accelerated origin to the point  

If we take the 1st and 2nd derivatives of the above formula with respect to time, we obtain: 

v = V + v’ a = A + a’ 

— V = Velocity of the accelerated system with respect to the inertial system 

— A = Acceleration of the accelerated system with respect to the inertial system

— v = Velocity of the point of interest with respect to the inertial frame

— a = Acceleration of the point of interest with respect to the inertial frame. 

These equations allow transformations between the 2 coordinate systems. Newton’s 2nd law can be rewritten as: 

F = m*a = m*A + m*a’

Inertia manifests from acceleration after being exerted by a force. Electric cars are designed to recharge its batteries during deceleration, which illustrates the physical strength of manifestation of inertia. Nevertheless, the manifestation of inertia doesn’t prevent positive or negative acceleration, but it occurs in response to change in velocity due to a force. From the perspective of a rotating frame of reference, the manifestation of inertia exerts a force (either in centrifugal direction, or in a direction orthogonal to an object's motion, the Coriolis effect). 

A common type of accelerating reference frame both rotates and and transforms e.g. A reference frame attached to a CD that plays while the player is carried. This leads to the equation: 

a = a’ + ω^.*r’ + 2*ω*v’ + ω*(ω*r’) + A₀

To evaluate the acceleration in the accelerated frame, we express it as: 

a’ = a - ω^.*r’ + 2*ω*v’ + ω*(ω*r’) + A₀

Then we multiply by mass m to get: 

F’ = F_physical + F’_Euler + F’_Coriolis + F’_Centripetal - m*A₀ 

— F’_Euler = -m*ω^.*r’ : Euler force 

— F’_Coriolis = -2*m*ω*v’ : Coriolis force 

— F’_Centripetal = -m*ω*(ω*r’) = m*(ω² * r’ - (ω*r’)*ω : Centrifugal force 

The next part delves into the details of force (inc. gravity), energy, work and relativity.