Illustrated Physics
physics, drawn
Concept

Collision — where it appears

An interaction brief enough that the bodies are free before and after, so momentum and energy can be counted at the two ends without following the middle. Momentum is always conserved; kinetic energy is not, and what is missing has gone into the internal state of what collided.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Six observers, six energies, one loss. The kinetic energy of the same collision before and after it, as measured by observers moving at 6 different speeds. No two of them agree about how much energy there was: the totals here range from 5.50 to 25.50 in the same units. Every one of them agrees about how much was lost — the gap between the two curves is 3.413 for all of them, varying by 6.2e-15. Energy is a quantity an observer owns; a change in it is not, and that is why heat, deformation and sound can be counted at all.

The energy that depends on the observer

A moving train has kinetic energy. Watched from a second train alongside it, it has none. Both statements are correct, neither can be corrected, and the whole of mechanics still works — because what conservation laws constrain is not how much energy there is but how much of it changes.

mechanics · Energy The floor does no work and the jumper leaves the ground. A 70-kilogram person pushing off the floor: the floor's force in units of body weight against time, with the centre of mass's height and speed drawn on the same axis, each scaled. The force reaches 2.6 body weights, the contact lasts 260 milliseconds, and the take-off speed that comes out of integrating it is 1.67 metres a second — a jump of 14 centimetres. Integrating the floor's force over the centre of mass's rise gives 247 joules. The work the floor does is zero, because the patch of floor under the foot never moves and work is a force times the displacement of its own point of application. Both numbers are correct and they are answers to different questions: the first is what Newton's second law integrated over the centre of mass gives, and the second is what crosses the boundary between the floor and the person, which is nothing.

The floor that does no work

A jumper leaves the ground with three hundred joules of kinetic energy, supplied by a floor that does exactly zero work — because work is a force times the displacement of its own point of application, and the patch of floor under the foot never moves. Newton's second law integrated over the centre of mass gives the right kinetic energy and is not the work-energy theorem, and telling the two apart is what the first law of thermodynamics is for.

mechanics · Energy A wall pulled away fast leaves the energy behind. The energy of a ball bouncing in a box whose wall is moved, against the length of the box, for 3 wall speeds — each a fraction of the ball's own starting speed — with the adiabatic prediction drawn dashed. Every collision is solved for exactly rather than stepped, so nothing here assumes the wall is slow. Moved slowly, the wall takes energy from the ball at the adiabatic rate, and the energy falls as the inverse square of the length. Moved as fast as the ball is moving, it takes almost nothing: the ball cannot catch a wall retreating faster than it travels, so the collisions stop and the energy stops falling. That is the difference between a gas pushing a piston and a gas expanding into a vacuum, and it is drawn here for one particle. At the end of the range the slowest wall leaves the energy at 0.058 of its starting value and the fastest at 1.000, against an adiabatic 0.065.

The wall that moves while the ball is in flight

Energy is conserved because the rules do not depend on the time. Move the walls of a box and the rules do depend on the time, so what is inside gains or loses without limit — and how much depends entirely on how fast. Moved slowly, a wall takes energy at exactly the rate the adiabatic law says; moved faster than the ball travels, it takes none at all, and the same box is a piston or a vacuum according to a speed.

mechanics · Energy A ball dropped from 1 m with e = 0.8, bouncing for ever and stopping anyway. The height of a ball dropped from 1 m onto a floor that returns 0.8 of its impact speed, against time. The first fall takes 0.452 s. Every flight after it is 0.8 times as long as the one before, so the flight times are a geometric series and their sum is finite: the ball has made infinitely many bounces by 4.064 s, t₁(1 + e)/(1 − e), and is at rest from then on. The stepped sum of the first 60 flights plus the geometric remainder agrees with the closed form to better than a part in a billion. The dashed line marks the moment it stops; the hops just before it are too small to draw.

The bounces that add up to a stop

A ball dropped on a hard floor bounces, and bounces again, each time a fixed fraction lower. The bounces never run out — there is always another, smaller one — and yet the ball is lying still four seconds later. The flight times form a geometric series with a finite sum, the floor's infinitely many kicks add up to exactly the ball's weight times that time, and the series is only broken where each flight becomes shorter than the impact that launched it.

mechanics · Momentum Restitution with nothing lossy in it. The coefficient of restitution of a perfectly elastic ball striking a large, perfectly elastic plate, against Zener's parameter λ, which grows as the square of the ball's radius over the plate's thickness and weakly with speed. At λ = 0 the ball comes back at its arrival speed. The rebound falls below half at λ ≈ 0.65, with no friction, plasticity or damping anywhere — the missing energy is travelling outward through the plate as bending waves. Marked: a steel ball 20 mm across at 1 m/s on steel plates 5 mm thick (λ = 0.94), 10 mm thick (λ = 0.23), 20 mm thick (λ = 0.06).

The bounce an elastic plate cannot give back

Drop a steel ball on a thick steel block and it bounces almost to where it started. Drop it on a steel plate five millimetres thick and it comes back at barely half its speed, although steel on steel is about as elastic as materials get and nothing in the impact is plastic, sticky or damped. The energy has not been turned into heat. It has been turned into waves, running out through the plate faster than the ball can take them back — and a ball striking a longer rod loses energy the same way, by leaving it ringing.

mechanics · Momentum

Named alongside it

The objects these essays reach for when they reach for this one.

EnergyMomentumCentre of massConservation of energyDissipationWorkActionAdiabatic invariantBending stiffnessBookkeepingCoefficient of restitutionConservation

All concepts