Illustrated Physics
physics, drawn
Concept

Restitution — where it appears

The ratio of the speed at which two bodies separate after a collision to the speed at which they approached. It is one for a perfectly elastic bounce and below one whenever energy is lost to heat, sound or deformation, and it decides whether repeated bounces end in finite time.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

A gas of grains that lose 51 per cent of their energy per collision, cooling into clumps. 2000 discs in a square box with periodic walls, covering 25 per cent of its area, started with random velocities and left alone. Each collision keeps a fraction e = 0.7 of the relative velocity along the line of centres. The panels are the same gas at three temperatures, where temperature means the mean kinetic energy of a grain: 0.85 T₀, 0.019 T₀, 0.00049 T₀. It starts uniform — the grain counts in a 10 × 10 grid have a variance 0.3 times their mean, more even than scattered points because discs cannot overlap — and ends in dense bands and clumps with empty space between, the variance 17 times the mean. Nothing attracts anything. The clumps form because a dense patch collides more often, cools faster, loses pressure and is squeezed denser by the hotter gas around it.

The gas that cools itself into clumps

Shake a box of grains hard enough and they fly about like the molecules of a gas. Stop shaking and the gas cools, because every collision destroys a little of the motion — and it does not cool evenly. A denser patch collides more often, cools faster, loses pressure, and is squeezed denser by the hotter gas around it. With nothing attracting anything to anything, the gas gathers itself into clumps and bands, and in the limit three grains can collide infinitely many times in a finite time.

fluids · Granular matter 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

Named alongside it

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

DissipationCollisionContact mechanicsEquation of stateGranular matterIdealisationImpulseInstabilityIsochronismKinetic theoryMean free pathMomentum

All concepts