Illustrated Physics
physics, drawn
Concept

Elastic collision — where it appears

A collision in which the total kinetic energy is the same afterwards as before, so both momentum and energy fix the outcome. Two equations and two unknowns leave exactly one answer, which is why an elastic one-dimensional collision can be solved without knowing anything about the forces involved.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

A collision with restitution 0.6. Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.

Collisions are easier than forces, and momentum is the reason

Nobody knows what happens inside a collision. Momentum conservation makes that ignorance irrelevant, which is the whole trick — and energy, deliberately, is not conserved.

mechanics · Momentum The wavelength shift against scattering angle. How much longer a scattered photon's wavelength is, against the angle it scattered through. The shift runs from nothing at 0 degrees to 4.853 picometres straight back, passing through the electron's Compton wavelength of 2.4263 picometres at 90 degrees. Nothing about the incident light or the target material appears anywhere on this axis.

A photon with a momentum, and a collision that proves it

X-rays bouncing off electrons come back with a longer wavelength. How much longer depends on the angle they turned through, and on nothing else — not the incident wavelength, not the target material, not the intensity.

quantum · Photon A collision with restitution 0.4. Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.

The point that keeps moving as if nothing had happened

Newton's third law makes every internal force cancel against its own partner, which leaves the external sum governing a single mass-weighted average of positions. In the collision below the total momentum stays at 4.00 kg·m/s while 63 per cent of the kinetic energy leaves, and the average travels at 1.00 m/s throughout, before and after.

mechanics · Momentum 5 balls whose contact force goes as the overlap to the three halves, touching. The velocity of every ball in a line of 5, against time in units of one binary contact, with the first arriving at unit speed. Nothing about collisions is assumed: neighbours push on each other with k times their overlap raised to the power 1.5, and the equations of motion are integrated. With the balls touching there is no such separation — several overlaps are non-zero at once and the disturbance crosses the line as a single compression wave. The far ball leaves at 0.989 of the striking speed and the others keep 0.011 between them, which is why a real cradle's balls do not quite come to rest. Momentum and energy are conserved to 4.4e-16 and 4.1e-8, so the difference between the two cases is the contact law and not the bookkeeping.

Five balls, and the law that does not choose

The usual account of a Newton's cradle says that momentum and energy conservation force one ball out at the striking speed. For three balls or more they do no such thing: the two laws leave a whole curve of possible outcomes, and what picks one is the shape of the force between two touching spheres.

mechanics · Momentum Equal masses, one at rest, struck head on at four energies. A particle striking an identical particle at rest and scattering elastically, with the two leaving at equal angles either side of the incoming line, at four incident kinetic energies measured in units of the particle's rest energy. At 0.001 mc² the two tracks open at 90.0°. At 0.5 mc² the two tracks open at 83.6°. At 2 mc² the two tracks open at 70.5°. At 10 mc² the two tracks open at 44.4°. Slowly, the angle between them is a right angle, as every billiards player knows. Fast, it closes, because the incoming particle's energy is part of its inertia and the pair is carried forward. A proton at 2 mc² — about 1.9 GeV — and an electron at 1 MeV are both well inside the narrowing.

The right angle a fast collision closes

When one billiard ball strikes an identical one at rest, the two always leave at right angles — a fact every snooker player uses without knowing its proof. The proof is Newton's, and it fails for fast particles. Two protons leaving a collision at 1.9 GeV open at 70 degrees, not 90, and the narrowing follows one law, tan θ₁ tan θ₂ = 2/(γ + 1), in which the right angle is the slow limit. The circle that forced the right angle has been squashed into an ellipse by the energy the incoming particle carries as inertia.

relativity · Relativistic dynamics The share of its energy a particle can hand to one electron. The largest fraction of its kinetic energy a muon, a pion, a proton or an alpha particle can give a free electron in one collision, against its Lorentz factor γ, on logarithmic axes. A slow proton can give at most 0.22 per cent, 4m/M, which is why a proton slows down through tens of thousands of gentle collisions rather than a few hard ones. The share rises as γ grows and reaches a half at γ = 916 for a proton and 101 for a muon — near M/2m, where the electron's energy in the centre-of-momentum frame has grown comparable to the projectile's, so that the two meet as equals and a head-on reversal can hand the electron most of the energy.

The kick a fast particle can give an electron

A slow proton crossing matter can give any one electron at most a fifth of a per cent of its energy, because an electron struck head-on by something two thousand times heavier rebounds at no more than twice the heavier one's speed. That rule of thumb fails at relativistic speeds, and fails upward: the electron's speed is capped below light, but its energy is not, and the largest kick grows as the square of the projectile's Lorentz factor until, above a teraelectronvolt, one collision can take more than half of everything a proton carries.

relativity · Relativistic dynamics The number of collisions for heavier and heavier blocks. The number of collisions, simulated one by one, for mass ratios of 1, 10², 10⁴, 10⁶, 10⁸, on a logarithmic axis: 3, 31, 314, 3141, 31415. Each factor of a hundred in the mass ratio adds one digit of π. Nothing in the setup mentions a circle — two blocks, a wall and the conservation of energy and momentum — but energy conservation makes the velocities live on a circle, and the count is how many equal steps of a small angle fit into half of it.

The collisions that count the digits of π

Slide a heavy block towards a light one resting near a wall. The light block is batted between the heavy one and the wall until the heavy block turns round and leaves. Count the collisions. With equal masses there are 3; with one block a hundred times heavier, 31; ten thousand times, 314; a million, 3141. Nothing in two blocks and a wall mentions a circle, but conservation of energy puts their velocities on one, and each collision steps round it by the same small angle. The count is how many such steps fit in half a turn.

mechanics · Momentum

Named alongside it

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

Energy conservationMomentum conservationCentre of massFour-momentumImpulseThe Lorentz factorBilliardsBoostCentre of momentum frameCoefficient of restitutionCompton wavelengthConservation laws

All concepts