Collisions

In a collision, two bodies push on each other for a very short time. Those forces are equal and opposite (third law), and nothing outside pushes, so the total momentum is always conserved. The total energy is conserved too, but some kinetic energy may turn into heat, sound or a dent.

Collisions are named by what happens to the kinetic energy:

• Elastic: the total kinetic energy after equals that before. Nearly true for steel balls or billiard balls.
• Inelastic: some kinetic energy is lost.
• Completely inelastic: the bodies stick together and move as one; this loses the most kinetic energy.

Example of sticking: a 2 kg trolley at 6 m s⁻¹ runs into a 4 kg trolley at rest, and they couple. Momentum: 2 × 6 = 6v, so v = 2 m s⁻¹. Kinetic energy before: 36 J; after: ½ × 6 × 2² = 12 J. So 24 J was lost.

For a head-on elastic collision with m₂ at rest, both momentum and kinetic energy are conserved, which gives:

• v₁f = (m₁ – m₂)v₁ᵢ/(m₁ + m₂) and v₂f = 2m₁v₁ᵢ/(m₁ + m₂).
• Example: 1 kg at 8 m s⁻¹ hits 3 kg at rest. v₁f = (1 – 3)/4 × 8 = –4 m s⁻¹ (it bounces back) and v₂f = 2/4 × 8 = 4 m s⁻¹. Check: 32 J before, 8 J + 24 J after.

Three special cases are worth remembering:

• Equal masses: they swap velocities. The moving ball stops dead and the other moves off at its speed, as in carrom.
• A light ball hitting a very heavy one at rest (a ball on a wall) bounces straight back with the same speed; the heavy one hardly moves.
• A very heavy body hitting a light one at rest carries on almost unchanged, and the light one flies off at twice its speed.

This is why a neutron loses the most energy when it hits a light nucleus of about its own mass. Reactors slow neutrons down with light-atom moderators such as heavy water.

In a glancing (two-dimensional) collision, momentum is conserved along x and along y separately. When two equal masses collide elastically and one was at rest, they always move off at right angles to each other.