Elastic Collision: Final Velocity of Body 1

An elastic collision conserves both momentum and kinetic energy, and solving those two equations together gives this closed form for the first body's rebound. Three cases are worth memorising. Equal masses with the target at rest: v₁ = 0 and the bodies swap velocities exactly, the behaviour of a Newton's cradle and of a well-struck cue ball. A light body striking a much heavier one: v₁ ≈ −u₁, a near-perfect bounce back, as when a ball hits a wall. A heavy body striking a light one: v₁ ≈ u₁, it barely notices.

Worked example: a 1 kg ball at 4 m/s hits a stationary 3 kg ball, giving v₁ = ((1 − 3)(4) + 0) ⁄ 4 = −2 m/s. It rebounds at half speed while the heavier ball moves off at 2 m/s. Truly elastic collisions are an idealisation for everyday objects (steel bearings come close, billiard balls reach about 95%), but they are exact for gas molecules and for neutrons scattering in a reactor moderator, which is precisely why moderators use light nuclei like hydrogen or carbon, where each collision strips the most energy.