Elastic collisions

Definition of an elastic collision

A collision is said to be elastic when, in addition to momentum, the total kinetic energy of the system is conserved:

p_total(before) = p_total(after) Ec_total(before) = Ec_total(after)

These collisions are an idealised model (billiards, collisions between elementary particles, an ideal gas in kinetic theory): no energy is dissipated as heat, sound or permanent deformation.

Special case in one dimension

For two masses m₁ and m₂ with initial velocities u₁ and u₂ (head-on collision, 1D), the simultaneous solution of the two conservation equations gives the final velocities:

v1 = ((m1 − m2) * u1 + 2 * m2 * u2) / (m1 + m2) v2 = ((m2 − m1) * u2 + 2 * m1 * u1) / (m1 + m2)

Notable cases

• If m1 = m2: v1 = u2 and v2 = u1. The two bodies simply exchange their velocities (the classic case of identical billiard balls).
• If m2 is at rest (u2 = 0) and m1 is much larger than m2: v1 ≈ u1 and v2 ≈ 2*u1 (the smaller mass is propelled at approximately twice the incident velocity).
• If m1 is very large and at rest, m2 rebounds with v2 ≈ -u2 (elastic rebound off a solid wall).

Example

A 1 kg ball travelling at 4 m/s elastically collides with a stationary 3 kg ball.

v1 = ((1–3) × 4 + 0) / 4 = –8/4 = –2 m/s v2 = ((3–1) × 0 + 2 × 1 × 4) / 4 = 8/4 = 2 m/s

Energy check: Ec_before = 0.5 × 1 × 4² = 8 J; Ec_after = 0.5 × 1 × 2² + 0.5 × 3 × 2² = 2 + 6 = 8 J. The lighter ball rebounds backwards.

Pitfall: assuming that an elastic collision implies that the two final velocities are equal; this is only true when m₁ = m₂ AND the bodies come to rest against each other, which is not the general rule.