Conservation of Momentum Calculator
Calculate final velocities after any 1D collision using momentum conservation and a coefficient of restitution from 0 (inelastic) to 1 (elastic).
Linear momentum is always conserved in a collision — no exceptions. What differs between collision types is whether kinetic energy is also conserved.
The coefficient of restitution e measures how much relative speed is preserved:
e = (v₂f − v₁f) / (v₁i − v₂i)
e = 1 — elastic: kinetic energy is fully conserved (billiard ball collisions approximate this) e = 0 — perfectly inelastic: objects stick together after impact (car crashes) 0 < e < 1 — partially inelastic: most real collisions, energy lost to heat and deformation
Final velocities for a 1D collision:
v₁f = [(m₁ − e·m₂)v₁i + (1 + e)m₂v₂i] / (m₁ + m₂) v₂f = [(1 + e)m₁v₁i + (m₂ − e·m₁)v₂i] / (m₁ + m₂)
The total momentum before and after is the same:
p = m₁v₁i + m₂v₂i = m₁v₁f + m₂v₂f
Sign convention: rightward velocities are positive, leftward are negative. A stationary object has velocity 0.
Typical e values: steel on steel ≈ 0.65, rubber ball on concrete ≈ 0.6–0.8, baseball bat ≈ 0.55, car crash ≈ 0.1–0.3.
The special case of e = 1 with equal masses is elegant: the two objects swap velocities. A moving ball hitting an identical stationary ball will stop completely, transferring all its speed. Newton’s cradle demonstrates this.