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.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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