Two laws for every collision
When two objects collide head-on with no outside force, their total momentum stays the same: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. What happens to their kinetic energy depends on the materials. Steel balls bounce back almost perfectly; clay sticks. The coefficient of restitution captures that difference in one number between 0 and 1.
The formulas
With masses m₁ and m₂, initial velocities u₁ and u₂ and a coefficient of restitution c:
v₁ = [m₁u₁ + m₂u₂ + m₂c(u₂ − u₁)] ÷ (m₁ + m₂)
v₂ = [m₁u₁ + m₂u₂ + m₁c(u₁ − u₂)] ÷ (m₁ + m₂)
- c = 1: perfectly elastic, kinetic energy conserved.
- 0 < c < 1: partly inelastic, some energy lost.
- c = 0: perfectly inelastic, both objects move off together at the velocity of the center of mass.
Worked example
A 4 lb cart rolling at 10 ft/s hits a 2 lb cart at rest, with springy bumpers (c = 1):
- Total momentum and center-of-mass velocity: (4 × 10) ÷ 6 = 6.67 ft/s.
- The first cart slows to 3.33 ft/s; the second shoots off at 13.33 ft/s.
- Kinetic energy before and after: 8.43 J (6.2 ft·lbf), unchanged.
With sticky bumpers (c = 0), both carts move at 6.67 ft/s, and one third of the kinetic energy is lost to heat and deformation.
Useful special cases
- Equal masses, elastic: the objects swap velocities. A cue ball hitting another ball dead center stops.
- Light object hits a very heavy one: it bounces back at nearly the same speed, like a ball off a wall.
- Heavy object hits a light one at rest: the light object leaves at almost twice the heavy one’s speed.
- Perfectly inelastic: the energy lost is the fraction m₂ ÷ (m₁ + m₂) when object 2 starts at rest.
Typical coefficients of restitution
- Billiard balls: about 0.9 to 0.95.
- Basketball on hardwood: about 0.85.
- Tennis ball on a hard court: about 0.75.
- Car crash at speed: close to 0; the crumple zones absorb energy by design.
You can estimate c for a ball by dropping it: c = √(bounce height ÷ drop height).
Signs and units
Pick a positive direction and give velocities moving the other way a negative sign. A negative result means the object moves in the negative direction after impact. The calculator works in one dimension; glancing collisions need separate equations for each direction. Momentum is shown in kg·m/s, the SI unit.