Understanding The Coefficient Of Restitution

30 September 2026Physics Try the interactive 3D model →

The coefficient of restitution (e) is a number between 0 and 1 that measures how bouncy a collision is. It is defined as the ratio of the relative speed of separation after impact to the relative speed of approach before impact, measured along the line of impact. A value of e = 1 means a perfectly elastic bounce with no energy lost; e = 0 means a perfectly inelastic impact where the objects stick together.

This single number matters because it captures everything complicated about a collision — the squashing, the heat, the sound — in one measurable value. Sports regulators use it to certify tennis balls, golf balls, and cricket bats; engineers use it to predict how high a dropped part will rebound on a factory floor; and accident investigators use it to estimate how much energy a crash absorbed. If you know e and the incoming speeds, you can compute the outgoing speeds without knowing any details of the impact forces.

Key Concepts

Relative speed of approach: how fast the two bodies are closing in on each other along the line of impact before the collision.
Relative speed of separation: how fast they are moving apart along the same line after the collision.
Newton's experimental law: e = (v₂ − v₁) / (u₁ − u₂) for a head-on collision, where u are velocities before and v after.
e = 1: perfectly elastic — separation speed equals approach speed; kinetic energy conserved.
e = 0: perfectly inelastic — zero separation speed; bodies move together.
0 < e < 1: real-world collisions, e.g. about 0.75–0.85 for a good tennis ball on a hard court.
Unitless: e is a pure ratio with no units, and it depends on the pair of materials, not just one.

Formulas

e = (relative speed of separation) / (relative speed of approach) — the defining relation.
Head-on: e = (v₂ − v₁) / (u₁ − u₂) — velocities along the line of impact with consistent signs.
Bouncing ball dropped from height h₁ rebounding to h₂: e = √(h₂ / h₁), since impact speed is √(2gh).
Kinetic energy retained: in a bounce off a fixed wall, the rebound KE is e² times the impact KE.

Assumptions: the formula applies along the line of impact; spinning balls need extra treatment. Limits: e is not a universal constant — it can vary slightly with impact speed and temperature; values above 1 would create energy and are impossible in passive collisions.

Worked Example

A ball is dropped from a height of 2 m onto a concrete floor and rebounds to 0.5 m. Find the coefficient of restitution.

Speed just before impact: u = √(2gh₁) = √(2 × 9.8 × 2) = √39.2 ≈ 6.26 m/s.
Speed just after rebound: v = √(2gh₂) = √(2 × 9.8 × 0.5) = √9.8 ≈ 3.13 m/s.
e = separation speed / approach speed = 3.13 / 6.26 = 0.5.
Shortcut check: e = √(h₂/h₁) = √(0.5/2) = √0.25 = 0.5. ✓ The ball keeps e² = 25% of its kinetic energy after each bounce.

Common Misconceptions

"e is a property of one object." It describes the collision between two specific surfaces — the same ball has a different e on concrete than on grass.
"A higher bounce always means e is closer to 1." Only when the drop height is fixed. Rebound height also depends on where you drop it from; e = √(h₂/h₁) needs both heights.
"e = 0 means the objects stop moving." It means zero relative speed — they can still move together at high speed, like two train cars coupling.

Real-World Applications

Sports equipment testing: tennis balls are dropped from 100 inches onto concrete and must rebound 53–58 inches (e ≈ 0.73–0.76) to be legal.
Car crash analysis: low e values confirm crumple zones absorbed energy instead of bouncing the car back.
Industrial handling: knowing e predicts rebound heights of parts dropped on conveyor belts and sorting lines.
Planetary science: lander designs use e of the surface material to predict how many times a probe will bounce on touchdown.

The coefficient of restitution e is the ratio of separation speed to approach speed along the line of impact: 1 for a perfect bounce, 0 for sticking together, and somewhere in between for every real collision. For a bouncing ball it simplifies beautifully to e = √(h₂/h₁), making it one of the easiest collision quantities to measure with just a ruler.