Coefficient of Restitution Calculator
Calculate the coefficient of restitution (e) from drop and bounce heights: e = √(h₁/h₀).
Shows energy retained per bounce and category from inelastic to elastic.
The coefficient of restitution (e) measures how much speed an object retains after a collision. It runs from 0 (perfectly inelastic, the objects stick) to 1 (perfectly elastic, no kinetic energy lost). Real-world collisions sit between those extremes.
Two equivalent formulas:
For a ball dropped onto a hard surface (the simplest measurement): e = √(h₁ / h₀)
where h₀ is the drop height and h₁ is the bounce height. This is what the calculator uses, because both values are easy to measure with a ruler.
For a general two-body collision: e = (v₂’ − v₁’) / (v₁ − v₂)
The ratio of relative velocity after to relative velocity before. The two forms are equivalent because energy is proportional to height (PE = mgh) and to velocity squared (KE = ½mv²).
Why the square root?
When you drop a ball, the speed at impact is v = √(2gh₀). The bounce-back speed is √(2gh₁). Their ratio is:
e = √(2gh₁) / √(2gh₀) = √(h₁ / h₀)
The square root comes from converting back from energies (which scale with h) to speeds (which scale with √h). Easy to forget when working from heights, easy to remember when you think in speeds.
Typical values for common materials
| Material / setup | Approximate e | Energy kept |
|---|---|---|
| Superball | 0.90 | 81% |
| Steel ball on steel | 0.85 to 0.90 | ~75% |
| Golf ball | 0.78 to 0.83 | ~65% |
| Rubber ball | 0.75 | 56% |
| Tennis ball (regulation) | 0.70 to 0.75 | 50% |
| Basketball (regulation) | 0.72 to 0.78 | 55% |
| Baseball on bat | 0.50 to 0.55 | ~28% |
| Clay or putty | 0.05 to 0.15 | ~1% |
| Wet snow | 0.00 to 0.05 | ~0% |
The values vary with surface, temperature, and ball pressure. Cold tennis balls bounce noticeably worse than warm ones because the air inside contracts and the rubber stiffens.
Sport regulation examples
The NBA specifies that a basketball, dropped from 6 feet onto a hardwood court, must bounce between 49 and 54 inches. That works out to an e of 0.72 to 0.75. The MLB likewise tests baseballs against a wooden surface at 60 mph; the rebound must hit 51 to 57 mph, corresponding to e in the 0.51 to 0.57 range — markedly lower than tennis or basketball because most of the impact energy goes into deforming the ball.
Multiple bounces: how high after n bounces?
After n bounces from drop height h₀: hₙ = h₀ × e^(2n)
A ball with e = 0.8 dropped from 1.0 m bounces to 0.64 m, then 0.41 m, 0.26 m, 0.17 m. The bounces get smaller fast because each one multiplies by e² = 0.64, not e.
When the calculator matters
- Sports engineering: comparing balls to regulation specs
- Materials testing: characterizing impact response of polymers, ceramics, and composites
- Vehicle crashworthiness: e is one of the parameters in bumper and airbag design
- Granular flow: rock impacts, pellet handling, and even powder mixing all use COR
- Demo physics labs: a single ruler and ball gives a clean experimental result
A note on the limits
The formula assumes a passive collision — no spring, no rocket, nothing that adds energy. With a perfectly stiff floor and a perfectly stiff ball, both with no internal damping, you would get e = 1 and the ball would bounce forever to the same height. Real materials always dissipate some energy to heat, sound, and permanent deformation, so e < 1 in practice. A bounce height that exceeds the drop height means something is wrong with the measurement (or the surface is doing work, which is no longer a pure collision).