Coefficient of Restitution Calculator

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

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 (measured to the bottom of the ball) must rebound 49 to 54 inches (measured to the top of the ball). Account for the ball’s roughly 9.5-inch diameter and that works out to an e of about 0.74 to 0.79. MLB tests baseballs by firing them at a wall of northern white ash at 58 feet per second; the rebound speed must be 51.4 to 57.8 percent of the incoming speed, an e of 0.514 to 0.578 — 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).