Coefficient of Restitution Calculator
Drop a ball, measure how high it bounces, and this coefficient of restitution calculator returns e — the 0-to-1 number that tells you how bouncy the collision is.
One bounce test, one number
Enter the height you drop from and the height of the bounce; the calculator returns e = √(bounce ÷ drop), a value between 0 and 1.
Same units, drop from rest
Use the same length unit for both heights and release the ball from rest — the bounce can never be higher than the drop.
What is the coefficient of restitution?
A 0-to-1 measure of bounciness
The coefficient of restitution (often written e) measures how much speed a bouncing object keeps after a collision — in plain terms, how bouncy it is. This coefficient of restitution calculator uses the simplest real-world test there is: drop a ball from rest, measure how high it bounces back, and read off e from the two heights. A value of 1 means a perfectly elastic bounce that loses no energy, while 0 means a perfectly inelastic thud that does not rebound at all. Every real ball sits somewhere in between, which makes e a quick, honest gauge of the material and the surface it hits.
Enter a drop height and a bounce height in the same unit to get the coefficient of restitution instantly, on a scale from 0 to 1.
For a ball released from rest, the coefficient of restitution is the square root of the bounce height divided by the drop height.
e = √(bounceHeight ÷ dropHeight)The square root appears because the heights track energy, while e compares speeds: the impact and rebound speeds are proportional to the square roots of the drop and bounce heights. Dividing the two heights and taking the square root cancels the units, so e is a pure ratio with no dimension — only the two heights need to share the same unit.
Suppose you drop a ball from a height of 1 m and it bounces back up to 0.6 m.
Divide the heights
0.6 ÷ 1 = 0.6 — the fraction of height the ball recovers.
Take the square root
√0.6 = 0.7746 — converting the energy ratio into a speed ratio.
Read the result
e ≈ 0.77 — a fairly bouncy ball, close to a basketball, that keeps about 77 % of its speed through the bounce.
The coefficient of restitution always lands between 0 and 1, and where it sits tells you a lot about the collision. A value near 1 means a very bouncy, nearly elastic rebound: a superball comes in around 0.9, returning to roughly 80 % of its drop height. A typical basketball sits near 0.75, which is exactly why league rules specify a bounce height range for an approved ball. Lower down, a tennis ball is around 0.7 and a baseball closer to 0.5. As e approaches 0 you reach the "dead" end of the scale — a lump of clay or a beanbag has e near zero, absorbing almost all the energy and barely rebounding at all. Because the heights enter as a square root, a ball that bounces to half its drop height (a height ratio of 0.5) has e ≈ 0.71, not 0.5 — the speed it keeps is always higher than the height fraction it recovers.
The bounce-test formula is simple and reliable, but it rests on a few assumptions worth keeping in mind.
Drop from rest, ignore air and spin
This formula assumes the ball is released from rest and falls and rebounds vertically, so it ignores air resistance and any spin — both of which can nudge the measured bounce height. It also treats e as a single constant, though real materials lose slightly more energy at higher impact speeds. And by physics the bounce height can never exceed the drop height: a value above 1 means a measurement or unit error, so the calculator rejects it.