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Bounce Rebound Height Calculator

Enter the drop height, restitution and number of bounces to get the rebound height.

How it is calculated

Speed after one bounce equals the impact speed times the coefficient of restitution e. Because height depends on the square of speed, each bounce multiplies the height by e², so after n bounces hn = h0 × e2n. The tool shows that height and, separately, the height after the first bounce, h0 × e², both to four decimals. The height comes out in whatever unit you used for the drop height. The coefficient must be between 0 and 1 (1 is a perfectly elastic ball, 0 does not bounce at all). If the bounces field is empty, 1 is used. The model assumes the same e on every bounce, ignores air drag and a ball that spins or lands at an angle.

Worked example

A ball is dropped from 2 m and has a coefficient of restitution of 0.8. Each bounce multiplies the height by 0.8² = 0.64. After the first bounce it rises 2 × 0.64 = 1.2800. After three bounces the factor is 0.86 = 0.262144, so the height is 2 × 0.262144 = 0.5243 (metres, as entered). Over three bounces the ball has dropped to about a quarter of its starting height, even though 0.8 sounds like a mild loss.

Try this example →

Common mistakes

  • Applying the coefficient to the height instead of to the speed. The height factor is e², so a ball with e = 0.8 keeps 64% of its height per bounce, not 80%.
  • Entering the coefficient as a percentage (80). The value must be between 0 and 1, and anything above 1 returns no result.
  • Counting the original drop as a bounce. Bounce 1 is the first rise after the first impact; entering 0 returns the drop height.
  • Expecting a single coefficient for all surfaces. The value belongs to the ball and surface together, so the same ball bounces differently on concrete and on carpet.
  • Using the formula for the time the ball keeps bouncing. This tool gives heights only, not timing or total distance.

About this calculator

Each time a ball bounces, it loses energy to deformation and sound, so its rebound height shrinks by a fixed fraction set by the coefficient of restitution (e) — the ratio of rebound to impact speed. Height after n bounces follows hₙ = h₀ · e^(2n), since kinetic energy (and therefore height) scales with velocity squared while e applies once per bounce to velocity.

Physics students verify this exponential decay experimentally with a dropped ball and a stopwatch or phone camera, while sports-equipment engineers use the same relationship (and standard bounce tests) to certify that basketballs, tennis balls, and other regulation balls meet a governing body's specified restitution range.

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