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The coefficient of restitution

Drop a ball and it never bounces back to quite the height it fell from — some kinetic energy is always lost in the impact, converted to heat and sound as the ball briefly deforms and springs back. The coefficient of restitution (COR), usually written e, quantifies exactly how much speed survives the bounce: it's the ratio of the ball's speed immediately after impact to its speed immediately before, for a bounce off a rigid, immovable surface.

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e = v_after / v_before                  // 0 = perfectly inelastic, 1 = perfectly elastic
h_n = h_0 * e^(2n)                       // height after n bounces
v_after = e * sqrt(2 * g * h_0)          // speed leaving the nth bounce

Because height depends on the square of speed (from energy conservation, mgh = 1/2 mv²), and speed itself loses a factor of e on every bounce, the rebound height shrinks by a factor of e² each time — a ball with e = 0.9 does not lose 10% of its height per bounce, it loses about 19% (1 − 0.9²), and that compounds fast: after ten bounces a 0.9-COR ball has lost over 85% of its original height.

Where the energy goes

During the fraction of a second a ball is in contact with the ground, it deforms — compressing like a spring — and then springs back, and the fraction of stored elastic energy that is actually returned rather than lost depends entirely on the internal material properties of the ball and the surface. A perfectly elastic collision (e = 1) would return every joule; real balls always fall short because deforming rubber, foam or leather is not perfectly reversible — some energy is lost to internal friction between polymer chains sliding past each other (viscoelastic damping), some to sound (the audible "thud" or "ping" of a bounce is literally acoustic energy leaving the system), and some to heat generated by that same internal friction.

Comparing real balls

A superball — made of a highly resilient synthetic rubber called polybutadiene — is deliberately engineered for an unusually high COR, often above 0.9, close to the elastic limit of any practical rubber compound. A basketball sits around 0.75–0.8, tuned by rule to bounce predictably for dribbling. A tennis ball's felt-covered rubber shell gives roughly 0.7–0.75. A soccer ball, larger and built more for controlled rolling and passing than for bouncing, comes in lower, around 0.6–0.7. A dense, essentially rigid steel ball bouncing off a hard steel or stone surface can be surprisingly elastic — well above 0.9 — because very little of the impact energy goes into permanent deformation, whereas the same steel ball dropped onto sand or grass loses almost all its energy into deforming the ground instead, giving an effective COR close to zero even though the ball itself barely deformed.

superball (polybutadiene)   e ~ 0.85-0.95
basketball                  e ~ 0.75-0.80
tennis ball                 e ~ 0.70-0.75
soccer ball                 e ~ 0.60-0.70
steel-on-steel               e ~ 0.90-0.95
steel-on-sand                e ~ 0.10-0.30  (energy lost deforming the ground, not the ball)

It isn't just the ball — the surface matters too

COR is really a property of the pair of materials in contact, not the ball alone — the same ball dropped on concrete, wood, grass and sand will show four different rebound heights, because energy can be absorbed by either surface deforming. This is why sports governing bodies specify not just ball construction but also standard test surfaces when setting official bounce regulations, and why a basketball that bounces predictably on a hardwood court behaves quite differently dropped onto grass or dirt.

Why the ball never quite stops falling silent

Because each bounce height shrinks geometrically (by e² each time) but never mathematically reaches exactly zero, an idealised bouncing ball technically bounces infinitely many times in a finite total time — the classic bouncing ball paradox, resolved by noting that the time between bounces also shrinks geometrically, so the infinite sequence of bounces sums to a finite duration, after which the ball is, for all practical purposes, at rest on the ground. Real balls stop audibly bouncing well before that limit simply because tiny bounces below a few millimetres are swallowed by air resistance and surface imperfections.

Frequently asked questions

What does a coefficient of restitution of 0.8 actually mean?

It means the ball leaves the ground at 80% of the speed it arrived with. Because rebound height depends on speed squared, an 0.8 COR ball only reaches 64% of its previous height on each bounce, not 80% — the energy loss compounds faster than the speed loss.

Why does a steel ball bounce differently on sand than on concrete?

The coefficient of restitution depends on both materials in contact, not just the ball. On rigid concrete, almost all the deformation and energy loss happens briefly in the ball itself and most speed is recovered. On sand, much of the impact energy goes into permanently deforming the ground instead of springing back, so far less speed survives even though the ball itself is unchanged.

Does a bouncing ball really bounce infinitely many times?

In the idealised physics model, yes — each bounce height and the time between bounces both shrink by a constant factor, forming a geometric series that sums to a finite total time despite having infinitely many terms. In reality, air resistance and imperfect elasticity make the bounces imperceptibly small well before that mathematical limit.

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The Bouncing Ball: What the Coefficient of Restitution Really Measures