The Coefficient of Restitution: What Makes a Ball Bouncy

The defining equation: a ratio of separation to approach

The coefficient of restitution compares how fast two bodies separate after a collision to how fast they approached before it. For a one-dimensional collision between masses m₁ and m₂ with pre-collision velocities v₁, v₂ and post-collision velocities v₁′, v₂′:

• e = (v₂′ − v₁′) / (v₁ − v₂) — the magnitude of the relative velocity of separation divided by the relative velocity of approach.
• By construction e is dimensionless and (for normal collisions) lies in the interval 0 ≤ e ≤ 1.
• e = 1 is a perfectly elastic collision: relative speed is unchanged, kinetic energy is conserved.
• e = 0 is a perfectly inelastic (plastic) collision: the bodies leave with zero relative velocity, i.e. stuck together at the common velocity fixed by momentum conservation.

This is Newton's experimental law of impact, stated in the Principia after he measured colliding pendulums of steel, cork, and wool. Crucially, e is an empirical property of the two surfaces and the impact speed — it is not derivable from Newton's laws of motion alone, because those laws conserve momentum but say nothing about how much kinetic energy the deformation stage dissipates. Momentum conservation (m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′) plus the restitution equation gives exactly two equations for the two unknown final velocities, so specifying e closes the problem completely.

Solving a collision with e: the two-equation system

Combine momentum conservation with the restitution relation and you can solve any 1-D two-body collision explicitly. The results are worth memorizing:

• v₁′ = [(m₁ − e·m₂)v₁ + (1 + e)m₂v₂] / (m₁ + m₂)
• v₂′ = [(m₂ − e·m₁)v₂ + (1 + e)m₁v₁] / (m₁ + m₂)

Set e = 1 and these collapse to the familiar elastic-collision formulas; set e = 0 and both give the same common velocity v′ = (m₁v₁ + m₂v₂)/(m₁ + m₂), the completely inelastic result. The kinetic energy lost in the collision has a beautifully compact form in terms of the reduced mass μ = m₁m₂/(m₁ + m₂) and the approach speed u = v₁ − v₂:

• ΔKE = ½ μ u² (1 − e²)

Every symbol earns its place: the energy that can possibly be lost is the kinetic energy in the center-of-mass frame, ½μu²; the factor (1 − e²) is the fraction of it that is dissipated. When e = 1 no energy is lost; when e = 0 the entire center-of-mass kinetic energy ½μu² becomes heat, sound, and permanent deformation. Notice the energy loss scales with u² — hit twice as fast and you dissipate four times the energy — which is exactly why high-speed impacts feel so much more violent.

The bounce test: why h′/h equals e²

The everyday way to measure e is to drop a ball and watch how high it returns. Take a ball hitting a massive floor (m₂ → ∞, v₂ = 0), which is the limiting case of the formulas above. The floor doesn't move, so e = v′/v: the rebound speed is simply e times the impact speed. Now connect speed to height with energy conservation during the (essentially loss-free) flight:

• Impact speed from a drop height h: v = √(2gh).
• Rebound speed: v′ = e·v.
• Rebound height from that speed: h′ = v′²/(2g) = e²v²/(2g) = e²·h.

So e = √(h′/h) — measure two heights and you have the coefficient. A ball with e = 0.90 dropped from 1.00 m returns to 0.81 m; on the next bounce it reaches 0.81 × 0.81 = 0.656 m, then 0.531 m, and so on. Each bounce height is e² times the previous, a geometric sequence. The total distance the ball travels is finite, h(1 + e²)/(1 − e²), and — a classic puzzle — the ball completes infinitely many ever-shorter bounces in a finite total time, 2√(2h/g)·[1/(1−e) − ½], because the bounce durations also form a converging geometric series.

Where the energy goes: the physics of a real bounce

A bounce is not a single event but a two-stage squeeze. During compression the ball flattens against the surface, converting kinetic energy into elastic strain energy (and some heat). During restitution that stored elastic energy pushes the ball back off — but not all of it comes back. The lost fraction (1 − e²) is dissipated by several mechanisms:

• Viscoelastic hysteresis — the dominant loss for polymers like rubber. The stress–strain curve traces a loop; its enclosed area is energy turned to heat inside the material. A cold squash ball has huge hysteresis (e ≈ 0.2); warmed to 45 °C in play its molecular chains recover faster and e climbs.
• Plastic deformation — permanent denting, the whole story for putty or a lead ball (e ≈ 0). Once atoms slide past one another irreversibly, that energy is gone.
• Vibration and sound — the audible "thock" and internal ringing carry off a few percent; a hollow ball rings more and bounces slightly less.
• Air and squeeze-film effects — negligible for a rigid ball but real for inflatables, where the trapped gas pressure sets much of the stiffness.

Because these mechanisms are rate- and temperature-dependent, e is not a fixed constant. It typically falls as impact speed rises (more of the material is driven past its elastic limit) — a golf ball measures e ≈ 0.86 at gentle speeds but the effective value drops toward 0.78 at a 45 m/s driver impact, which is why the USGA regulates the club-face "characteristic time" rather than trusting a single e.

Controlling variables: what a designer actually tunes

If you want a bouncier ball — or a deader one — these are the knobs, all traceable to the (1 − e²) energy-loss term:

• Material and cross-link density. Polybutadiene (superball rubber) is highly resilient with low hysteresis, giving e ≈ 0.85–0.90. Adding fillers or plasticizers raises internal friction and lowers e.
• Temperature. Near a polymer's glass transition the loss modulus peaks and e plummets; well above it, chains snap back elastically. This is why tennis balls are stored pressurized and why a squash match "warms up" the ball.
• Impact speed. Higher u pushes more material past yield, so e generally decreases with velocity — the reason automotive crash tests, which need reliable energy absorption, run at fixed standardized speeds.
• Surface stiffness on both sides. e is a property of the pair. The same superball gives e ≈ 0.90 on granite but far less on a plush carpet, which absorbs energy in its own fibers. A steel ball on steel keeps e ≈ 0.92 because neither surface deforms plastically.
• Inflation pressure. For balls whose restoring force is gas pressure, e rises with pressure — a flat basketball (e ≈ 0.5) versus a properly inflated one at ~55 kPa gauge (e ≈ 0.76). The NBA specifies the rebound height directly: dropped from 1.80 m, the ball must bounce to 1.20–1.40 m, i.e. e between about 0.82 and 0.88 on a hard floor.

From baseball bats to Mars landers: e in the wild

The coefficient of restitution is a design parameter across engineering and sport:

• Baseball. The bat–ball collision has e ≈ 0.50 (MLB regulates the ball's "COR" at 0.51–0.58 against a wall at 26.8 m/s). Because e enters the batted-ball speed formula, a change from 0.50 to 0.55 adds several km/h of exit velocity — the difference between a warning-track out and a home run. Aluminum bats were restricted partly because their trampoline effect raises the effective e.
• Golf. The USGA/R&A limit the driver "spring-like effect" to keep e (technically the COR of the club face) at or below 0.83; thin titanium faces flex and give energy back, otherwise pushing e higher and lengthening drives beyond course design.
• Spacecraft landing. Airbag landers (Mars Pathfinder, 1997) and crumple structures are engineered for low e — you want a near-plastic collision so the vehicle does not bounce off the surface. Conversely, particle dampers and vibration mounts exploit near-zero-e granular collisions to bleed off energy.
• Granular physics. Sand, gravel, and planetary rings are enormous collections of colliding grains with e ≈ 0.3–0.9. Their collective behavior — clustering, "inelastic collapse" when e is low — is governed by the same restitution coefficient scaled up to Avogadro-like numbers of impacts.
• The Newton's cradle owes its clean click-clack to steel's e ≈ 0.92 and the near-perfect momentum-and-energy transfer of elastic collisions between equal masses.

Subtleties, limits, and common misconceptions

The tidy 0-to-1 picture hides several genuinely tricky points:

• e is not a material constant. The single biggest misconception. It depends on impact speed, temperature, geometry, and the other surface. Quoting "the coefficient of restitution of rubber" without specifying the counter-surface and speed is meaningless.
• Oblique impacts split into two coefficients. For a ball hitting at an angle, the normal component obeys the usual e while the tangential component is governed by friction and spin. A ball can even "gain" tangential speed by grip, and a well-cut spin makes it kick sideways — behavior the single-number model ignores.
• e can exceed 1 in special cases. If a collision releases stored energy — an internal spring, a chemical or explosive interaction, or a hollow ball whose walls buckle and snap through — the separation speed can beat the approach speed. These "superelastic" collisions are rare but real.
• Rotation matters. A dropped ball with backspin, or one that lands on a compliant, tilted surface, converts translational and rotational energy back and forth; naive height measurement then mis-reports e.
• The instantaneous-collision assumption. The restitution model treats impact as instantaneous. Real contacts last hundreds of microseconds to milliseconds (a golf ball is on the club face for ~450 μs, a footfall lasts tens of ms), and during that window the contact force can peak at thousands of newtons — which is why we treat the collision as an impulse and only track before/after states.

Coefficient of restitution and energy retention across materials and collisions

Collision	e (approx.)	Rebound height h′/h = e²	KE lost per bounce
Superball on steel	0.90	0.81	19%
Steel ball on steel	0.92	0.85	15%
Basketball on wood	0.76	0.58	42%
Tennis ball on court	0.73	0.53	47%
Baseball on bat	0.50	0.25	75%
Putty / clay (plastic)	≈ 0.05	≈ 0.003	≈ 99.7%