🏀 Bouncing Ball
Drop any ball and watch energy disappear with each bounce. Different balls lose energy at different rates — why?
Drop any ball and watch energy disappear with each bounce. Different balls lose energy at different rates — why?
Drop a ball from a chosen height and watch how much energy survives each bounce. Every impact is governed by the coefficient of restitution — the ratio of rebound speed to impact speed — whilst gravity of 9.81 m/s² pulls the ball back down. A high-coefficient ball such as a superball keeps most of its energy and returns almost as high as before; a low-coefficient ball such as steel loses most of it to heat and sound within a few bounces.
Six ball presets — Basketball, Tennis, Rubber, Soccer, Steel and Superball — each with its own coefficient of restitution, from 0.55 to 0.92. Live readouts track current height, bounce number, coefficient and energy remaining, whilst dashed peak markers and a colour-coded energy bar show the bounce envelope decaying.
Pick a ball type from the presets, set the Drop height slider (0.5-3.5 m) and press Drop to release it. Use the Speed slider (1×-4×) to slow things down for a closer look or speed through many bounces quickly, and press Reset to return the ball to its starting height.
Real superballs, invented in 1965 from a cross-linked rubber called Zectron, have a coefficient near 0.9 and rebound to about 80% of their drop height. Steel deforms very little on impact yet dissipates most of its energy as heat and sound, giving it one of the lowest common coefficients, around 0.5.
The ratio of a ball's rebound speed to its impact speed, from 0 (no bounce) to 1 (perfectly elastic). Each preset here has a fixed value, shown as COR, from 0.55 to 0.92.
Every impact converts part of the ball's kinetic energy into heat, sound and deformation. The energy remaining after n bounces equals the coefficient squared, raised to the power n, shown live as a percentage.
Height after the nth bounce follows h(n) = h0 × COR^(2n), shown in the info box. Because COR is under 1, each bounce is smaller, continuing in theory forever whilst shrinking towards zero.
The coefficient reflects how efficiently a material stores and returns elastic energy rather than losing it to internal friction. Superball rubber returns energy efficiently (0.92); steel dissipates far more as heat (0.55).
It is a playback multiplier from 1× to 4× applied to the physics time-step, not a change to gravity or the coefficient of restitution. It simply lets you watch the same drop unfold faster.
When a ball bounces, it undergoes a collision with the ground that is characterised by the coefficient of restitution e — the ratio of the speed after impact to the speed before: e = v_after/v_before. A perfectly elastic ball (e = 1) would bounce forever, reaching the same height each time. A real ball (0 < e < 1) loses a fraction (1 − e²) of its kinetic energy per bounce as heat, sound, and internal deformation. The height after n bounces is h_n = e^(2n) · h₀, and the ball makes infinitely many bounces in a finite total time T = h₀^(1/2) · (1 + e) / ((1 − e) · (g/2)^(1/2)), a classic result from Zeno-like geometric series.
This simulation lets you drop a ball from a chosen height and adjust the coefficient of restitution, gravity, and ball radius. Watch the real-time bouncing animation, the height-versus-time graph showing the decaying envelope, and the energy bar chart decomposing total, kinetic, and potential energy.
What is the coefficient of restitution?
The coefficient of restitution e = v_rebound/v_impact is a dimensionless ratio between 0 and 1. For e = 1 (perfectly elastic), kinetic energy is conserved. For e = 0 (perfectly inelastic), the ball does not bounce at all. Typical values: superball ~0.9, tennis ball ~0.73, basketball ~0.76, golf ball ~0.68, dead rubber ~0.15. The coefficient depends on ball material, temperature (cold rubber has lower e), and impact speed (higher speeds give lower e for most materials).
How much energy is lost per bounce?
Each bounce retains a fraction e² of the kinetic energy from the previous bounce (since KE ∝ v² and v_after = e·v_before). The fractional energy loss per bounce is 1 − e². For a tennis ball (e = 0.73), energy loss = 1 − 0.73² = 1 − 0.533 ≈ 47% per bounce. For a superball (e = 0.9), loss = 1 − 0.81 = 19% per bounce. The bounce height is proportional to energy, so height after n bounces is h₀·e^(2n).
Why does the ball eventually stop bouncing?
Each bounce takes a time proportional to the bounce height, so successive bounce times form a geometric series: t_n = 2v_n/g = 2e^n·v_0/g. The total bouncing time is T = (2v_0/g)·Σe^n = (2v_0/g)·(1/(1−e)) — a finite sum even though there are infinitely many bounces. The ball mathematically completes infinitely many bounces in finite time, after which all kinetic energy has been dissipated. In practice, the final micro-bounces are too small to observe.
In an elastic collision, kinetic energy is conserved (e = 1): both momentum and kinetic energy are unchanged before and after. In an inelastic collision (0 < e < 1), kinetic energy is partly converted to internal energy (heat, sound, deformation), but momentum is still conserved. In a perfectly inelastic collision (e = 0), the objects stick together and share the maximum kinetic energy loss consistent with momentum conservation. Real collisions are always somewhere between elastic and perfectly inelastic.
Gravity sets the scale of bounce times and heights but does not change the coefficient of restitution or the energy fraction lost per bounce. On the Moon (g ≈ 1.62 m/s²), a ball dropped from 1 m would take √(2/1.62) ≈ 1.11 s to fall and reach a much longer total bounce time T = (1+e)/((1−e)·√(g/2h₀)). The total bounce number is unchanged, but each bounce is taller and longer. On Jupiter (g ≈ 24.8 m/s²), bounces are shorter and faster.
The sound of a bounce is caused by the rapid deformation and recovery of the ball and the ground during impact, which radiates pressure waves (sound). The pitch of the bounce sound increases as bounces get smaller — because the contact time shortens and the dominant frequency of the impact force (which scales as 1/contact_time) rises. The characteristic "thwack" of a basketball on a court contains frequencies from 200 Hz to over 2,000 Hz, peaking around 400–800 Hz.
The simplest measurement drops a ball from a known height h₀ and measures the rebound height h₁. Then e = √(h₁/h₀). For sports equipment, standardised tests specify drop height and surface: e.g., FIFA regulations require a football dropped from 2 m onto a steel plate to rebound to between 1.20 m and 1.45 m (e between 0.775 and 0.851). More precise measurement uses high-speed cameras (1,000+ fps) to capture pre- and post-impact velocities directly.
A superball, invented by Norman Stingley in 1965, is made from a highly cross-linked synthetic rubber (polybutadiene) called Zectron with a coefficient of restitution of about 0.9 — much higher than a normal rubber ball (~0.7). Its exceptional bounce comes from the very high elastic modulus of the vulcanised rubber, which stores and releases elastic energy with very low hysteresis losses. A superball can bounce to 90% of its drop height and, due to its high friction coefficient, can store and return spin in remarkable ways.
A single ball cannot bounce higher than its drop height without added energy input (violating energy conservation). However, with two stacked balls, a remarkable phenomenon occurs: if a small ball (m) sits atop a large ball (M) and both are dropped together, upon impact the large ball bounces up and imparts extra momentum to the small ball. For a superball on top of a basketball, the small ball can bounce to about 9 times the drop height. This is the "stacked ball drop" experiment, which conserves total energy but concentrates it in the smaller ball.
A spinning ball stores angular momentum, which is partially converted to or from translational momentum upon collision, depending on the friction between ball and surface. A ball with topspin bounces more forward (lower rebound angle) because the spin adds to the forward velocity component during contact. A ball with backspin can bounce nearly vertically or even backwards. A ball with sidespin curves after the bounce. These effects are crucial in cricket, tennis, snooker, and table tennis, where spin control is a core skill.