Coefficient of Restitution Simulator — Decay of a Bouncing Ball

Q: Why is the total bounce time T finite when the number of bounces is infinite?

This is the classical-mechanics version of Zeno's paradox. Each bounce duration t_n = (2 v_0 e^n)/g is a geometric series with ratio e, and its infinite sum T = (2 v_0/g)(1+e)/(1-e) converges to a finite value when e < 1. With h = 2 m, e = 0.8, g = 9.81 m/s^2 the total time is about 11.5 s, after which all motion has ceased. The number of bounces is mathematically infinite, but in practice the ball stops once the bounce height reaches the molecular vibration scale (10^-10 m). Push e toward 0.99 in this tool and watch T grow rapidly; push e toward 0 and motion ends almost instantly.

Q: Why does h_n = h e^(2n) carry the exponent 2n instead of n?

Each collision multiplies the speed by e, but height is proportional to the square of the speed (h = v^2/(2g)). One bounce therefore reduces height by a factor of e^2, and after n bounces the factor is (e^2)^n = e^(2n). For e = 0.8 the per-bounce height factor is 0.64, so after five bounces 0.64^5 = 0.107 of the original height remains, taking 2 m down to 0.215 m. The same factor of e^2 follows from energy conservation (each impact retains a fraction e^2 of the kinetic energy). Vary e and n in this tool to watch the geometric decay collapse the height.

Q: Which physical effects does this tool ignore?

The simulator uses Newton's definition of e and ignores air drag, ball spin, finite contact time, temperature dependence, contact friction, and internal vibration modes. In practice a golf ball striking grass drops to e about 0.4, hard balls heat up during a single impact and the e value drifts with time, air resistance reduces the impact velocity from a 2 m drop by about 0.3%, and many rubber compounds lose restitution at low temperature. The tool is more than enough for understanding the basic geometry and for back-of-the-envelope estimates; for accurate measurements consult standard tests such as ASTM F2117 or ISO 8124.

Parameters

Drop height h

Coefficient e

—

Gravity g

m/s²

Bounce count n

bounces

Defaults are h = 2.0 m, e = 0.80 (an elastic rubber ball), g = 9.81 m/s² (Earth) and n = 5. For e < 1 the total time T and total path D converge to finite values as the sums of geometric series.

Results

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Impact velocity v₀

—

Bounce height after n

—

Total bounce time T

—

Total vertical path D

Bounce trajectory (n bounces on the floor)

Brown band = floor / blue parabolas = each bounce (left to right, n bounces) / yellow dots = apex of each bounce / white labels = apex height [m] / x-axis is purely a layout offset (real horizontal motion is zero)

Geometric decay of bounce height

X = bounce count n (0 to 30) / Y = bounce height h_n = h e^(2n) [m] / blue curve = current e / faint curves = e = 0.5, 0.7, 0.9 reference / yellow dot = current n marker

Theory & Key Formulas

Newton defined the coefficient of restitution $e$ as the ratio of relative speeds before and after a collision; it captures how much kinetic energy a single impact dissipates.

Impact velocity (free fall):

$$v_0 = \sqrt{2gh}$$

Apex height after the $n$-th bounce and the impact velocity at bounce $n$:

$$h_n = h\,e^{2n},\qquad v_n = v_0\,e^{n}$$

Geometric-series totals for total bounce time $T$ and total vertical path $D$ (convergent for $e<1$):

$$T = \frac{2v_0}{g}\cdot\frac{1+e}{1-e},\qquad D = h\cdot\frac{1+e^2}{1-e^2}$$

Here $h$ is the initial drop height [m], $g$ the gravitational acceleration [m/s²], and $e$ the coefficient of restitution ($0 \le e \le 1$). $e=1$ is perfectly elastic, $e=0$ is perfectly inelastic. A typical rubber ball gives $e\approx 0.8$ and a golf ball $\approx 0.78$.

What the Coefficient of Restitution Simulator does

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I learned about the "coefficient of restitution e" in high school, but what can you actually compute when you drop a real ball?

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A great place to start. Newton defined e in the 17th century as the ratio of the relative speed after a collision to the relative speed before. For a ball dropped onto a floor you immediately get the impact velocity v_0 = sqrt(2 g h), the rebound speed v_1 = e v_0, and the rebound height h_1 = e^2 h. Try the defaults h = 2 m, e = 0.80, g = 9.81 and n = 5 in this tool. The results panel should read v_0 = 6.26 m/s, height after five bounces 0.215 m, total bounce time 11.49 s and total path 9.11 m.

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Wait — the ball bounces an infinite number of times, but the total time is only 11.49 s?

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That is the lovely part. Each bounce duration t_n = (2 v_0 e^n)/g forms a geometric series with ratio e, and its infinite sum is T = (2 v_0/g)(1+e)/(1-e), which is finite. With e = 0.80 the ratio (1+0.8)/(1-0.8) is 9, so T is nine times the first round-trip 1.28 s, about 11.5 s. Infinitely many bounces in finite time — Zeno's paradox in classical mechanics. Push e to 0.99 and watch T explode.

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What e values do real sports balls have? I'd like a feel for tennis, golf and baseball.

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Standard test values are about 0.78 for a golf ball on a hard floor, 0.73-0.76 for a tennis ball on concrete, 0.55 for a baseball on a wooden bat, 0.76 for a basketball on an official court and 0.92 for a "super ball" rubber sphere. A leather American football is surprisingly low at about 0.4. Switch e between 0.55 (baseball) and 0.92 (super ball) in this tool: after five bounces the latter retains nearly 1 m of height while the former drops below 5 cm. Sports equipment standards strictly bound the allowable e.

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In the right-hand chart, the e = 0.5 and e = 0.9 curves look totally different. What does that say about energy conservation?

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Important point. One impact multiplies the speed by e, so the kinetic energy retained is e^2 — the fraction (1 - e^2) becomes heat, sound and deformation. With e = 0.5 you lose 75% of the energy in a single bounce; with e = 0.9 only 19%. The exponent 2n in h_n = h e^(2n) follows from this "speed -> energy -> height" double-square chain. Sweep n in the right chart to compare the gentle decay at e = 0.9 with the steep collapse at e = 0.5 — geometric series at work.

Frequently Asked Questions

The coefficient of restitution e is a dimensionless number defined by Newton as the ratio of the relative speed after a collision to the relative speed before. e = 1 is a perfectly elastic collision (energy is conserved), e = 0 is perfectly inelastic (the bodies stick), and real materials sit between, with 0 < e < 1. A golf ball on a hard floor gives e about 0.78, a tennis ball about 0.75 and a baseball on a wooden bat about 0.55. Slide e from 0.0 to 0.99 in this tool to see how dramatically the bounce-height decay changes.

This is the classical-mechanics version of Zeno's paradox. Each bounce duration t_n = (2 v_0 e^n)/g is a geometric series with ratio e, and its infinite sum T = (2 v_0/g)(1+e)/(1-e) converges to a finite value when e < 1. With h = 2 m, e = 0.8, g = 9.81 m/s^2 the total time is about 11.5 s, after which all motion has ceased. The number of bounces is mathematically infinite, but in practice the ball stops once the bounce height reaches the molecular vibration scale (10^-10 m). Push e toward 0.99 in this tool and watch T grow rapidly; push e toward 0 and motion ends almost instantly.

Each collision multiplies the speed by e, but height is proportional to the square of the speed (h = v^2/(2g)). One bounce therefore reduces height by a factor of e^2, and after n bounces the factor is (e^2)^n = e^(2n). For e = 0.8 the per-bounce height factor is 0.64, so after five bounces 0.64^5 = 0.107 of the original height remains, taking 2 m down to 0.215 m. The same factor of e^2 follows from energy conservation (each impact retains a fraction e^2 of the kinetic energy). Vary e and n in this tool to watch the geometric decay collapse the height.

The simulator uses Newton's definition of e and ignores air drag, ball spin, finite contact time, temperature dependence, contact friction, and internal vibration modes. In practice a golf ball striking grass drops to e about 0.4, hard balls heat up during a single impact and the e value drifts with time, air resistance reduces the impact velocity from a 2 m drop by about 0.3%, and many rubber compounds lose restitution at low temperature. The tool is more than enough for understanding the basic geometry and for back-of-the-envelope estimates; for accurate measurements consult standard tests such as ASTM F2117 or ISO 8124.

Real-World Applications

Sports equipment regulations: The ITF requires a tennis ball dropped from 254 cm onto concrete to rebound to between 135 and 147 cm — equivalent to e about 0.73-0.76. MLB requires an official baseball dropped from 8 ft (2.44 m) onto a marble slab to retain 0.514-0.578 of its speed (e of 0.514-0.578). The USGA caps the COR (Coefficient of Restitution) of golf driver/ball combinations at 0.83; anything higher is banned in tournament play. Switch e to 0.55, 0.78 and 0.92 in this tool to feel the difference between sport categories.

Automotive crash safety: Front-end collision design uses e to size the crushable zone of the bumper and frame. A nearly inelastic e about 0 absorbs the impact in vehicle deformation and minimises occupant deceleration; e closer to 1 would launch the occupant backwards and dramatically raise neck-injury risk. NCAP frontal tests target e about 0.1 to 0.2, dissipating 96 to 99% of the kinetic energy (1 - e^2) in body deformation and tearing.

Materials and granular testing: Pellet-drop tests measure e by recording the rebound height at a known drop height, characterising surface roughness, particle size and humidity. Dry silica gel pellets give e about 0.65; wet sand grains drop to e about 0.05. The difference governs wall-wear rates in silos and pneumatic-conveying lines. The Leeb hardness test sends a steel ball at the surface and infers material hardness from the ratio of rebound to impact speed (essentially an e measurement).

Robotic gripper control: Industrial robot arms picking up fragile items must account for the e of the contact pair. Glass parts (e about 0.95) bounce strongly and require suction pads or low-loss-factor rubber grippers (e about 0.2). Collaborative robot certification (ISO/TS 15066) limits the momentum exchange on contact with a human, and the allowable approach speed depends directly on the combined e of the robot surface and the body tissue.

Common Misconceptions and Pitfalls

The most common misconception is that "e depends on the mass or the drop height of the ball". Newton's definition e = v_after/v_before is purely a velocity ratio and does not contain mass; for a fixed pair of materials and contact conditions e is a property of the pair, so a golf ball on a marble slab gives the same e from a 1 m drop as from a 5 m drop. Only at very high impact speeds (>50 m/s) does plastic deformation reduce e, which is why standards like ASTM specify the test velocity. Sweep h from 0.1 to 10 m in this tool and confirm that the height ratio h_n/h does not change — h enters only via v_0 and the totals.

Next is the belief that "with e = 1 a real ball would bounce forever". The slider in this tool stops at 0.99, but even at e = 1 a real ball loses energy through internal friction, air drag and vibration energy transmitted into the floor. Perfectly elastic collisions are an idealisation, approached only in special systems such as superconducting magnetically-levitated spheres. Setting e = 0.99 in this tool yields a very long T (hundreds of seconds), which is what Newton's model predicts but which always falls short of reality due to the additional dissipation channels.

Finally, people often assume that "the same formula works for oblique impacts". For an oblique collision e applies only to the normal component of the velocity; the tangential component decays separately through the friction coefficient mu and the angular momentum. Baseball launch angles and ping-pong spin behaviour both require this two-component decomposition. This tool is restricted to the idealised one-dimensional case of "vertical free-fall plus normal impact"; real sport calculations use six-degree-of-freedom models including bounce angle, spin and friction (Hertzian contact theory and friends).

Coefficient of Restitution Simulator — Decay of a Bouncing Ball

What the Coefficient of Restitution Simulator does

Frequently Asked Questions

Real-World Applications

Common Misconceptions and Pitfalls

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