Experiment 4: The Bouncing Ball — Energy Loss

Drop a ball and it never bounces back to your hand at the release height. Each bounce is a tiny theft: some energy leaves as heat, sound and squash, and the bounce heights shrink in a precise geometric pattern. This experiment measures that pattern with a metre stick, links it to the coefficient of restitution e, and uses the sim to show why heights fall as h, e²h, e⁴h… — a decay you can predict bounce by bounce.

The physics behind the practical

When a ball dropped from height h₀ hits the floor, it rebounds with a slightly smaller speed. The coefficient of restitution is the ratio of those speeds: e = (speed just after impact) ÷ (speed just before impact). Because height is proportional to speed squared (from v² = 2gh), the rebound height is h₁ = e² × h₀, and the fraction of kinetic energy surviving the bounce is also e². A superball with e ≈ 0.9 keeps about 81% of its energy per bounce; a tennis ball with e ≈ 0.75–0.8 keeps about 60%; a lump of clay, e ≈ 0, keeps nothing and does not bounce at all.

Repeated bounces multiply: after n bounces the height is h₀ × (e²)ⁿ. That is why bounce heights collapse so quickly — a ball with e = 0.8 dropped from 1.00 m reaches 0.64 m, then 0.41 m, then 0.26 m, then 0.17 m. The energy is never destroyed; it is transferred to internal energy of the ball and floor (they warm microscopically), sound, and a little air resistance on the way up and down.

Variables and controls

In the virtual lab

Set bounciness (restitution e) to 0.80 and drop height to 2.0 m and press Drop ball. The readouts show the bounce count and the last apex height, and the panel tells you the key number: Energy kept per bounce = 64% — that is e² = 0.80². Watch the bar chart in the Data Visualization panel: apex heights fall geometrically, each bar 0.64× the previous. Before dropping, predict the 4th apex from 2.0 m: 2.0 × 0.64³ ≈ 0.52 m — then check. Compare e = 0.95 (superball: 90% energy kept, many bounces) with e = 0.50 (25% kept, dead in a few bounces). Log (e, drop height, bounces counted) in the Observation Log.

Hands-on: catching the bounce

Tape a metre stick vertically to a wall. One partner releases the ball from rest with its bottom at 1.00 m; the other crouches with eyes level with the expected bounce top and calls the height. Judge to the same point on the ball every time. Three trials per ball, then swap roles. Better still, film in slow motion against the stick and read the apex frame by frame — video removes most of the guesswork. Keep the drop zone clear; balls wander.

A worked investigation

Tennis ball, dropped from 1.00 m. First-bounce heights: 63, 65, 64 cm → mean 64 cm.

Step 1 — Find e. e = √(h₁/h₀) = √(64/100) = √0.64 = 0.80.

Step 2 — Energy kept. Fraction surviving each bounce = e² = 0.64, so 64% kept, 36% transferred to heat, sound and deformation per bounce.

Step 3 — Predict and test. Predicted second bounce = 64 × 0.64 = 41 cm; measured 40 cm — agreement within judgement error. Predicted third bounce = 41 × 0.64 ≈ 26 cm.

Step 4 — Compare balls (all dropped from 1.00 m):

The ranking is a materials story: springy rubber stores and returns elastic energy; clay deforms permanently and returns almost none.

Step 5 — Uncertainty. Judging the apex by eye is good to about ±2 cm, so h₁ = 64 ± 2 cm gives e between √0.62 = 0.79 and √0.66 = 0.81: e = 0.80 ± 0.02. Slow-motion video shrinks this to about ±0.005.

Conclusion and evaluation

The bounce chain obeys h × (e²)ⁿ with a single constant e per ball, and the measured second bounce (40 cm) matches the prediction (41 cm) within uncertainty. The main error source is apex judgement by eye; the main systematic risk is inconsistency — reading the top of the ball on one trial and the bottom on another shifts every height by a ball diameter. Dropping (not throwing) from rest matters too: any downward push adds energy that masquerades as extra bounciness.

Safety

Keep the drop area clear of feet and faces, use balls that cannot damage the room, and never substitute glass or hard heavy objects for balls.

Common mistakes

• Inconsistent reference point on the ball (top vs bottom) between the drop and the bounce readings.
• Parallax — reading the metre stick from above or below instead of eye level with the apex.
• Releasing with a push or spin, which adds energy the calculation never included.
• Averaging bounce 2 with bounce 1 — they are different quantities; the chain multiplies.
• Soft flooring for one ball only — the floor is part of the collision and must stay the same.

Exam Experience

Question 1 — calculation [4 marks]. A ball is dropped from 1.50 m and the first bounce reaches 0.96 m. Calculate the coefficient of restitution, the percentage of energy kept in the bounce, and the height of the third bounce.

Scoring guide:

• e = √(0.96 ÷ 1.50) = √0.64 = 0.80 [1]
• Energy kept = e² = 0.64 = 64% [1]
• Bounce heights multiply by 0.64: h₂ = 0.96 × 0.64 = 0.61 m [1]
• h₃ = 0.61 × 0.64 = 0.39 m [1]

Question 2 — evaluation and design [4 marks]. A student measures bounce height by watching the ball and estimating where it stops rising. Evaluate this method and describe an improved procedure for comparing the bounciness of two balls fairly.

Scoring guide:

• Eye judgement has parallax and reaction problems; the apex is fast and hard to read to better than ±2 cm [1]
• Improvement: film in slow motion against a fixed scale and read the apex frame by frame [1]
• Fairness controls: same drop height (bottom of ball), same floor, release from rest without push [1]
• Repeat each ball at least three times and average; use the same reference point on the ball throughout [1]

Multiple choice 1. A ball with e = 0.90 keeps what fraction of its kinetic energy at each bounce?

• A. 90%
• B. 81%
• C. 45%
• D. 10%

Answer: B. Energy kept = e² = 0.81. The speed ratio is 0.90, but energy scales with speed squared.

Multiple choice 2. The energy "lost" by a bouncing ball mainly becomes:

• A. Gravitational potential energy
• B. Heat and sound in the ball and floor
• C. Chemical energy in the ball
• D. It is destroyed, which is why the ball stops

Answer: B. Energy is conserved overall; it leaves the mechanical form as internal energy (heat) and sound, which is why the apex shrinks.

Going deeper: the bounce as an energy audit

Follow one tennis ball through its whole bouncing life. Dropped from 1.00 m with e = 0.80, its apex heights run 100, 64, 41, 26, 17, 11 cm… — each is the previous × 0.64 (e²). Since apex height is proportional to the energy remaining, the same chain is an energy ledger: 100%, 64%, 41%, 26%… Each bounce spends 36% of whatever energy was left, on sound (the *thock*), heat in the squashed ball and felt, and a whisper of vibration in the floor. After five bounces the ball keeps barely a tenth of its original energy — which is why dropped balls stop so decisively instead of bouncing forever in smaller and smaller hops.

The chain also sums beautifully. Total distance travelled before rest = drop + 2 × (sum of all later apexes) = 1.00 + 2 × (0.64 + 0.41 + 0.26 + …) = 1.00 + 2 × [0.64 ÷ (1 − 0.64)] = 1.00 + 2 × 1.78 = 4.56 m — an infinite geometric series with a finite answer, hiding in a bouncing ball. And the time to finish is finite too, which is why the ball's last hops blur into a buzz and then silence: each bounce is quicker than the last, by the same ratio logic. Whenever a process multiplies by a fixed factor below 1 — bouncing balls, radioactive decay, a cooling cup's temperature gap — this exponential fade is the shape to expect, and spotting it in your data is a genuine scientific skill.

One more angle: how long until the bouncing stops?

The geometric chain also answers the playground question "how many bounces?" Apex after n bounces is 100 × 0.64ⁿ cm; demand an apex under 1 cm and solve 0.64ⁿ < 0.01. Taking logarithms: n > ln(0.01)/ln(0.64) = (−4.61)/(−0.45) ≈ 10.3 — so the 11th bounce is the first whose apex stays under a centimetre, after which the ball is effectively rolling. Change the surface and you change the count: carpet might give e ≈ 0.5 (apex factor 0.25, so only 0.25ⁿ — under 1 cm by the 4th bounce), while a steel plate pushes e towards 0.9, and the ball is still visibly bouncing after twenty hops. The exponent is the personality of the surface pair.