Mathematics 7–10 · Year 10
Bouncing ball: rebound heights as an exponential decay
Number and algebra
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The idea
Each bounce returns a fixed fraction of the height it fell from, so successive rebound heights form a geometric sequence and the graph of height against bounce number is an exponential decay rather than a straight line.
What you need
- A ball that bounces cleanly (tennis ball, super ball or basketball) per group
- A 2 m tape measure fixed vertically to a wall with a bold mark every 5 cm (a metre rule is too short for the 125 cm drop)
- A phone recording video at 30 frames per second or more, on a tripod or books, level with the lower part of the wall
- Hard flat floor
How to do it
- Drop the ball from 125 cm (bottom of the ball level with the mark), with no throw, while a partner records video.
- Step through the video to read the peak height of the first five rebounds against the wall scale, to the nearest centimetre.
- Repeat the drop three times and average each rebound height.
- Compute the ratio of each rebound height to the previous height; check whether the ratios are close to constant, and call the average r.
- Fit h_n = 125 × r^n with r averaged from the first three ratios, plot the measured points and the model on the same axes, predict the height of the fifth rebound, and compare it with the measured fifth rebound.
- Repeat with a different ball or a different floor surface and compare r.
What you should see
The ratios come out close to one constant for a given ball and floor, so the heights fall on a curve that flattens towards zero, not on a straight line. If the ratio is 0.6, as in the NRICH problem, the heights after the first, second and third bounces are 75, 45 and 27 cm and the fifth rebound is predicted at 9.7 cm; a ball with a ratio of 0.8 gives 100, 80 and 64 cm and a fifth rebound of 41.0 cm. The constant ratio r is the square of the coefficient of restitution e (r = 0.6 gives e = 0.775), because the rebound speed is e times the impact speed and height is proportional to speed squared. The learner knows it worked when the fifth-rebound prediction made from the first three ratios agrees with the measured fifth rebound within the reading error of the video against the wall scale.
What changes
- What you change
- bounce number (and ball or surface in the extension)
- What you measure
- rebound height
- What you keep the same
- same drop height
- same ball and floor within a trial
- ball released, not thrown
- camera position fixed
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- The ball loses the same number of centimetres each bounce.
- An exponential decay reaches zero after a fixed number of steps.
- A heavier ball bounces lower because of its weight alone.
Safety card
Hazards
- ball striking people or windows
Controls
- drop away from windows and faces
- one drop at a time
- the video frames the wall scale only, with no learner in shot, and is deleted from the phone once the rebound heights have been read
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K–10 Syllabus (2022), Stage 5 Non-linear relationships A; page read 2026-09-22MA5-NLI-C-01
- Mathematics K–10 Syllabus (2022), Stage 5 Non-linear relationships B; page read 2026-09-22MA5-NLI-C-02
- Australian Curriculum v9AC9M10A03AC9M10A04
Sources
The pages the author read to write this activity.