Mathematics 11–12 · Year 12

Bouncing ball: bounce heights as a geometric sequence with a limiting sum

Sequences and series: Geometric sequences and series (Mathematics Advanced, Year 12); Algebraic relationships: Exponential relationships (Mathematics Standard 2, Year 12)

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The idea

Each bounce returns a fixed fraction of the previous height, so the heights form a geometric sequence and the total distance travelled is a finite limiting sum.

What you need

A basketball or bouncy ball on a hard floor
A phone running the phyphox (In)elastic collision experiment, which times successive bounces by their sound
A 2 m tape fixed to the wall and a second phone filming as a check

How to do it

Drop the ball from rest at 1.00 m with the phone listening; phyphox records the time between successive bounces.
Convert each interval to a bounce height with h = g t^2 / 8 (the ball rises for half the interval and falls for the other half).
Tabulate h0, h1, h2, ... and compute the ratio r = h(n+1) / h(n) for each pair; check whether it stays constant.
Write h(n) = h0 r^n and the total distance D = h0 + 2 h0 r / (1 - r) from the limiting sum.
Compute the total bouncing time from sqrt(2 h0 / g) and the geometric series of the intervals. phyphox times bounces by their sound and the release is silent, so subtract the first fall sqrt(2 h0 / g) and compare the rest with the time from the first bounce until the sound stops.

What you should see

Bounce intervals of 0.40, 0.50 and 0.60 s correspond to heights of 0.196, 0.306 and 0.441 m. For h0 = 1.00 m and r = 0.64 (coefficient of restitution 0.8) the heights are 1.00, 0.64, 0.41, 0.26 and 0.17 m, the total distance is 4.56 m and the total time from release is 4.07 s, of which the silent first fall takes 0.452 s, so the bouncing heard from the first impact lasts 3.61 s. The learner's r is a property of their ball and floor, and the same r reappears as the gradient in the drop-height against bounce-height entry. The learner knows it worked when the ratios agree to within 0.05 and the predicted 3.61 s from the first impact matches the recorded bouncing time within about 0.3 s.

What changes

What you change: bounce number
What you measure: bounce height and interval
What you keep the same: same ball and floor
drop from rest
same starting height

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

A ball that bounces forever travels an infinite distance; the geometric series converges.
The height ratio is the coefficient of restitution; it is its square.

Safety card

Low riskLearners carry it out

Hazards

ball rebounding towards faces

Controls

drop the ball, do not throw it
clear the space around the drop point

Note

No chemicals and no 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 Advanced 11–12 Syllabus (2024), Year 12 focus area Sequences and series; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-12-03
Mathematics Standard 11–12 Syllabus (2024), Year 12 Standard 2 focus area Algebraic relationships; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MST-12-S2-01
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

All Concept Studio activities