How far does a bouncing ball really travel?

AA SLAI SLAA HLAI HL Accessible Also in: Sport, Pure maths

Research question

Do the rebound heights of a ball form a geometric sequence, and does the geometric-series prediction of total distance travelled match the measured total?

Adapt it: change the place, the data or the comparison until the question is yours.

Why it makes a good exploration

A ball that loses the same fraction of height each bounce travels a finite total distance even though it bounces “forever”. Testing that idea with real balls connects sequences, series and a physical constant (the coefficient of restitution).

The mathematics you'll need

Geometric sequences and the ratio r
Sum to infinity of a geometric series
Exponential model h = h₀rⁿ and logarithms to find r
Comparison of balls using the fitted ratio

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Film balls (tennis, basketball, ping-pong) dropped from a fixed height beside a metre rule; read each rebound height frame by frame in Tracker.

Tracker video analysis — Free tool to track an object frame by frame in a video and export its x–y coordinates.
phyphox (RWTH Aachen) — Free app that turns your phone's accelerometer, microphone, barometer and gyroscope into data loggers with CSV export.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Explain the geometric model and what r means physically.
Measure 6–10 bounces for three balls, three drops each.
Estimate r by averaging ratios and by a log-linear fit; compare.
Predict total distance with S∞ and compare with the distance to the last visible bounce.
Discuss why r is not constant (speed-dependent losses, spin, surface).

Pitfalls that cost marks

Reading heights by eye rather than frame by frame.
Summing to infinity without discussing that the ball actually stops.
Not repeating drops to estimate uncertainty.

Showing personal engagement

Test a prediction: does r change on grass, wood and concrete?
Use a ball from your own sport and link r to the rules (e.g., ball-testing standards).
Investigate whether r depends on drop height.

See Criterion C: personal engagement for what examiners look for.

Taking it further

Model total time in the air as well as distance — it is also a geometric series — and compare with the video's timestamps.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).