Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

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?

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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

Modelling the data, step by step

This idea usually needs an exponential model. See it worked step by step, with a criterion tip at every step: Exponential y = a e^(kx) + c and y = a·bˣ.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for 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.

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.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Geometric sequences and the ratio r; Sum to infinity of a geometric series
AA HL	Fits, but add an HL technique	Geometric sequences and the ratio r; Sum to infinity of a geometric series
AI SL	Good fit	Geometric sequences and the ratio r; Sum to infinity of a geometric series
AI HL	Fits, but add an HL technique	Geometric sequences and the ratio r; Sum to infinity of a geometric series

Level: Accessible. A good first extended piece of maths, with room to go deeper. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: reading heights by eye rather than frame by frame; and summing to infinity without discussing that the ball actually stops. Say how each one affects your answer.

Use of mathematics (E)

SL: 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, each explained in your own words and carried out correctly.

How modelling explorations reach the top bands in general: Modelling with functions IA ideas. Guides: A · B · C · D · E · self-grade sheet.

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.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL modelling exploration adds.

See a complete IA, marked

How long does a skydiver take to reach terminal velocity? (AA HL) is a complete modelling exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL, AI SL, AA HL and AI HL. We rate it Accessible: a good first extended piece of maths, with room to go deeper.

What mathematics does this IA use?

Geometric sequences and the ratio r, sum to infinity of a geometric series, exponential model h = h₀rⁿ and logarithms to find r and comparison of balls using the fitted ratio.

Where can I get data for this IA?

Film balls (tennis, basketball, ping-pong) dropped from a fixed height beside a metre rule; read each rebound height frame by frame in Tracker. Checked sources and tools: Tracker video analysis and phyphox (RWTH Aachen).

Turn this idea into your IA

The planner opens with “How far does a bouncing ball really travel?” filled in: the question, the maths and the data sources.

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