IB Maths IA idea · Science, nature and society

Bouncing ball: how far does it travel in total?

A ball bounces for ever in the maths and stops in your hands. You're going to measure the gap between those two stories.

Science, nature and society
SL ideal
HL ideal
Maths AA Maths AI Your own data
Syllabus: AA 1.3, AA 1.5, AA 1.8, AA 2.9, AA 1.15

By Photon Academy · Updated 13 November 2026

These ideas are starting points, not an IA to copy. Never copy an idea blindly: make the question, method and analysis your own, and check your plan with your teacher.

The big idea

Each bounce reaches a fixed fraction r of the one before. That makes the heights a geometric sequence and the total distance a geometric series, which converges when r is below 1. Measure the peak heights, fit $h_{n}$ = $h_{0}$ $r^{n}$ , and test whether r really is constant. Then predict the total distance and compare it with a measured one. It is a gentle topic, but the series needs justifying, not quoting: why does it converge, and does the model hold on every bounce? At HL, prove the formula by induction and take on a harder series such as total time or energy.

The maths you will use

Geometric sequences. Model the peak heights as $h_n = h_0 r^n$ and test the constant ratio from $\frac{h_{n+1}}{h_n}$ for every bounce.
Infinite geometric series. Sum the distance $h_0 + 2h_0(r + r^2 + \cdots) = h_0\frac{1+r}{1-r}$ (4.0 times $h_0$ for $r = 0.6$ ) and say why it converges.
Logarithms and linearising. Take $\ln h_n = \ln h_0 + n\ln r$ , plot a straight line and read $r$ from the gradient with an uncertainty.
Square-root series. Use that a fall of height $h$ takes $\sqrt{2h/g}$ to build a second series with ratio $\sqrt r$ and sum the total time.
Proof by induction. Prove the formula for the partial sum $S_n$ by induction, then take the limit as $n \to \infty$ (HL).

Key equations

D = h_0 + 2h_0\sum_{n=1}^{\infty} r^{n} = h_0\,\frac{1+r}{1-r}

T = \sqrt{\frac{2h_0}{g}}\frac{1+\sqrt{r}}{1-\sqrt{r}}

The maths in action

Two panels. Left: three straight lines of peak height against bounce number on a logarithmic axis, steeper for smaller r. Right: three rising curves of total distance that level off at dashed limits of 3.0, 5.7 and 12.3 times the drop height. — (a) Peak heights for three values of $r$ fall on straight lines on a log axis, with gradient $\ln r$ . (b) The distance travelled settles at $h_0(1+r)/(1-r)$ : 3.0, 5.7 and 12.3 times $h_0$ ; the closer $r$ is to 1, the slower the approach.

Example research question

To adapt, never to copy

How well does a constant-ratio model of bounce heights predict the total distance and the total time a ball travels, for a tennis ball dropped from 1.0 m onto two surfaces, and how does the restitution ratio relate to the energy lost per bounce?

Change the context, the data or the method so the question is yours. Our Maths IA guide shows how to check it.

Check your research question

Names the ball, the surface and the drop height (or heights).
Says how heights will be measured and to what resolution.
Names the model being tested (geometric) and the prediction being checked: total distance, total time, or bounces to a given height.
Says what the prediction is compared with.
Is narrow: at least 6 bounces recorded, with repeated drops.

Which course it fits

AA SL ideal
AA HL ideal
AI SL ideal
AI HL ideal

Syllabus links in Maths AA:

AA 1.3 Geometric sequences and series
AA 1.5 Exponents and an introduction to logarithms
AA 1.8 Infinite geometric series
AA 2.9 Exponential and logarithmic functions
AA 1.15 Proof by induction, contradiction and counterexample (HL only)

Steps

Read and predict. Learn the geometric sequence and series. Sketch what the heights should do, and guess r with a reason.
Trial run (1 lesson). Drop and film one ball. Can you read each peak? Does the ball stay in the frame? Is the ruler visible and square-on?
Plan the experiment. Choose balls, surfaces, drop heights and repeats. Write down why for each.
Collect the data. Record peak heights for at least 6 bounces per drop, with resolution noted.
Test constant r. Plot each height against the one before it. Does the ratio stay put, or drift?
Fit the model. Fit $h_{n}$ = $h_{0}$ $r^{n}$ and find r with an uncertainty. Compare the two methods.
Predict the distance. Build the distance as a series (up and down), sum it, and measure the total for comparison.
Go further. SL: a second surface or ball. HL: prove the formula and treat time or energy with a harder series.
Write, then cut. Write in layout order. Show one worked sum, then a table; put long data in an appendix.
Check before your teacher's draft. Check each symbol is defined, graphs have titled axes with units, and reflection runs through the piece.

Make it yours

Compare surfaces. Floor, carpet, grass, a gym mat.
Compare balls. Tennis, basketball, squash, a bouncy ball.
Test the constant ratio. Plot each height against the previous one and look for a line through the origin.
Predict the time. Time between bounces depends on the square root of the height, giving a second series.
Bring in energy. Link r to the energy lost per bounce.
Film it. Compare reading peaks by eye with a frame-by-frame method.

HL extension

Prove the partial-sum formula by induction, derive the total time from a second series with ratio $\sqrt r$ , link $r$ to the coefficient of restitution by $r = e^{2}$ , and test whether $r$ is really constant by plotting $h_{n+1}$ against $h_n$ .

The data you need

Drop a ball from a measured height and read the peak of each bounce from slow-motion video against a ruler, for several surfaces or balls; repeat each drop at least three times.

Traps

Quoting the infinite sum formula without saying why the series converges.
Forgetting the ball goes up and down: after the first drop each bounce counts twice.
Reading peaks to the nearest 5 cm, then trusting heights of 3 cm.
Fitting an exponential in software without testing whether the ratio is constant.
Assuming the ball stops exactly when the formula says: the model has a practical end.
Mixing bounce number and time on the same axis.

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