Maths IA · Modelling · topic 35 of 40

Bouncing ball: how far does it travel in total? · IB Maths IA idea: guide, sample graph and step-by-step plan

gentle, but be exact about the series

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

Difficulty●○○ Gentle
KitA ball, a ruler, a phone, a spreadsheet
TimeAbout 12 to 15 hours
FitsAA and AI, SL and HL
  • ✓ you like quick experiments with a clear model
  • ✓ you can film in slow motion
  • ✓ you want a series that actually converges

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.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

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Compare surfaces

Floor, carpet, grass, a gym mat.

→ Each surface has its own r. Soft surfaces may break the constant-ratio model, which is worth discussing.
Compare balls

Tennis, basketball, squash, a bouncy ball.

→ r depends on the ball, so the question becomes what r tells you.
Test the constant ratio

Plot each height against the previous one and look for a line through the origin.

→ A direct test of the model, more informative than fitting an exponential in software.
Predict the time

Time between bounces depends on the square root of the height, giving a second series.

→ A new series with ratio √r, and a check that the ball stops in finite time.
Bring in energy

Link r to the energy lost per bounce.

→ A physical meaning for the parameter. Be careful with units; HL can derive the link.
Film it

Compare reading peaks by eye with a frame-by-frame method.

→ Measurement uncertainty is real and discussed: a good thread for reflection.

Your Maths IA research question: run the check

write it FIRST, then check it

Write your own question first. Then make sure it ticks every box:

  • 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.

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The picture to draw

Heights shrink by the same fraction each bounce. Sketch yours with a ruler in shot, and mark which heights you will read.

h0h1drop from a measured heighteach peak is a fixed fraction r of the lastphone video: read each peakruler in shot, camera square-oninfinite bounces, finite distance?

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Introduction and aim~1.5 pages
Your question; why the ball and surface you chose; what you expect r to be, roughly, and why.
Sequences and series needed~2.5 pages
The geometric sequence; the sum to n terms; the sum to infinity and when it exists. HL: prove the formula.
The experiment~1.5 pages
Set-up, how you read each peak, repeats; one clear table.
Fitting the model~3 pages
Regression on the heights, and a direct test of constant r; r with its uncertainty.
Predicting total distance~2 pages
The series for the distance (the ball goes up and down); the number; compared with a measured total.
A second series~1.5 pages
Time or energy: a different ratio, a different sum, and what it adds.
Conclusion~1 page
Answer with numbers: how well does the model predict, and where does it fail?
Reflectionbuilt in, plus ~0.5 page
Not one paragraph at the end: a comment after each result or choice (a poor fit, a surprise, a decision you changed), then a specific evaluation of the model's limits.
Bibliographynot counted
Every source for data, theory and any code or software used.

What your Maths IA graph could look like

error bars on EVERY point ✓

Made-up data to show the form of a strong graph, not the answer: a ball dropped from 1.0 m. Yours will look different. example only

012345670.00.10.20.30.40.50.60.70.80.91.01.1Bounce number nPeak height h_n / m (± 0.02 m)model: hₙ = h₀rⁿa straight line (poor model)your measured peaksconstant ratio: checkh(n+1)/h(n) for every nILLUSTRATIVE · MADE-UP DATA
  • Bounce number on the x-axis, height in metres with the reading uncertainty.
  • A straight line is drawn on purpose: heights fall, but not by a constant amount, and the graph should show it.
  • Error bars are the same size on every point because they come from the ruler or video, not the bounce.
  • Late bounces are small compared with the uncertainty: say where you stop trusting them.
  • Another graph to consider: h(n+1) against h(n). If the model holds it is a straight line through the origin with gradient r.

How much of the distance comes early?

where can you stop measuring?

A partial sum shows how fast the series converges, and how many bounces you actually need to measure. The ratio here is made up; compute it for your own r and say where that tells you to stop.

After 8 bounces99.1%After 6 bounces97.2%After 4 bounces91.7%After 2 bounces75.3%most of it comes early% of the predicted total distance travelled so far (r = 0.58)ILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

film it once before the real run

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Read and predictLearn the geometric sequence and series. Sketch what the heights should do, and guess r with a reason.
  2. 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?
  3. Plan the experimentChoose balls, surfaces, drop heights and repeats. Write down why for each.
  4. Collect the dataRecord peak heights for at least 6 bounces per drop, with resolution noted.
  5. Test constant rPlot each height against the one before it. Does the ratio stay put, or drift?
  6. Fit the modelFit h_n = h_0 r^n and find r with an uncertainty. Compare the two methods.
  7. Predict the distanceBuild the distance as a series (up and down), sum it, and measure the total for comparison.
  8. Go furtherSL: a second surface or ball. HL: prove the formula and treat time or energy with a harder series.
  9. Write, then cutWrite in layout order. Show one worked sum, then a table; put long data in an appendix.
  10. Check before your teacher's draftCheck each symbol is defined, graphs have titled axes with units, and reflection runs through the piece.

Maths you'll need

Traps to avoid in this Maths IA

  • 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.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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