IA idea · Kinematics & physics-style modelling

The path of a bike valve: parametric equations of a cycloid

AA HLAI HL Ambitious Also in: Art & music, Calculus

Research question

What path does the valve on a rolling bike wheel trace, why is the top of the wheel moving at twice the bike's speed while the bottom is momentarily still, and does a tracked video agree?

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

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

Parametric equations and calculus explain a surprising fact you can see in a video. The comparison with tracked data makes it an investigation, not a derivation from a book.

The mathematics you'll need

Parametric equations x = r(θ − sin θ), y = r(1 − cos θ)
Velocity by differentiating parametric equations
Speed at the top and bottom
Arc length of one arch (8r) by integration
Fitting the model to tracked positions

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 a bike wheel rolling with a marked valve against a scale and track it in Tracker.

Tracker video analysis — Free tool to track an object frame by frame in a video and export its x–y coordinates.
GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.

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

Derive the parametric equations.
Track the valve and compare.
Find velocity and speed and check at top and bottom.
Find the arc length of one arch.
Reflect on tracking error and wheel slip.

Pitfalls that cost marks

Camera not perpendicular to the wheel.
Using the formulas without deriving them.
Ignoring that the valve isn't at the rim.

Showing personal engagement

Use your own bike.
Track a point not on the rim (a curtate cycloid).
Explain why photos of spokes look blurred at the top.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	Parametric equations x = r(θ − sin θ), y = r(1 − cos θ); Velocity by differentiating parametric equations
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	Parametric equations x = r(θ − sin θ), y = r(1 − cos θ); Velocity by differentiating parametric equations

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Film or record the motion yourself, choose what to vary, and say what you expected before you analysed it.

Reflection (D)

Compare your model with the data and explain each mismatch (air resistance, friction, tracking error); reflect on how noise affects numerical derivatives. For this idea, start with: camera not perpendicular to the wheel — say how it affects your answer.

Use of mathematics (E)

SL: Displacement, velocity and acceleration related by differentiation and integration, fitted models compared with residuals, and the physics assumptions turned into stated mathematical assumptions.

HL: Vectors for motion in two dimensions, a differential equation for a resisted motion solved and compared with data, or numerical differentiation with an error analysis.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Model a point inside the rim and compare its path and speeds.

Extending it for HL

Add air resistance or friction as a differential equation, solve it (analytically or with Euler's method) and compare with the simpler model.

See a complete IA, marked

Our annotated exemplar How long does a skydiver take to reach terminal velocity? (AA HL) asks a different question, but shows how a complete kinematics exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).