IA idea · Differential equations & dynamics

How much of cycling effort goes on air resistance? A coast-down test

AA HLAI HL Ambitious Also in: Sport, Modelling

Research question

By coasting on a flat road and recording speed against time, can I estimate the rolling-resistance and air-drag coefficients in dv/dt = −a − bv², and what do they imply about the power needed at 25 km/h?

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

Why it makes a good exploration

Engineers use coast-down tests to measure a vehicle's drag. Doing one on your own bike, with your phone logging speed, turns a differential equation into a measurement of something real.

The mathematics you'll need

Setting up dv/dt = −a − bv²
Solving by separation of variables (AA HL) or fitting with Euler's method (AI HL)
Parameter estimation by least squares
Power = force × velocity

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

On a flat, quiet road (safely, with permission), coast from about 30 km/h and log speed with phyphox GPS or a bike computer; repeat in both directions to cancel wind.

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 forces and derive the DE.
Collect several coast-downs in both directions.
Estimate a and b (e.g., by fitting dv/dt against v²).
Validate by predicting another run.
Compute the power split between rolling and air resistance at different speeds.

Pitfalls that cost marks

Riding on a slope (even 1% dominates the result).
Ignoring wind; average both directions.
Unsafe data collection — plan it with an adult.

Showing personal engagement

Compare upright and crouched positions.
Compare two bikes or tyre pressures.
Link to a race time or commute you do.

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

Taking it further

Estimate the speed advantage of drafting behind another rider using a published drag reduction and your model.

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).