IA idea · Differential equations & dynamics
How much of cycling effort goes on air resistance? A coast-down test
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?
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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
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.
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.
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 | Setting up dv/dt = −a − bv²; Parameter estimation by least squares |
| 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 | Setting up dv/dt = −a − bv²; Solving by separation of variables (AA HL) or fitting with Euler's method |
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
Reflection (D)
For this idea, start with the weak points: riding on a slope (even 1% dominates the result); and ignoring wind; average both directions. Say how each one affects your answer.
Use of mathematics (E)
HL: setting up dv/dt = −a − bv²; parameter estimation by least squares; power = force × velocity; solving by separation of variables (AA HL) or fitting with Euler's method, each explained in your own words and carried out correctly.
How differential equations explorations reach the top bands in general: Differential equations & dynamics IA ideas. Guides: A · B · C · D · E · self-grade sheet.
Taking it further
Estimate the speed advantage of drafting behind another rider using a published drag reduction and your model.
Extending it for HL
This idea already has HL mathematics in it: Solving by separation of variables (AA HL) or fitting with Euler's method. Make that the centre of the exploration and use it with full rigour.
See a complete IA, marked
Is it cheaper to leave the heating on low overnight? A two-temperature model of my bedroom (AI HL) is a complete differential equations exploration, marked on every criterion.
Frequently asked questions
Which IB Maths course is this IA idea for?
It works for AA HL and AI HL. We rate it Ambitious: it suits confident students, who should expect to learn some mathematics on their own.
What mathematics does this IA use?
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 and power = force × velocity.
Where can I get data for this IA?
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. Checked sources and tools: phyphox (RWTH Aachen).
Turn this idea into your IA
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