IA idea · Kinematics & physics-style modelling

Modelling a bus journey between two stops from a phone's sensors

AA SLAI SLAI HL Solid Also in: Calculus, Modelling

Research question

Can the speed-time graph of a bus between two stops be modelled by a piecewise function, and does the area under it match the distance between the stops on a map?

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

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Why it makes a good exploration

An everyday journey gives a rich speed-time graph. Building a piecewise model and checking its area against a map is clear, testable calculus.

The mathematics you'll need

Piecewise functions for accelerate, cruise and brake phases
Area under a speed-time graph by integration
Numerical integration of recorded data
Comparing with a map distance
Fitting each phase

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.

Modelling the data, step by step

This idea usually needs a piecewise model. See it worked step by step, with a criterion tip at every step: Piecewise models.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

Where the data comes from

Record speed (GPS) or acceleration with phyphox on a bus journey you make anyway; measure the stop-to-stop distance on OpenStreetMap.

phyphox (RWTH Aachen) — Free app that turns your phone's accelerometer, microphone, barometer and gyroscope into data loggers with CSV export.
OpenStreetMap — Free map with exact coordinates of schools, hospitals, shops and stations; export or read off coordinates.
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

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

Record several journeys between the same stops.
Build and fit a piecewise model.
Integrate it and the raw data.
Compare with the map distance.
Reflect on GPS error and traffic.

Pitfalls that cost marks

GPS speed is noisy in towns; say how you dealt with it.
Phases chosen by eye with no rule.
Only one journey.

Showing personal engagement

Use your own route to school.
Compare drivers or times of day.
Estimate the energy wasted in braking.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Piecewise functions for accelerate, cruise and brake phases; Area under a speed-time graph by integration
AA HL	Not a natural fit	The mathematics is mainly from the AI course; at AA HL the exploration would need an AA-level approach (calculus, proof or probability theory) to reach the top of Criterion E.
AI SL	Good fit	Piecewise functions for accelerate, cruise and brake phases; Area under a speed-time graph by integration
AI HL	Good fit	Piecewise functions for accelerate, cruise and brake phases; Area under a speed-time graph by integration

Level: Solid. Needs some independent work beyond class examples. 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: gps speed is noisy in towns; say how you dealt with it — 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

Find the smoothest speed profile that covers the distance in the same time.

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).