IA idea · Kinematics & physics-style modelling

How high did the lift go? Integrating a phone's accelerometer

AA SLAA HLAI HL Solid Also in: Calculus, Numerical methods

Research question

If you record a lift's acceleration with a phone, how accurately does integrating it twice give the height travelled, and why does the error grow over time?

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

It uses real sensor data and calculus in the most natural way: acceleration to velocity to displacement. The drift in the answer is a genuine and explainable problem.

The mathematics you'll need

Acceleration, velocity and displacement related by integration
Numerical integration (trapezoidal rule) of sampled data
Removing the offset (gravity and sensor bias)
Error growth: a constant bias gives a quadratic displacement error
Comparing with the measured height between floors

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

Record the lift with phyphox (acceleration without g), measure the floor height with a tape or building plans.

phyphox (RWTH Aachen) — Free app that turns your phone's accelerometer, microphone, barometer and gyroscope into data loggers with CSV export.
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 rides between known floors.
Integrate once and check velocity returns to zero.
Integrate again for height.
Correct for bias and compare.
Reflect on why drift happens and how to reduce it.

Pitfalls that cost marks

Not correcting for sensor offset.
Using one ride only.
Integrating without explaining the method.

Showing personal engagement

Use the lift in your own building.
Predict the error before correcting.
Test on stairs as well.

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	Acceleration, velocity and displacement related by integration; Numerical integration (trapezoidal rule) of sampled data
AA HL	Good fit	Acceleration, velocity and displacement related by integration; Numerical integration (trapezoidal rule) of sampled data
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	Acceleration, velocity and displacement related by integration; Numerical integration (trapezoidal rule) of sampled data

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: not correcting for sensor offset — 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

Fit a piecewise model to the acceleration (speed up, cruise, slow down) and integrate it exactly.

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.

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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).