IA idea · Kinematics & physics-style modelling

How quickly does a swing die down? Exponential or linear decay?

AA SLAA HLAI HLAI SL Solid Also in: Modelling, Differential equations

Research question

Does the amplitude of a swinging pendulum or playground swing decrease exponentially, linearly or in some other way, and what does the better model suggest about the kind of friction acting?

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

Two simple models make different predictions, and the data decides. The link between the type of resistance and the shape of the decay gives depth.

The mathematics you'll need

Exponential and linear models for amplitude
Log-linearisation to test exponential decay
Comparing models with residuals
Damped sinusoidal model A e^(−kt) cos(ωt)
HL: differential equation for damped motion (stated and solved)

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 sinusoidal, exponential or straight-line models. See it worked step by step, with a criterion tip at every step: Sinusoidal y = a sin(b(x − c)) + d · Exponential y = a e^(kx) + c and y = a·bˣ · Straight line y = mx + c · Choosing and comparing 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

Film a pendulum or swing and track it, or use phyphox's gyroscope on a swing.

Tracker video analysis — Free tool to track an object frame by frame in a video and export its x–y coordinates.
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

Collect amplitude over many swings.
Fit linear and exponential models.
Compare with residuals and a log plot.
Fit a damped sinusoid to the full motion.
Reflect on what the model says about the friction.

Pitfalls that cost marks

Measuring amplitude inconsistently.
Fitting only one model.
Too short a recording.

Showing personal engagement

Use a swing in your local park.
Compare two pendulum bobs.
Predict how long until the swing nearly stops.

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	Exponential and linear models for amplitude; Log-linearisation to test exponential decay
AA HL	Good fit	Exponential and linear models for amplitude; Log-linearisation to test exponential decay
AI SL	Good fit	Exponential and linear models for amplitude; Log-linearisation to test exponential decay
AI HL	Good fit	Exponential and linear models for amplitude; Log-linearisation to test exponential decay

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: measuring amplitude inconsistently — 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

Combine linear and quadratic damping in a model and fit both parameters.

Extending it for HL

This idea already has HL mathematics in it: differential equation for damped motion (stated and solved). 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).