Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

IA idea · Differential equations & dynamics

Linear or quadratic drag? Testing air resistance with falling coffee filters

AA HLAI HL Solid Also in: Modelling, Kinematics

Research question

Is the air resistance on a falling paper cone proportional to its speed or to its speed squared, judged by how terminal velocity changes as the mass is doubled and tripled?

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Why it makes a good exploration

Stacking coffee filters changes the mass without changing the shape. If drag is proportional to v, terminal speed doubles with mass; if proportional to v², it rises by √2. A kitchen experiment decides between two differential-equation models.

The mathematics you'll need

Force models and the DE m dv/dt = mg − kvⁿ
Terminal velocity from the steady state
Power models and log-log regression to estimate n
Solving or simulating the DE and comparing with position–time data

Modelling the data, step by step

This idea usually needs a power model. See it worked step by step, with a criterion tip at every step: Power y = a·xⁿ (log–log).

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

Drop stacks of 1–6 filters in front of a metre rule; film at high frame rate and track position with Tracker.

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

A possible outline

Derive terminal velocity for linear and quadratic drag.
Measure terminal speed for each stack (repeat drops).
Fit v_T against m on log axes and estimate n.
Solve the chosen DE and compare with the full position–time data.
Reflect on air currents, flutter and the transition region.

Pitfalls that cost marks

Measuring speed before terminal velocity is reached.
Too few repeats to see the difference between √2 and 2.
Drafts in the room.

Showing personal engagement

Predict n before the experiment.
Try muffin cases or paper of different stiffness.
Link to the Reynolds number idea without over-claiming.

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	Force models and the DE m dv/dt = mg − kvⁿ; Terminal velocity from the steady state
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	Force models and the DE m dv/dt = mg − kvⁿ; Terminal velocity from the steady state

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

Reflection (D)

For this idea, start with the weak points: measuring speed before terminal velocity is reached; and too few repeats to see the difference between √2 and 2. Say how each one affects your answer.

Use of mathematics (E)

HL: force models and the DE m dv/dt = mg − kvⁿ; terminal velocity from the steady state; power models and log-log regression to estimate n; solving or simulating the DE and comparing with position–time data, 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

Fit both models to the full motion, not just the terminal speed, and compare sums of squared residuals.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL differential equations exploration adds.

See a complete IA, marked

How long does a skydiver take to reach terminal velocity? (AA HL) explores a question like this one, 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 Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Force models and the DE m dv/dt = mg − kvⁿ, terminal velocity from the steady state, power models and log-log regression to estimate n and solving or simulating the DE and comparing with position–time data.

Where can I get data for this IA?

Drop stacks of 1–6 filters in front of a metre rule; film at high frame rate and track position with Tracker. Checked sources and tools: Tracker video analysis.

Turn this idea into your IA

The planner opens with “Linear or quadratic drag? Testing air resistance with falling coffee filters” filled in: the question, the maths and the data sources.

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