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

IA idea · Differential equations & dynamics

How long does a skydiver take to reach terminal velocity?

AA HLAI HL Ambitious Also in: Sport, Calculus

Research question

Using a differential equation with quadratic air resistance, how long does a skydiver take to reach 95% of terminal velocity, and how different is the answer from a linear-drag model and from Euler's method?

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Why it makes a good exploration

“Terminal velocity” is approached but never reached, so the question “how long does it take?” needs a careful definition. Solving the quadratic-drag equation exactly (partial fractions, tanh) and comparing it with simpler models is ideal AA HL mathematics; AI HL students can do it with Euler's method and slope fields.

The mathematics you'll need

Setting up m dv/dt = mg − kv² from Newton's second law
Separation of variables and partial fractions (AA HL)
Hyperbolic tangent solution v = v_T tanh(gt/v_T) — explain it
Euler's method and step-size error (AI HL)
Comparing with the linear-drag model

Modelling the data, step by step

This idea usually needs a reciprocal model. See it worked step by step, with a criterion tip at every step: Reciprocal y = a/x + b and a/(x − h) + k.

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

Use published typical values (mass, belly-to-earth terminal speed around 55 m/s) and state them; optionally test the model on a falling object you film.

Tracker video analysis — Free tool to track an object frame by frame in a video and export its x–y coordinates.
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Derive the DE from forces and state assumptions.
Find the terminal velocity from dv/dt = 0.
Solve the DE exactly (AA HL) or numerically (AI HL).
Find the time to reach 95% and 99% of v_T.
Compare with linear drag and discuss which is realistic at these speeds.

Pitfalls that cost marks

Using a drag constant with no justification.
Asking when v = v_T (never) instead of defining a threshold.
Euler's method with one step size and no error discussion.

Showing personal engagement

Compare belly-down and head-down positions using their different terminal speeds.
Test your model on a coffee filter or shuttlecock filmed falling.
Calculate the distance fallen before the parachute opens in a real jump profile.

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 m dv/dt = mg − kv² from Newton's second law; Separation of variables and partial fractions
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 m dv/dt = mg − kv² from Newton's second law; Hyperbolic tangent solution v = v_T tanh(gt/v_T) — explain it

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: using a drag constant with no justification; and asking when v = v_T (never) instead of defining a threshold. Say how each one affects your answer.

Use of mathematics (E)

HL: setting up m dv/dt = mg − kv² from Newton's second law; separation of variables and partial fractions; hyperbolic tangent solution v = v_T tanh(gt/v_T) — explain it; comparing with the linear-drag model; euler's method and step-size error, 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

Include air density changing with altitude — the DE no longer separates, so solve it with Euler's method and compare.

Extending it for HL

This idea already has HL mathematics in it: Separation of variables and partial fractions; Euler's method and step-size error. Make that the centre of the exploration and use it with full rigour.

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 Ambitious: it suits confident students, who should expect to learn some mathematics on their own.

What mathematics does this IA use?

Setting up m dv/dt = mg − kv² from Newton's second law, separation of variables and partial fractions (AA HL), hyperbolic tangent solution v = v_T tanh(gt/v_T) — explain it, euler's method and step-size error (AI HL) and comparing with the linear-drag model.

Where can I get data for this IA?

Use published typical values (mass, belly-to-earth terminal speed around 55 m/s) and state them; optionally test the model on a falling object you film. Checked sources and tools: Tracker video analysis and Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “How long does a skydiver take to reach terminal velocity?” filled in: the question, the maths and the data sources.

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