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?

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

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

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

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.

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

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.

Taking it further

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

See it done

Our annotated exemplar How long does a skydiver take to reach terminal velocity? (AA HL) explores a question like this one, with an examiner's comment on every criterion.

Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Desmos workspacePlot your data, fit models and keep your graphs with your plan. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, a practising IA examiner, reads your draft and talks it through with you (€100).