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

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

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

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.

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

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

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.

Taking it further

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