Which mug keeps tea hot longest? Newton's law of cooling, compared

Why it makes a good exploration

Cooling is a very common IA, so the twist matters: comparing cups turns a textbook model into a consumer test with an answer you can use every morning.

The mathematics you'll need

• Exponential decay T = Tₐ + (T₀ − Tₐ)e^(−kt)
• Linearising with ln(T − Tₐ)
• Regression and comparison of fitted constants
• Solving exponential equations
• HL: deriving the model from dT/dt = −k(T − Tₐ)

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 a digital thermometer or temperature probe, same starting temperature and volume, readings every 30 s for 30 minutes, room temperature recorded.

• 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

1. State the assumptions of Newton's law.
2. Collect data for 4 cups with controlled conditions.
3. Linearise and fit k for each cup; check residuals.
4. Rank the cups by time above 60 °C.
5. Reflect where the model fails (evaporation, the lid, early fast cooling).

Showing personal engagement

• Predict the ranking first and explain your reasoning.
• Test the effect of a lid or of milk.
• Compare with the manufacturer's claim for an insulated cup.

See Criterion C: personal engagement for what examiners look for.

Taking it further

HL: model two-stage cooling or evaporation with a modified differential equation and compare fits.