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

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

AA SLAA HLAI SLAI HL Accessible Also in: Differential equations, Health & biology

Research question

How does the cooling constant k in Newton's law of cooling differ between ceramic, glass, insulated and paper cups, and how long does each keep a drink above 60 °C?

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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)

Modelling the data, step by step

This idea usually needs an exponential model. See it worked step by step, with a criterion tip at every step: Exponential y = a e^(kx) + c and y = a·bˣ.

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 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.

A possible outline

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

Pitfalls that cost marks

Forgetting to subtract room temperature before taking logs.
Different volumes or starting temperatures between cups.
Over-claiming from one trial per cup.

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.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Exponential decay T = Tₐ + (T₀ − Tₐ)e^(−kt); Linearising with ln(T − Tₐ)
AA HL	Fits, but add an HL technique	Exponential decay T = Tₐ + (T₀ − Tₐ)e^(−kt); Linearising with ln(T − Tₐ)
AI SL	Good fit	Exponential decay T = Tₐ + (T₀ − Tₐ)e^(−kt); Linearising with ln(T − Tₐ)
AI HL	Fits, but add an HL technique	Exponential decay T = Tₐ + (T₀ − Tₐ)e^(−kt); Linearising with ln(T − Tₐ)

Level: Accessible. A good first extended piece of maths, with room to go deeper. 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: forgetting to subtract room temperature before taking logs; and different volumes or starting temperatures between cups. Say how each one affects your answer.

Use of mathematics (E)

SL: exponential decay T = Tₐ + (T₀ − Tₐ)e^(−kt); linearising with ln(T − Tₐ); regression and comparison of fitted constants; solving exponential equations, each explained in your own words and carried out correctly.

HL adds: deriving the model from dT/dt = −k(T − Tₐ), each explained in your own words and carried out correctly.

How modelling explorations reach the top bands in general: Modelling with functions IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

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

Extending it for HL

This idea already has HL mathematics in it: deriving the model from dT/dt = −k(T − Tₐ). Make that the centre of the exploration and use it with full rigour.

See a complete IA, marked

Is it cheaper to leave the heating on low overnight? A two-temperature model of my bedroom (AI 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 SL, AA HL, AI SL and AI HL. We rate it Accessible: a good first extended piece of maths, with room to go deeper.

What mathematics does this IA use?

Exponential decay T = Tₐ + (T₀ − Tₐ)e^(−kt), linearising with ln(T − Tₐ), regression and comparison of fitted constants, solving exponential equations and hL: deriving the model from dT/dt = −k(T − Tₐ).

Where can I get data for this IA?

Use a digital thermometer or temperature probe, same starting temperature and volume, readings every 30 s for 30 minutes, room temperature recorded. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Which mug keeps tea hot longest? Newton's law of cooling, compared” filled in: the question, the maths and the data sources.

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