Maths IA · Modelling · topic 33 of 40

Cooling drinks: does Newton's law of cooling fit? · IB Maths IA idea: guide, sample graph and step-by-step plan

simple kit, real maths. good start :)

Everyone says a hot drink cools exponentially. You're going to test that with real readings, and measure how wrong it is.

Difficulty●○○ Gentle
KitA cup, a probe thermometer or phone sensor, a spreadsheet or GDC
TimeAbout 12 to 15 hours
FitsAA and AI, SL and HL
  • ✓ you like testing a claim with real data
  • ✓ you can log a temperature every minute
  • ✓ you want exponentials and residuals to mean something

The big idea

← the claim you'll test

Newton's law says the gap between a drink's temperature and the room's decays exponentially: dT/dt = −k(T − T_R). The maths is tidy; the experiment is not. Cups also lose heat by evaporation and radiation, and the room is never perfectly steady. Predict which cup cools fastest and why, fit the model properly to get k for each, then use residuals to see where the law fails. At HL, regression alone is not enough: solve the differential equation yourself and justify every step, because that is the maths criterion E rewards.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Compare cups

Ceramic, glass, steel flask, paper; lid on and off.

→ You get a k for each cup and a reason to compare them: the question becomes what k means physically.
Log the room temperature

Record the room every few minutes instead of assuming one value.

→ Your model becomes T − T_R(t); you must decide what to do if the room drifts.
Fit it two ways

Straighten the data with ln(T − T_R) and fit a line; then fit the exponential directly.

→ Two estimates of k to compare, and a reason why linearising weights the points differently.
Test a second model

Add a model with a constant term, or two exponentials (fast, then slow).

→ Residuals, not R² alone, decide which model is better: a stronger evaluation.
Change the start temperature

Cool from 90 °C and from 60 °C.

→ Tests whether k depends on temperature (the law says not, for small gaps): deeper analysis.
Solve the equation yourself

Separate variables to get T(t), then add a changing room temperature or a radiation term.

→ Takes the maths past regression. At HL this is where criterion E is earned.

Your Maths IA research question: run the check

write it FIRST, then check it

Write your own question first. Then make sure it ticks every box:

  • Names what you measure over time (temperature of what, where in the cup) and how often you read it.
  • Names what changes between runs (cup, lid, starting temperature), or says you change nothing and test the model.
  • Says what the data is compared with: Newton's law, a second model, or both, and how 'fit' will be judged.
  • Makes clear the quantity you are after (the cooling constant k) and what you will do with it.
  • Fits in 10 to 15 hours, with at least 25 readings per run and repeats.

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The model

A cup, a probe and a clock give you T against t. The model says the curve levels off at room temperature, not at zero. Sketch yours with the cup you really use.

TtTRprobe: middle, same depthsame volume + start temperatureroom temperature: measure itis it exponential?

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Introduction and aim~1.5 pages
Your question; why cooling matters to you (a flask, a hot meal, a cup on a desk); a prediction made before any fitting.
The model~2 pages
Newton's law as a differential equation and its solution; every symbol with units; the assumptions behind it (list them, you will test some). HL: show the derivation.
Collecting the data~1.5 pages
Cups, volumes, probe resolution, reading interval, how you recorded the room temperature, and why you chose each. Keep one clear table; bulk readings go in an appendix.
The first fit~2.5 pages
One worked example by hand of the straightened fit (ln(T − T_R) against t): gradient, intercept and what they mean; then the software fit; k with units.
Residuals and model check~2 pages
Residual plot; where the model fails and why; a measure of fit that is not R² alone.
Comparing cups~2 pages
k for each cup in one table; are the differences bigger than the spread between repeats?
Going further~2 pages
SL: a second model or a varying room temperature. HL: derive and solve the equation, and justify the method.
Conclusion~1 page
Answer your question with numbers; say whether the law fits and for which cups.
Reflectionbuilt in, plus ~0.5 page
Not one paragraph at the end: a comment after each result or choice (a poor fit, a surprise, a decision you changed), then a specific evaluation of the model's limits.
Bibliographynot counted
Every source for data, theory and any code or software used.

What your Maths IA graph could look like

show T_R as its own line ✓

Made-up data to show the form of a strong graph, not the answer: a drink cooling in a 22 °C room. Yours will look different. example only

01020304050602030405060708090Time t / min (± 0.2 min)Temperature T / °C (± 0.5 °C)model: exponential decay to T_Rroom temperature T_R = 22 °Cyour readings, with error barsdoes the data level off at T_R,not at zero?ILLUSTRATIVE · MADE-UP DATA
  • Axis titles say what is plotted, with units and the reading uncertainty.
  • The room temperature is drawn as its own line, so the reader sees the curve heads for it.
  • The y-axis starts near the room temperature, not at zero. Say so in the text.
  • Both the model and the data are shown, so a gap between them is visible straight away.
  • Another graph to consider: ln(T − T_R) against t. If the law holds it is a straight line with gradient −k. Put a residual plot underneath.

Which source of error moves k most?

Rank by the effect on your result k, not on the raw readings. A quick simulation like this, with your own numbers, shows what to measure carefully. Here the assumed room temperature matters far more than the clock.

Room temperature assumed 1.5 °C too low10.7%Probe reading ± 0.5 °C (random)1%Clock timing ± 15 s (random)0.15%biggest effect: discuss this first!% change in the fitted k (a reading every 2 min for 60 min)ILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

don't skip the trial run!

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Read and predictRead the law and its assumptions. Before measuring, predict the shape of T against t and which cup will cool fastest, each with a reason.
  2. Trial run (1 lesson)Cool one cup for 30 minutes. Is the probe at the same depth each time? Is the room steady? Is your reading interval too slow at the start?
  3. Decide the planChoose cups, volumes, starting temperature, reading interval, run length and repeats. Write down why for each choice; this becomes your method justification.
  4. Collect the dataRecord straight into a table with units, and log the room temperature too. Note anything odd (a draught, a lid left off).
  5. Fit one run by handStraighten the data and fit a line yourself so you know where k comes from. Decide what to do about readings close to room temperature.
  6. Fit with softwareFit the exponential directly. Compare k from both methods and explain any difference.
  7. Check the residualsPlot residuals against time and describe any pattern. Decide whether the model is good enough, with a number to back it up.
  8. Compare and go furtherCompare k across cups and test one alternative model. SL: justify your choices and limits. HL: derive and solve the equation as well.
  9. Write, then cutWrite in the order of the layout above. Show one worked example, not five, and keep repeated tables in an appendix.
  10. Check before your teacher's draftCheck notation (no calculator syntax like e^), axis titles with units, each symbol defined once, and reflection all the way through. Your teacher reads one draft, so make it count.

Maths you'll need

Traps to avoid in this Maths IA

  • Treating room temperature as zero: fitting Ae^(−kt) to raw temperatures drags the curve towards 0 °C.
  • R² as the only evidence: a high R² can hide a clear pattern in the residuals.
  • Taking ln(T − T_R) near the end of the run, where noise can make the difference zero or negative.
  • Stopping too early, so there is no data near the tail and k is poorly pinned down.
  • Comparing cups with different volumes or starting temperatures, then blaming the material.
  • Calculator notation in the text (e^(-kt), 1.2E-3) instead of proper mathematical notation.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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