Mathematics 11–12 · Years 11–12
Cooling curve of a cup of hot water: Newton's law of cooling
Further applications of calculus: Further rates of change (Mathematics Extension 1, Year 12); Graph transformations (Mathematics Advanced, Year 11)
School laboratory, not for home
In a school laboratory, with a teacher supervising, under the school's risk assessment. Not for home.
This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.
The idea
The rate at which a hot drink cools is proportional to how far its temperature sits above the room, which gives an exponential approach to room temperature.
Safety card
Setting: In a school laboratory, with a teacher supervising, under the school's risk assessment. Not for home.
Hazards
- scalds from water near 80 degrees C
- a hot cup breaking
Controls
- the adult pours with the cup standing on the bench
- keep the kettle and cord away from the bench edge
- do not move the cup while it is hot
Note
Hot water near 80 degrees Celsius but no chemicals: the NSW Department of Education Chemical Safety in Schools package does not apply; follow the school's RiskAssess (riskassess.com.au) assessment for hot liquids.
What you need
- A 250 mL ceramic cup, a kettle, a digital thermometer or datalogger probe reading to 0.1 degree
- A stopwatch, a room thermometer, a recording sheet
How to do it
- Record the room temperature P. Pour water at about 80 degrees Celsius into the cup on the bench and insert the probe.
- Record the temperature every minute for 30 minutes without stirring or moving the cup.
- Plot T against t and ln(T - P) against t and check whether the second plot is a straight line.
- Find k from the gradient of the log plot, or from the 10-minute reading with k = ln((T0 - P)/(T10 - P)) / 10.
- Use T = P + A e^(-kt) to predict the 20-minute and 30-minute readings and the time to reach 40 degrees Celsius; compare with the data.
- Describe the model as the graph of y = e^(-kt) dilated vertically by the factor T0 - P and translated up by P, and say which feature of the cup each transformation represents.
What you should see
Worked check with stated assumptions T0 = 80 degrees C, P = 22 degrees C and a 10-minute reading of 60 degrees C: k = 0.0423 per minute, predicted 46.9 degrees C at 20 minutes and 38.3 degrees C at 30 minutes, with 40 degrees C reached at 27.7 minutes. The curve is y = e^(-kt) dilated vertically by 58 and translated up by 22, so its horizontal asymptote is the room temperature. The learner's own k replaces the assumed one. The learner knows it worked when the log plot is straight and the 20- and 30-minute predictions land within 2 degrees of the readings.
What changes
- What you change
- time since pouring
- What you measure
- water temperature
- What you keep the same
- cup and volume
- room temperature
- no lid, no stirring
- probe position
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- Hot water cools by a steady number of degrees per minute; the rate falls as the gap to the room closes.
- The water cools towards zero; it approaches room temperature and stays there.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics Extension 1 11–12 Syllabus (2024), Year 12 focus area Further applications of calculus; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22ME1-12-05
- Mathematics Advanced 11–12 Syllabus (2024), Year 11 focus area Graph transformations; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-11-03
- Australian Curriculum v9No Australian Curriculum v9 code is listed.
Sources
The pages the author read to write this activity.