Newton’s Cooling as an ODE

Lesson notes

Overview

The rate of cooling is proportional to the temperature difference. This topic is rated intermediate and is number 175 of 245 in Mathematics. Builds on the basics with relationships, methods and trade-offs.

What you are seeing

The panel above shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.

In depth

Newton’s law of cooling states that an object’s temperature changes at a rate proportional to the difference between its temperature and the surroundings: dT/dt = −k(T − Tₐ). The solution is an exponential approach to room temperature, T = Tₐ + (T₀ − Tₐ)e^(−kt), fast at first and then slowing. It applies to a cup of coffee or a cooling engine reasonably well, as long as the temperature difference is not too large.

Reading the relationship

The expression "dT/dt = −k(T − Tₐ)" is a compact summary of the idea. It uses the symbols d, T, t, k, Tₐ; name what each one stands for before you start. Notice that it contains a ratio, so increasing the denominator makes the result smaller. Then use the sliders to see how changing one quantity changes the others.

A guided exploration

1. Set every slider to its middle value and note the shape of the curve. 2. Move one slider at a time and describe whether the curve shifts, stretches or turns. 3. Find the value where the behaviour changes sharply, then connect it to the formula.

Try this

Predict the outcome first, then test extreme slider values to find where the pattern changes or breaks.

Common pitfalls

A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.

How it connects

It builds on “Differential Equation Solutions”, the topic before it in Mathematics. Once you are comfortable, continue with “Logistic Differential Equation”.

Why it matters

Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.