---
tag: 02HB5F40VR
kind: example
name: Newton's law of cooling
msc: [80A19]
instance_of: [07JH131FVP, 074E9A8ZJK]
uses: [075VBY4K4X, 00SVP6W3CB]
sources:
  - url: https://en.wikipedia.org/wiki/Newton%27s_law_of_cooling
    locator: First-order transient response of lumped-capacitance objects
same_as:
  wikidata: Q2902868
  wikipedia: Newton's law of cooling
---

## Example

Let $T(t)$ be the temperature of a body at time $t$. Let $T_a$ be the constant temperature of the surroundings, and let $k > 0$ be a constant.
Newton's law of cooling is the equation
$$T'(t) = -k \left(T(t) - T_a\right).$$
With the initial condition $T(0) = T_0$, its unique solution is
$$T(t) = T_a + (T_0 - T_a)\, e^{-k t}.$$

## Remarks

The difference $D = T - T_a$ satisfies $D' = -k D$. This is [[exponential decay|00SVP6W3CB]], so $D(t) = D(0) e^{-kt}$.
As $t \to \infty$, $T(t) \to T_a$. If $T_0 > T_a$ the body cools, and if $T_0 < T_a$ it warms.
The law is a good model when the temperature difference is small and the body has a nearly uniform temperature.

## Explanations

### L1 human

A hot cup of coffee cools fast at first and then more slowly.
The law says that the speed of cooling is proportional to the difference between the coffee and the room.
So the difference halves after each fixed time interval.