Newton's Law of Cooling

Physical Law
The rate at which a hot body loses heat is proportional to the difference in temperature between it and its surroundings.

Let $H$ be the temperature at time $t$, and let $H_a$ be ambient temperature.

Let $H_0$ be the temperature at time $t = 0$.

Then the temperature of a body $H_a$ at time $t = a$ is given by:


 * $H = H_a - \left({H_0 - H_a}\right) e^{-k t}$

where $k$ is some positive constant.

Solution
We have the differential equation:
 * $\displaystyle \frac{\mathrm{d}{H}}{\mathrm{d}{t}} \propto - \left({H - H_a}\right)$

that is:
 * $\displaystyle \frac{\mathrm{d}{H}}{\mathrm{d}{t}} = - k \left({H - H_a}\right)$

where $k$ is some constant.

This is an instance of the Decay Equation, and so has a solution:


 * $H = H_a + \left({H_0 - H_a}\right) e^{-k t}$

He applied this law to make an estimate of the temperature of a red-hot iron ball. Although this approximation was somewhat crude, it was better than anything else up till then.