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:
 * $$\frac{\mathrm{d}{H}}{\mathrm{d}{t}} \propto - \left({H - H_a}\right)$$

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

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.