Temperature of Body under Newton's Law of Cooling

Theorem
Let $B$ be a body in an environment whose ambient temperature is $H_a$.

Let $H$ be the temperature of $B$ at time $t$.

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

Then:
 * $H = H_a - \paren {H_0 - H_a} e^{-k t}$

where $k$ is some positive constant.

Proof
By Newton's Law of Cooling:
 * The rate at which a hot body loses heat is proportional to the difference in temperature between it and its surroundings.

We have the differential equation:
 * $\dfrac {\d H} {\d t} \propto - \paren {H - H_a}$

That is:
 * $\dfrac {\d H} {\d t} = - k \paren {H - H_a}$

where $k$ is some positive constant.

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


 * $H = H_a + \paren {H_0 - H_a} e^{-k t}$

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.