Newton's Law of cooling/warming

Pour a $150^\circ$ cup of coffee onto a table in an $80^\circ\mathrm C$ room, and it cools until it reaches room temperature; take a $35^\circ$ glass of water from the fridge into the same room, and it warms up until it reaches room temperature. Both observations are captured by a single law.

Newton's Law of Cooling/Warming. The rate at which a body's temperature changes is proportional to the difference between the body's temperature and the temperature of the surrounding medium (the ambient temperature). If $T(t)$ is the body's temperature at time $t$ and $T_m$ is the (constant) ambient temperature,

where $k$ is the constant of proportionality. In either case — cooling or warming, with $T_m$ held constant — it stands to reason that $k<0$: the temperature gap always shrinks towards zero over time.

Solution. Separating variables, $\dfrac{dT}{T-T_m}=k\,dt$, and integrating gives

The constant $C$ is fixed by the temperature at $t=0$ (usually the first reading), and $k$ is fixed from a second reading at some later time; once both constants are known, $T(t)$ can be evaluated at any time, or the equation can be inverted to find the time at which a target temperature is reached. …