Newton's Law of Cooling and Related Contexts

Hot coffee loses heat fast at first because the gap between its temperature and the room is huge, and that gap is what drives the cooling. Newton's Law of Cooling writes this as a differential equation: dT/dt = -k(T - Ts), where T is the object's temperature, Ts is the surrounding temperature, and k is a positive constant describing how fast heat transfers. When T is far above Ts, the right side is a large negative number, so temperature drops quickly. As T creeps toward Ts, the gap T - Ts shrinks toward zero, so the rate of change shrinks too. The object never overshoots room temperature and never truly stops approaching it, which is why the graph bends and flattens instead of running in a straight line to Ts.

You solve this by separation of variables, same as any other separable differential equation from this unit. Separate T and t, integrate, and solve for T to get T(t) = Ts + (T0 - Ts) e^−kt, where T0 is the starting temperature. The trick that makes this an exponential decay problem is subtracting Ts: let y = T - Ts, and dy/dt = -ky, the exact equation that produces y = y0 e^−kt. Shifting by Ts turns a cooling problem into the same exponential decay shape you already know from population and radioactive decay models.

This topic is really a slope-field and separable-equation problem wearing a thermometer. It sits at the end of the unit because it asks you to combine the separation technique with a real initial condition and interpret the constant k as a cooling rate rather than a growth rate.