AP Calculus AB and BC glossary

Newton's Law of Cooling

Also called: Law of cooling

Newton's law of cooling says an object's temperature changes at a rate proportional to the gap between it and the ambient temperature, with a negative constant of proportionality. It is a separable differential equation, and every non-equilibrium solution approaches the ambient temperature without ever reaching it.

\frac{dT}{dt}=k\bigl(T-T_a\bigr),\ k<0

Separating the variables is routine once you notice what is decaying. The substitution $y=T-T_a$ turns the equation into $dy/dt=ky$ , so the gap between the object and its surroundings is the exponential quantity. Adding $T_a$ back gives the temperature itself, with $T_0$ the starting value and $k<0$ in every case, cooling or warming: it is the sign of $T_0-T_a$ , not the sign of $k$ , that says which way the temperature moves.

T(t)=T_a+(T_0-T_a)e^{kt}

Because $k<0$ the exponential decays, so $T_a$ is a horizontal asymptote, and it is also the equilibrium solution, since $dT/dt=0$ exactly when $T=T_a$ . Coffee left on a desk gets arbitrarily close to room temperature but never equals it; the only object ever at room temperature is one that started there, which is the constant solution $T=T_a$ . Warming runs on the same equation with $T_0<T_a$ , so the gap is negative and grows toward zero from below.

The mistake

Fitting the temperature itself to a decaying exponential. What decays is the difference $T-T_a$ , so the model has to be written as ambient plus a shrinking term. Leaving $T_a$ out sends the temperature to zero rather than to the temperature of the room.

Appears in: Unit 7: Differential Equations