Cooling and Heating

Differential Equations · Axiom Academy

LESSON Newton's Law of Cooling and Heating Deriving the differential equation that governs temperature change, and seeing why the same law describes both cooling and heating. 1. From Heat Flow to a Differential Equation Consider an object at temperature T(t) sitting in an environment held at a constant temperature . Newton's physical insight: heat flows at a rate proportional to the temperature difference between the object and its surroundings — a large gap drives fast heat flow, a small gap drives slow heat flow. 2. Solving by Separation of Variables This is a separable first-order ODE. Substitute (so u' = T' , since is constant), separate the variables, and integrate both sides. Exponentiating and folding the constant into A gives . Applying the initial condition T(0) = T_0 pins down , so the temperature curve is: 3. One Equation, Two Phenomena The beauty of Newton's Law: the same formula models both cooling and heating. Only the sign of the initial gap differs — everything else is identical. The gap starts positive, so dT/dt < 0 : temperature falls. The exponential term shrinks to zero, and from above. The gap starts negative, so dT/dt > 0 : temperature rises. The exponential term still shrinks to zero, and from below. You've derived Newton's Law of Cooling and Heating from a physical principle, solved it exactly, and seen why one formula governs both directions of temperature change. Scroll up to revisit any step.

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