Newton's Law of Cooling

Newton's Law of Cooling

The rate of heat loss of an object is proportional to the difference between the object's temperature and the ambient (surrounding) temperature:

$\frac{dT}{dt} = -k \cdot (T - T_{\text{env}})$

• $T$: object temperature
• $T_{\text{env}}$: ambient temperature (constant)
• $k > 0$: cooling constant (depends on material and environment)

Exact Solution

$T(t) = T_{\text{env}} + (T_0 - T_{\text{env}}) \cdot e^{-kt}$

The object approaches $T_{\text{env}}$ exponentially. This is just exponential decay applied to the temperature difference $(T - T_{\text{env}})$.

Applications

• Forensics: estimating time of death from body temperature
• Engineering: thermal management of electronics
• Food science: how quickly food cools to safe serving temperature
• Metallurgy: quenching and tempering of metals

Worked Example

A cup of coffee at 90°C in a 20°C room with $k = 0.1$:

$T(t) = 20 + 70 \cdot e^{-0.1t}$ $T(10) = 20 + 70 \cdot e^{-1} \approx 20 + 25.7 = 45.7°C$ $T(30) \approx 20 + 3.5 = 23.5°C$

Your Task

Implement cooling(k, T_env, T0, t_end, n) using Euler's method. Return the final temperature.