= Newton's law of cooling
{c}
{title2=$y'=-k(y-T_a)$}
{wiki}
= Newton law of cooling
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{synonym}
<Newton's law of cooling> models an object's <temperature> $y$ using a positive constant rate $k$ and ambient <temperature> $T_a$. The displayed <ordinary differential equation> assumes one <temperature> represents the object, the environment remains at $T_a$, and the proportional rate stays fixed over the modeled range. It also describes warming when $y<T_a$. These are modeling assumptions: https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/1ce2cf783d5f1d15ee411ec2e5f928ec_MIT18_03SCF11_s1_6text.pdf[MIT's notes on modeling by first-order linear equations]{ref} explain the proportional law through a <tangent line> approximation and caution about large <temperature> differences.
<Dimensional analysis> requires $k$ to have units of inverse time. The product $k(t-t_0)$ in the <exponential function> solution is therefore dimensionless. The <equilibrium point> $y=T_a$ attracts both warmer and colder starting states: the excess <temperature> undergoes <exponential decay>. See <initial value problem> for computed cooling times, <direction field> for the family of trajectories, and <separable differential equation> for a derivation.
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