Newton's Law of Cooling
Newton's Law of Cooling
From experimental observations it is known that (up to a
``satisfactory'' approximation) the surface temperature of an object
changes at a rate proportional to its relative temperature. That is,
the difference between its temperature and the temperature of the
surrounding environment. This is what is known as Newton's law of
cooling. Thus, if 
is the temperature of the object at time
t, then we have
where S is the temperature of the surrounding environment. A
qualitative study of this phenomena will show that k >0. This is
a first order linear differential equation. The solution, under the
initial condition 
, is given by
Hence,
,
which implies
This equation makes it possible to find k if the interval of time
is known and vice-versa.
Example: Time of Death Suppose that a corpse
was discovered in a motel room at midnight and its temperature was
. The temperature of the room is kept constant at
. Two hours later the temperature of the corpse dropped to
. Find the time of death.
Solution: First we use the observed
temperatures of the corpse to find the constant k. We have
.
In order to find the time of death we need to remember that the
temperature of a corpse at time of death is 
(assuming the
dead person was not sick!). Then we have
which means that the death happened around 7:26 P.M.
One of our interested readers, E.P. Esterle, wrote a program that helps find the time of death based on the above notes. Click HERE to download it. Have fun with it.
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Author: Mohamed Amine Khamsi