Newton's Model of Cooling says that the rate at which an object loses or gains heat is proportional to the difference between its temperature and the ambient temperature. This model is most naturally described in the language of differential equations by the differential equation
dT
-- = k (A - T)
dt
where T denotes the temperature of the object, k is a positive constant, and A is the ambient temperature. In the simplest situation the ambient temperature is constant but we also study situations in which the ambient temoperature is varying.
The value of the constant k is determined by the physical characteristics of the object. If the object is large and well-insulated then it loses or gains heat slowly and the constant k is small. If the object is small and poorly-insulated then it loses or gains heat more quickly and the constant k is large.
When the ambient temperature is constant the solution of this differential equation is
-kt
T(t) = A + Ce
where the value of the constant C is determined by the initial condition -- note that T(0) = A + C.
If we take measurements of the object's temperature at fixed intervals -- that is,
a(0) = T(a)
a(1) = T(a + h)
a(2) = T(a + 2 h)
.
.
a(n) = T(a + n h)
then
-kh
a(n + 1) = a(n) e
and
a(n + 1) -kh
-------- = e
a(n)
is a fixed constant.