When a small object of temperature Tis placed in a large heat bath of temperature TB, the temperature of the object will eventually approach the temperature of the bath. An approximate way of describing the change in the object's temperature as a function of time is given by Newton's law of cooling:
(a) If the temperature of the object at the time t = 0 (when it is immersed in the bath) is T(0) = T0, use the Euler method to approximately find the temperature of the bath at times tn = n Δt for n = 1, and 2, assuming Δt is small.
(b) Since the rate of change of T is proportional to T, the solution is bound to be an exponential, looking something like . Without going through the process of solving the equation, figure out what A, B, and α have to be and explain why you think so.
(c) In asking part (a), we said "assuming Δt is small." Compared to what? Explain your choice.
(d) In stating the problem, we said that Newton's law of cooling was "an approximate way of describing the change in an object's temperature..." Why approximate? What are some things we have left out that might be relevant for a real-world example?
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Last revision 16. October, 2005.