From: "JACK L. URETSKY (C) 1996; HEP DIV., ARGONNE NATIONAL LAB, ARGONNE, IL 60439" <JLU@hep.anl.gov>
Date: Sun, 8 Dec 1996 19:54:50 -0600 (CST)
Hi all-
I think that intuituion and the art of estimation walk hand-in-hand.
The exponential cooling problem provides a nice example, in my opinion.
So try this:
The cooling of a radiating (black) body of temperature T,
immersed in a radiation field of black-body temperature T_0, is
given by the equation:
dT/dt = -k[T^4 -(T_0)^4]
where k is some constant.
1. Show that if T is close to T_0, then the cooling is
exponential in time with inverse time constant 4k(T_0)^3, and
calculate the correction to this expression proportional to (T-T_0).
2. Show that if T is much larger than T_0, then the
temperature decrease is inversely proportional to the time and the
proportionality constant is -k(T_i)^4. Correction: I mean the
decrease is proportional to the time (sorry, but this is a lousy editor).
Regards,
Jack