Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

••• In his Feynman Lectures in Physics (Vol. 1), Richard Feynman gives the following proof of the equipartition theorem for the special case of molecules in an ideal gas: Consider two molecules of masses m1 and m2 with initial velocity vectors v1 and v2,prior to collision. The velocity of the center of mass vcm is defined by the equation

 (m1 + m2) vcm = (m1v1 + m2v2).

The relative velocity w of molecules is defined as w = (v1 − v2). One can argue that if the system is in thermal equilibrium, the direction of the center-of-mass motion is completely uncorrelated to the direction of the relative velocity, so that 〈w · vcm〉 = 0, where the brackets represent an average over all pairs of molecules. (Since the directions are random, the dot-product is positive as often as negative and it averages to zero.) Use this relation to argue that