The relaxation of the most important heavy-particle velocity averages of a Rayleigh gas is studied both exactly in the Maxwell model and, by using the approximate theory based on the usual Fokker-Planck equation, in the Rayleigh limit. It is pointed out that this usual (approximate) kinetic equation yields, in the Maxwell model, the exact relaxation of the heavy-particle mean velocity, and a very good description of the relaxation of the heavy-particle mean square speed, irrespectively of the fact that in several physical situations the same kinetic equation cannot correctly describe the relaxation of the heavy-particle velocity distribution. It is found, moreover, that in certain situations the heavy-particle mean-square velocity components relax, in the approximate theory, in an unphysical way, unless an appropriate limitation on the initial heavy-particle mean velocity is imposed.