Using a flexible model of scattering for atomic collisions and the continuous energy loss approximation for electronic stopping, we have obtained exact solutions of the time-energy distribution of atoms slowing down in cascade from a primary knock on. The time moments are constructed from which the slowing down time and relative variance may be calculated. Some results of Sanders and Winterbon are generalized and the connection between the backward equation of conventional damage theory and the partial adjoint of the forward equation are pointed out. We also discuss two complementary methods for calculating energy deposition by atoms as they slow down.