The drift velocity of a particle under a driving force has its roots in the theory of electrical conduction. Although it has been studied for over 100 years, it still yields surprises. At the heart of a particle’s drift velocity is an interplay of classical, quantum, and statistical mechanics. Irreversibility and energy loss have been assumed as essential features of drift velocities and very little effort has been made to isolate the aspects of particle transport that are due to elastic mechanisms alone. In this paper, we remove energy loss and quantum mechanics to investigate the classical and statistical factors that can produce a drift velocity using only elastic scattering. A Monte Carlo simulation is used to model a particle in a uniform force field, subject to randomly placed scatterers. Time-, space-, and energy-dependent scattering models, with varied ranges of scattering angles, are investigated. A constant drift velocity is achieved with the time scattering model, which has a constant average time between scattering events. A decreasing drift velocity is observed for space and energy-dependent models. The arrival of a constant drift velocity has to do with a balance of momentum gained between collisions and momentum lost after a collision.