We study the mean exit time of a free inertial random process from a region in space. The acceleration alternatively takes the values +[ital a] and [minus][ital a] for random periods of time governed by a common distribution [psi]([ital t]). The mean exit time satisfies an integral equation that reduces to a partial differential equation if the random acceleration is Markovian. Some qualitative features of the behavior of the system are discussed and checked by simulations. Among these features, the most striking is the discontinuity of the mean exit time as a function of the initial conditions.