We study a simple one-and-a-half degree-of-freedom impact oscillator in which the dynamical behavior is characterized by two regimes — modeling contact and non-contact dynamics, subject to a description of the impactive event. The contact is modeled as a linearly elastic element and the time interval over which contact occurs is of finite duration. This model allows for a coefficient of restitution to be defined which depends on not only the system parameters, but the relative velocity at the point of impact. For sufficiently small impact velocities the colliding bodies remain in contact while for larger relative speeds, the coefficient of restitution is positive. With this model for the collision, an impact oscillator is described and is shown to contain rich dynamical behavior, including periodic and non-periodic motions.