Impact problems arise in many practical applications. The need for obtaining an accurate model for the inelastic impact is a challenging problem. In general, two approaches are common in solving the impact problems: the impulse-momentum and the compliance based methods. The former approach included the coefficient of restitution which provides a mechanism to solve the problem explicitly. While the compliance methods are generally tailored to solve elastic problems, researchers in the field have proposed several mechanisms to include inelastic losses. In this paper, we present correlations between the coefficient of restitution in the impulse-momentum based method and the contact stiffness in the compliance methods. We conducted numerical analysis to show that the resulting solutions are indeed identical for a specific range of impact conditions. The impulse-momentum based model is considered as a reference case to compare the post impact velocities. The numerical results showed that, the impulse-momentum and the compliance based methods can produce similar outcomes for specific range of coefficient of restitution if they satisfied a set of end conditions. The correlations lead to introduce a new contact force model with hysteresis damping for low coefficient of restitution impact.