In this paper the extended penalty function approach to dynamic contact problems recently proposed by Asano is further discussed. The equation of motion, in which velocities as well as displacements are constrained by the physical consideration, coincides with the one based on the virtual work principle given by Asano. The optimal penalty numbers in this approach are discussed by using a simple two-element model. The obtained results are applied to the finite element analysis of contact-impact problem of two elastic rods and the validity of the present method is confirmed through the comparison of the numerical results with the exact solution based on the one-dimensional wave propagation theory. The present method is also applied to the friction-excited vibration of a two-dimensional elastic block on a sliding belt, and it is shown that the kinematic coefficient of friction can be lower than the static one in the camplete system even when they are assumed to be equal to each other on the interface between the block and the belt.