This paper deals with a numerical analysis of a one-dimensional dynamic purely elastic (i.e. hyperbolic) model with dry friction. Since we consider a Coulomb friction law with a slip velocity dependent coefficient, generally, the problem has more than one solution. A mass perturbation approach is developed to regain the uniqueness and to perform the numerical analysis. This study can be viewed as a first step in the numerical analysis of more elaborated dynamic purely elastic problems with dry friction.