We consider a system that belongs to the class of well-known Painlev´e problems; to solve it, we use the method of opening spatio-temporal singularities proposed by the authors. This method makes it possible to unambiguously solve the problem of finding the speeds of a body after hitting a surface with dry friction, i.e., to resolve a situation where traditional approaches either yield an ambiguous answer or lead to an infeasible system of relations (the Painlev´e paradox).