In this paper the important for the design of shock-absorbers case of a perfectly rigid body impact with a viscoelastic incompressible body is considered. Mathematical apparatus for calculating the parameters of impact of a rigid body with a highly- elastic rod is developed, taking into account the peculiarities of the behavior of elastomers under fast loading. An analytical solution was obtained for longitudinal impact on a vertically disposed rod. The solution was fulfilled by means of the Bubnov- Galerkin variational method, reducing the problem to solving of the integro-differential equation for given boundary and initial conditions The solution is received for the exponential relaxation kernel for the Maxwell model of a highly-elastic material, it describes the process of damped longitudinal vibration, taking into account the effect of creep caused by the instantaneous impact loading. Based on the equation of vibratory motion the equations of velocity and acceleration are received, which are used for stress-strain behavior analysis of rubber damper. A numerical example of an axial tensile impact on a viscoelastic shock-absorber in the form of a cylindrical rod is given with the plots of time dependence of displacement, velocity, and acceleration of the impact end of the rod are provided, the possibility of using them for analyzing the shock absorber is demonstrated.