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The impact of a rigid body and a rod of finite length made of a viscoplastic incompressible material moving towards each other at constant velocity is studied in the article. One-dimensional motion is considered, i.e., velocity, stress, and other parameters are considered to be averaged over the rod section. According to the model of a viscous-plastic material, the rod is divided into two parts: the region of plastic strain and the region that moves as a solid body after impact. The boundary between these regions is unknown and needs to be determined. As a result, we arrive at a problem with a free boundary, a well-known example of which is the Stefan problem. However, the formulation of the problem under consideration differs significantly from the Stefan problem. In addition, the function to be determined that describes the unknown boundary, in contrast to the Stefan problem, is not explicitly present in the boundary conditions. There are various numerical methods for solving problems with an unknown moving boundary. However, as is known, implementing these methods is associated with significant difficulties. In this article, the method of integral relations is used - a modification of the Karman-Pohlhausen method known in the boundary layer theory. The problem is reduced to the Cauchy problem for a system of nonlinear ordinary differential equations proposed to be solved by successive approximations and the Runge-Kutta method. Numerical calculations were performed. The influence of changes in the mass and magnitude of the velocities of the rod and rigid body on the size of the plastic strain region and the change in stresses on the contact surface is revealed.