We consider motion of a system consisting of a viscoelastic rod of finite length, described by the fractional Zener model and a body attached to its free end (the other end of a rod is fixed). The motion of a rod is described by the system of partial differential equations of integer and fractional order. The motion of a body is translatory along the axis of the rod and it represents the boundary condition for the equations describing the motion of a rod. The initial-boundary value system is solved by the use of the Laplace transform and is given in the convolution form. Solutions are tested numerically.