A spatial motion of a large passive satellite (space debris) and a space tug connected by an elastic tether is considered. The motion of the system is excited by a thrust force acting on the space tug. Major attention is given to a derivation of the equations of the motion based on the Lagrange formalism. Correctness of the mathematical model is proved by the theorem on variation of angular momentum and by the numerical simulations. The influence of the initial conditions and the system parameters on the behavior of the passive satellite is studied. The possibility of critical modes of the system motion leading to entanglement of the tether is shown by means of the numerical simulations.