A simplified mechanical model of vehicle towing is constructed. The towing procedure is modeled by means of two blocks connected with a linear spring that represents the elastic drawbar. The driver in the towing car controls the velocity of towing, while the driver of the towed vehicle tries to maintain a constant tension in the drawbar. Both control forces are subjected to a constant time delay that represents the reaction time of the operators. Utilizing the translational symmetry of the system, Routh reduction is carried out to make the governing equations as compact as possible. Stability charts are constructed in the space of the control parameters and the fastest decay of signals is identified. As time delay increases, the stable domain disappears and reappears periodically. Guidelines are provided for the design of the elasticity of the drawbar, which takes into account the masses of the vehicles and the average reaction time of the drivers.