An approximate approach to the description of the deep penetration of a plastic medium by elongated bodies [1] is developed. The results obtained in [2–5] are generalized to another geometry and to a wider range of velocities, which is subdivided into intervals. An attempt is made to provide an averaged description of the motion in each of the intervals assuming quasistationarity, and the formula for the depth of penetration is analysed. Lavrent'yev's idea on the closeness of the velocity and stress fields during the high-velocity motions of bodies in a solid medium and in a liquid, which has been translated into the language of asymptotic representations [5], has been further developed in an hypothesis on the closeness of the configurations of bodies of minimum resistance in a plastic medium and a liquid subject to the condition of the smallness of the ratio of the strength (resistance) force to the hydrodynamic drag force. The advantage of tubular bodies compared with solid bodies with respect to mass and depth of penetration is indicated. The bounds of the limiting penetration depths are estimated.