In the present paper, a physical model is proposed for reducing the problem of the drag reduction of an attached bow shock around the nose of a high-speed vehicle with on-board discharge, to the problem of a balance between the magnetic pressure and gas pressure of plane shock of a partially ionized gas consisting of the environmental gas around the nose of the vehicle and the on-board discharge-produced plasma. The relation between the shock strength and the discharge-induced magnetic pressure is studied by means of a set of one-fluid, hydromagnetic equations reformed for the present purpose, where the discharge-induced magnetic field consists of the electron current (produced by the discharge)-induced magnetic field and the partially ionized gas flow-induced one. A formula for the relation between the above parameters is derived. It shows that the discharge-induced magnetic pressure can minimize the shock strength,successfully explaining the two recent experimental observations on attached bow shock mitigation and elimination in a supersonic flow during on-board discharge [Phys. Plasmas 9 (2002) 721 and Phys. Plasmas 7 (2000) 1345]. In addition,the formula implies that the shock elimination leaves room for a layer of higher-density plasma rampart moving around the nose of the vehicle, being favourable to the plasma radar cloaking of the vehicle. The reason for it is expounded.