 Mass-loading power at fixed velocity

A body moving at constant velocity $V$ incorporates material at rate $\dot{M}$ with zero initial momentum along its motion. The required external force is $F=\dot{M}V$, so the supplied power is $P=\dot{M}{V}^2$. Only $\dot{M}{V}^2/2$ becomes the added material's translational kinetic energy; the other half is dissipated during inelastic incorporation. This conclusion concerns the component along the body's motion; an incoming transverse velocity can add further dissipation.