 Constant-area capture of stationary dust

A body of initial mass $M_0$ and speed $v_0$ collects all stationary dust swept out by constant area $A$ in a uniform cloud of mass density $\rho$. Then $M=M_0+\rho Ax$ and $Mv=M_0{v}_0$ by conservation of momentum. Integration gives the displayed mass law, $v=M_0{v}_0/M$ and $x=(M-M_0)/(\rho A)$. Sticking dissipates kinetic energy at rate $\dot{M}{v}^2/2$. The model assumes no external longitudinal force and negligible relativistic corrections.