Past exam of the physics 1 paper of the University of Oxford / 2013 / 11 / e

For constant-area capture of stationary dust, let $x=0$ at cloud entry, $M(0)=M_p$, $v(0)=v_{\infty }$, and define $k=\rho A$ with fixed collecting area $A$. Every swept volume $Adx$ adds stationary mass, soThere is no external longitudinal force, and the incoming dust has zero momentum in the cloud frame. The variable-mass system therefore satisfies $d(Mv)/dt=0$, rather than $M\dot{v}=0$. Using $Mv=M_p{v}_{\infty }$ givesHenceThe kinetic energy decreases during sticking: $dK/dt=-\frac{1}{2}\dot{M}{v}^2$. The lost energy becomes internal energy. Radiation, gravity, sputtering and changes in area are excluded by the ideal complete-collection model. The zero-density limit recovers $M=M_p$, $v=v_{\infty }$ and $x=v_{\infty }t$.