Question #43712

A spacecraft of mass $M$ (in $\text{kg}$) is traveling in deep space along the positive $x$-direction with a velocity $v_0$ (in $\text{m/s}$). It is propelled by a rocket engine that ejects exhaust gas at a constant mass rate $\mu = -\frac{dM}{dt}$ (in $\text{kg/s}$) with a constant relative speed $u$ (in $\text{m/s}$) in the negative $x$-direction. At $t=0$, the spacecraft encounters a localized 'fusion barrier' that exerts an instantaneous repulsive impulse of magnitude $J$ (in $\text{N}\cdot\text{s}$) in the negative $x$-direction. Assuming the mass $M$ remains constant during the infinitesimally short duration of the impulse, what is the instantaneous change in the rate of change of the spacecraft's kinetic energy, $\Delta \left( \frac{dK}{dt} \right) = \left. \frac{dK}{dt} \right|_{t=0^+} - \left. \frac{dK}{dt} \right|_{t=0^-}$?