Question

A cannon of mass $M$ starts sliding freely down a smooth inclined plane at an angle $\alpha$ to the horizontal. After the cannon covered the distance $l$, a shot was fired, the shell leaving the cannon in the horizontal direction with a momentum $\overrightarrow{p}$. As a consequence, the cannon stopped. Assuming the mass of the shell to be negligible,as compared to that of the cannon, determine the duration of the shot.

Answer 1

Velocity of cannon as well as that of shell equals $\sqrt{2gl\sin \alpha }$ down the inclined plane taken as the positive x-axis. From the linear impulse momentum theorem in projection form along x-axis for the system (connon $+$ shell) i.e. $\Delta p_x=F_x\Delta t$,
$p\cos \alpha -M\sqrt{2gl\sin \alpha }=Mg\sin \alpha \Delta t$ (as mass of the shell is negligible)
or, $\Delta t=\frac{p\cos \alpha -M\sqrt{2gl\sin \alpha }}{Mg\sin \alpha }$