Question

A plate of mass M is held in air by firing bullets each of mass m with same velocity in vertically upward direction. If n bullets are fired per minute and each bullet strikes the plate with speed v and rebounds with half the speed with which it strikes the plate, then the speed with which bullets are striking the plate is [Take g = 10m/s^2]

Answer 1

Dear Student,
Answer to your question is given below:

$$\begin{gathered} Givenmassofbullet=m \\ andmassofPlate=M \\ Nowchangeinmomentumofbulletwillgivetheimpulseimpartedtoplate, \\ InitialVelocityofbullet=vinm/s \\ Finalvelocityofbullet=-\frac{v}{2}m/s \\ Changeinmomentumofbullet=mv-(-m\frac{v}{2}) \\ \Rightarrow dP=\frac{3mv}{2} \\ Nowchangeinmomentum,Fdt=dP \\ \Rightarrow F=\frac{n}{60}\times \frac{3mv}{2} \\ andFshouldbalancetheweightofPlate \\ Mg=\frac{n}{60}\times \frac{3mv}{2} \\ \Rightarrow v=\frac{40Mg}{nm} \\ Therforethespeedwithwhichbulletsarestrikingtheplateis\frac{40Mg}{nm} \end{gathered}$$
Regards,