$A$ buggy of mass $100\, kg$ is free to move on a frictionless horizontal track. Two men,each of mass $50\, kg$,are standing on the buggy,which is initially stationary. The men jump off the buggy with a velocity of $10\, m/s$ relative to the buggy. In one situation,the men jump one after the other. In another situation,the men jump simultaneously. What is the ratio of the recoil velocities of the buggy in the two cases?

$A$ rifleman,who together with his rifle has a mass of $100\,kg$,stands on a smooth surface and fires $10$ shots horizontally. Each bullet has a mass of $10\,g$ and a muzzle velocity of $800\,ms^{-1}$. The velocity which the rifleman attains after firing $10$ shots is $..........\,ms^{-1}$.

$A$ person of mass $80 \ kg$ is standing on a trolley of mass $320 \ kg$. The trolley is at rest on a frictionless horizontal rail. If the person walks on the trolley at a speed of $1 \ m/s$,what will be their displacement relative to the ground after $4 \ s$?

$A$ bullet is fired from a rifle,and the rifle experiences recoil. What will be the kinetic energy of the recoiling rifle?

$A$ ball of mass $10 \, kg$ moving with a velocity $10 \sqrt{3} \, m/s$ along the $X$-axis,hits another ball of mass $20 \, kg$ which is at rest. After the collision,the first ball comes to rest and the second one disintegrates into two equal pieces. One of the pieces starts moving along the $Y$-axis at a speed of $10 \, m/s$. The second piece starts moving at a speed of $20 \, m/s$ at an angle $\theta$ (in degrees) with respect to the $X$-axis. The configuration of pieces after the collision is shown in the figure. The value of $\theta$ to the nearest integer is:

$A$ ball of mass $200\,g$ rests on a vertical post of height $20\,m$. $A$ bullet of mass $10\,g$,travelling in a horizontal direction,hits the centre of the ball. After the collision,both travel independently. The ball hits the ground at a distance of $30\,m$ and the bullet at a distance of $120\,m$ from the foot of the post. The value of the initial velocity of the bullet will be $............m/s$ (if $g = 10\,m/s^2$).