$A$ 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?

An explosion breaks a rock into three parts in a horizontal plane. Two of them go off at right angles to each other. The first part of mass $1 \, kg$ moves with a speed of $12 \, m s^{-1}$ and the second part of mass $2 \, kg$ moves with $8 \, m s^{-1}$ speed. If the third part flies off with $4 \, m s^{-1}$ speed,then its mass is ......... $kg$.

At high altitude, a body at rest explodes into two equal fragments. One fragment receives a horizontal velocity of $10 \,ms^{-1}$. The time taken by the two position vectors connecting the point of explosion to the fragments to make an angle of $90^{\circ}$ is (in $\,s$)

$A$ shell of mass $0.020 \; kg$ is fired by a gun of mass $100 \; kg$. If the muzzle speed of the shell is $80 \; m s^{-1}$,what is the recoil speed in $m/s$ of the gun?

$A$ stationary particle explodes into two particles of masses $x$ and $y$,which move in opposite directions with velocities $v_1$ and $v_2$. The ratio of their kinetic energies $(E_1: E_2)$ is

Two masses $A$ and $B$ of mass $M$ and $2M$ respectively are connected by a compressed ideal spring. The system is placed on a horizontal frictionless table and given a velocity $u \hat{k}$ in the $z$-direction as shown in the figure. The spring is then released. In the subsequent motion,the line from $B$ to $A$ always points along the $\hat{i}$ unit vector. At some instant of time,mass $B$ has an $x$-component of velocity as $V_x \hat{i}$. The velocity $\vec{V}_A$ of mass $A$ at that instant is: