An object of mass $3\,m$ at rest splits into three equal fragments. Two fragments have velocities $v\hat j$ and $v\hat i$. The velocity of the third fragment is

$A$ block of mass $10 \ kg$ moving with a velocity of $5 \hat{i} \ m \ s^{-1}$ on a frictionless horizontal surface suddenly explodes into two pieces. If one piece with mass $4 \ kg$ moves with a velocity of $10 \hat{i} \ m \ s^{-1}$,then the velocity of the second piece is:

$A$ man of mass $m_1$ is standing on a platform of mass $m_2$ kept on a smooth horizontal surface. If the man starts moving on the platform with a velocity $v$ relative to the platform,then the recoil velocity of the platform is:

$A$ stationary body explodes into two parts of masses $M_{1}$ and $M_{2}$. They move in opposite directions with velocities $v_{1}$ and $v_{2}$. The ratio of their kinetic energies is

$A$ projectile moving vertically upwards with a velocity of $200\, ms^{-1}$ breaks into two equal parts at a height of $490\, m$. One part starts moving vertically upwards with a velocity of $400\, ms^{-1}$. How much time will it take,after the break up,for the other part to hit the ground? .............. $s$

Two persons of mass $m_1$ and $m_2$ are standing at the two ends $A$ and $B$ respectively,of a trolley of mass $M$ as shown. When both the persons jump simultaneously with $u_{rel}$ with respect to the trolley,then the velocity of the trolley is