$A$ bomb of mass $3m \text{ kg}$ explodes into two pieces of mass $m \text{ kg}$ and $2m \text{ kg}$. If the velocity of the $m \text{ kg}$ mass is $16 \text{ m/s}$,the total kinetic energy released in the explosion is ................. $m \text{ J}$.

$A$ 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?

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 only the person standing at $B$ jumps from the trolley towards the right while the person at $A$ keeps standing,then

$A$ man of $60\,kg$ is running on the road and suddenly jumps into a stationary trolley car of mass $120\,kg$. Then,the trolley car starts moving with a velocity of $2\,ms^{-1}$. The velocity of the running man when he jumped into the car was . . . . . . $ms^{-1}$.

$A$ gun fires a small bullet with kinetic energy $K$. Then the kinetic energy of the gun while recoiling is

$A$ bomb moving with velocity $(40 \hat{i} + 50 \hat{j} - 25 \hat{k}) \text{ m/s}$ explodes into two pieces of mass ratio $1:4$. After the explosion,the smaller piece moves away with velocity $(200 \hat{i} + 70 \hat{j} + 15 \hat{k}) \text{ m/s}$. The velocity of the larger piece after the explosion is: