$A$ shell of mass $M$ initially at rest suddenly explodes into three fragments. Two of these fragments are of mass $M/4$ each, which move with velocities $3 \text{ m/s}$ and $4 \text{ m/s}$ respectively in mutually perpendicular directions. The magnitude of the velocity of the third fragment is: (in $\text{ m/s}$)

At high altitude,a body at rest explodes into two equal fragments,with one fragment receiving a horizontal velocity of $10 \, m/s$. The time taken by the two radius vectors connecting the point of explosion to the fragments to make an angle of $90^o$ is ............ $s$.

$A$ trolley of mass $200\; kg$ moves with a uniform speed of $36\; km/h$ on a frictionless track. $A$ child of mass $20\; kg$ runs on the trolley from one end to the other ($10\; m$ away) with a speed of $4\; m/s$ relative to the trolley in a direction opposite to its motion,and jumps out of the trolley. What is the final speed of the trolley? How much has the trolley moved from the time the child begins to run?

$A$ body at rest breaks into three fragments. Two fragments of equal mass fly off perpendicular to each other with a speed of $30 \; m/s$. The mass of the third fragment is three times that of the other two fragments. What will be the direction and magnitude of the velocity of the third fragment immediately after the explosion?

$A$ bomb at rest explodes into three fragments $X, Y$ and $Z$. Each one of them has the same mass. Which of the following correctly describes the motion of the fragments?

$A$ gun fires a bullet of mass $50 \ g$ with a velocity of $30 \ m \ s^{-1}$. Due to this,the gun recoils with a velocity of $1 \ m \ s^{-1}$. What is the mass of the gun in $kg$?