$A$ projectile moving with velocity $v$ in space bursts into $2$ parts with masses in the ratio $1 : 3$. The smaller part becomes stationary. What is the velocity of the other part?

$A$ cannon shell fired breaks into two equal parts at its highest point. One part retraces the path to the cannon with kinetic energy $E_1$ and the kinetic energy of the second part is $E_2$. The relation between $E_1$ and $E_2$ is:

$A$ bomb at rest explodes into three equal fragments. Two of the fragments move at right angles to each other with velocities of $9 \ m s^{-1}$ and $12 \ m s^{-1}$. The magnitude of the velocity of the third fragment is ....... $m s^{-1}$.

$A$ body of mass $M$ at rest explodes into three pieces,in the ratio of masses $1: 1: 2$. Two smaller pieces fly off perpendicular to each other with velocities of $30 \, m/s$ and $40 \, m/s$ respectively. The velocity of the third piece will be ............... $\, m/s$.

$A$ bomb of mass $10\, kg$ explodes into two pieces of masses $4\, kg$ and $6\, kg$. If the kinetic energy of the $4\, kg$ piece is $200\, J$,find the kinetic energy of the $6\, kg$ piece.

An artillery piece of mass $M_1$ fires a shell of mass $M_2$ horizontally. Instantaneously after the firing,the ratio of kinetic energy of the artillery and that of the shell is: