$A$ body moving with velocity $v$ has momentum and kinetic energy numerically equal. What is the value of $v$ in $m/s$?

$A$ shell is fired from a cannon with velocity $v \text{ m/s}$ at an angle $\theta$ with the horizontal direction. At the highest point in its path,it explodes into two pieces of equal mass. One of the pieces retraces its path to the cannon. The speed in $\text{m/s}$ of the other piece immediately after the explosion is:

$A$ ball of mass $m$ is dropped from a height $s$ on a horizontal platform fixed at the top of a vertical spring. The platform is depressed by a distance $h$. The spring constant is ($g=$ acceleration due to gravity):

$A$ bullet of mass $10 \ g$ pierces through a plate $A$ of mass $500 \ g$ and then gets embedded into a second plate $B$ of mass $1.49 \ kg$ as shown in the figure. Initially,the two plates $A$ and $B$ are at rest and move with the same velocity after the collision. The percentage loss in the initial kinetic energy of the bullet when it is between the plates $A$ and $B$ is . . . . . . (Neglect any loss of material of the plates during the collision).

Is it possible for the kinetic energy of an object not to increase when work is done on it? If so,when?

Four particles $A, B, C$ and $D$ of equal mass $m$ are placed at four corners of a square. They move with equal uniform speed $v$ towards the intersection of the diagonals. After collision,$A$ comes to rest,$B$ traces its path back with the same speed $v$,and $C$ and $D$ move with equal speeds $v'$. What is the velocity of $C$ after collision?