Two balls $A$ and $B$ having masses $1 \ kg$ and $2 \ kg$,moving with speeds $21 \ m/s$ and $4 \ m/s$ respectively in opposite directions,collide head-on. After the collision,$A$ moves with a speed of $1 \ m/s$ in the same direction. The coefficient of restitution is:

$A$ smooth sphere moves on a horizontal surface with a velocity vector $2\hat{i} + 2\hat{j}$ before it strikes a vertical wall. The wall is parallel to the $\hat{j}$ vector,and the coefficient of restitution between the sphere and the wall is $e = 1/2$. What will be the velocity vector of the sphere after it strikes the wall?

$A$ ball is dropped from a height of $20 \ m$. If it loses $20\%$ of its energy upon collision with the ground,what is the coefficient of restitution?

$A$ body of mass $m$ moving with velocity $v$ collides inelastically with another body of the same mass. After the collision,the first body moves with velocity $\frac{v}{\sqrt{3}}$ in a direction perpendicular to the initial direction of motion. What will be the speed of the second body after the collision?

$A$ sphere of mass $m$ moving with velocity $u$ collides with another sphere of the same mass $m$ at rest. If the coefficient of restitution is $e = \frac{1}{2}$, what is the ratio of their velocities after the collision?

$A$ bullet of mass $0.01\, kg$ travelling at a speed of $500\, m/s$ strikes and passes horizontally through a block of mass $2\, kg$ which is suspended by a string of length $5\, m$. The centre of gravity of the block is found to rise a vertical distance of $0.1\, m$. What is the speed of the bullet after it emerges from the block? (The time of passing of the bullet is negligible,$g = 9.8\, m/s^2$)