$A$ ball is dropped from a height of $80 \ m$ on a surface which is at rest. Find the height attained by the ball after the $2^{nd}$ collision if the coefficient of restitution is $e = 0.5$.

$A$ ball $A$ of mass $m$ moving along the positive $x$-direction with kinetic energy $K$ and momentum $p$ undergoes an elastic head-on collision with a stationary ball $B$ of mass $M$. After the collision,the ball $A$ moves along the negative $x$-direction with kinetic energy $K/9$. The final momentum of $B$ is:

$A$ body of mass $m$ moving with velocity $v$ collides with a stationary body of mass $2m$. The fraction of kinetic energy lost by the body of mass $m$ is:

$A$ body of mass $m$ is at rest. Another body of the same mass moving with velocity $V$ makes a head-on elastic collision with the first body. After the collision,the first body starts to move with velocity:

$Assertion$ : If collision occurs between two elastic bodies,their kinetic energy decreases during the time of collision.
$Reason$ : During collision,intermolecular space decreases and hence elastic potential energy increases.

$A$ particle of mass $m$ with an initial velocity $u\hat{i}$ collides perfectly elastically with a mass $3m$ at rest. It moves with a velocity $v\hat{j}$ after collision. Then, $v$ is given by: