Two wooden blocks of mass $M_1$ and $M_2$ rest on a frictionless table. $A$ bullet of mass $m$ is fired at $M_1$ with speed $v$,which gets embedded in it,and the two together finally collide with $M_2$. Find the velocity of $M_2$ after the collision. (Assume the collision between the $(M_1+m)$ system and $M_2$ is elastic and treat the problem as one-dimensional.)

$A$ ball is allowed to fall from a height of $10 \, m$. If there is $40 \%$ loss of energy due to impact,then after one impact the ball will rise to a height of ........ $m$.

$A$ small bob $A$ of mass $m$ is attached to a massless rigid rod of length $1 \ m$ pivoted at point $P$ and kept at an angle of $60^{\circ}$ with the vertical as shown in the figure. At a distance of $1 \ m$ below point $P$,an identical bob $B$ is kept at rest on a smooth horizontal surface that extends to a circular track of radius $R$ as shown in the figure. If bob $B$ just manages to complete the circular path of radius $R$ up to a point $Q$ after being hit elastically by bob $A$,then the radius $R$ is . . . . . . $m$.

$A$ ball loses $15.0\%$ of its kinetic energy when it bounces back from a concrete wall. With what speed must you throw it vertically down from a height of $12.4\, m$ to have it bounce back to the same height (ignore air resistance)? ............. $m / s$

Two particles $P$ and $Q$ each of mass $3m$ lie at rest on the $X$-axis at points $(-a, 0)$ and $(+a, 0)$,respectively. $A$ third particle $R$ of mass $2m$ initially at the origin moves towards the particle $Q$ with velocity $v$. If all the collisions of the system of $3$ particles are elastic and head-on,the total number of collisions in the system is

$A$ sphere of mass $m$ moving with a constant velocity $u$ hits another stationary sphere of the same mass. If $e$ is the coefficient of restitution,then the ratio of the velocity of the two spheres after the collision will be