$A$ metal ball falls from a height of $32 \ m$ on a steel plate. If the coefficient of restitution is $0.5$,to what height will the ball rise after the second bounce?

Ball $1$ collides with another identical ball at rest. For what value of coefficient of restitution $e,$ the velocity of the second ball becomes two times that of the first ball after the collision?

Two pendulums with identical bobs and lengths are suspended from a common support such that in rest position the two bobs are in contact (figure). One of the bobs is released after being displaced by $10^{\circ}$ so that it collides elastically head-on with the other bob.
$(a)$ Describe the motion of two bobs.
$(b)$ Draw a graph showing variation in energy of either pendulum with time,for $0 \leqslant t \leqslant 2T$,where $T$ is the period of each pendulum.

$A$ ball strikes a smooth horizontal ground at an angle of $45^o$ with the vertical. What cannot be the possible angle of its velocity with the vertical after the collision? (Assume $e \leq 1$ ).

$A$ uniform rod of mass $M$ and length $L$ is placed on a smooth horizontal surface. $A$ particle of mass $m$ strikes the rod at one end with a velocity $v$ perpendicular to the rod. If the particle comes to rest after the collision,the velocity of the center of mass of the rod after the collision is:

$A$ ball strikes a horizontal surface as shown in the figure. If the coefficient of restitution is $e = 1/\sqrt{3}$,then the angle $\theta$ is ............... $^o$.