$A$ ball of mass $1.2 \ kg$ moving with a velocity of $12 \ ms^{-1}$ makes a one-dimensional collision with another stationary ball of mass $1.2 \ kg$. If the coefficient of restitution is $\frac{1}{\sqrt{2}}$,then the ratio of the total kinetic energy of the balls after the collision to the initial kinetic energy is:

$A$ bullet of mass $4.2 \times 10^{-2} \text{ kg}$, moving at a speed of $300 \text{ m/s}$, gets stuck into a block with a mass $9$ times that of the bullet. If the block is free to move without any kind of friction, the heat generated in the process will be (in $\text{ cal}$)

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 $P$ of mass $0.5 \ kg$ moving with a velocity of $10 \ ms^{-1}$ collides with another ball $Q$ of mass $1 \ kg$ at rest. If the coefficient of restitution is $0.4$,the ratio of the velocities of the balls $P$ and $Q$ after the collision is

$A$ cricket ball hits the ground with a speed of $5 \,m \,s^{-1}$ making an angle of $30^{\circ}$ with the horizontal. If the coefficient of restitution is $0.2$, the speed of the reflected ball in $m \,s^{-1}$ is

$A$ ball falling freely from a height of $4.9 \, m$ hits a horizontal surface. If the coefficient of restitution is $e = \frac{3}{4}$,then the ball will hit the surface for the second time after .............. $s$.