$A$ sphere of mass $m$ travelling at constant speed $v$ strikes another sphere of the same mass. If the coefficient of restitution is $e$,then the ratio of the velocities of both spheres just after the collision is:

When a ball is freely fallen from a given height,it bounces to $80\%$ of its original height. What fraction of its mechanical energy is lost in each bounce?

$A$ pendulum of length $2 \; m$ consists of a wooden bob of mass $50 \; g$. $A$ bullet of mass $75 \; g$ is fired towards the stationary bob with a speed $v$. The bullet emerges out of the bob with a speed $\frac{v}{3}$ and the bob just completes the vertical circle. The value of $v$ is $\dots \; ms^{-1}$. (if $g = 10 \; m/s^2$)

$A$ ball of mass $10\, kg$ moving with a velocity $10 \sqrt{3} \, m/s$ along the $x$-axis,hits another ball of mass $20\, kg$ which is at rest. After the collision,the first ball comes to rest while the second ball disintegrates into two equal pieces. One piece starts moving along the $y$-axis with a speed of $10 \, m/s$. The second piece starts moving at an angle of $30^{\circ}$ with respect to the $x$-axis. The velocity of the ball moving at $30^{\circ}$ with the $x$-axis is $x \, m/s$. The configuration of pieces after the collision is shown in the figure. The value of $x$ to the nearest integer is:

$A$ bullet of mass $0.012\;kg$ and horizontal speed $70\;m\;s^{-1}$ strikes a block of wood of mass $0.4\;kg$ and instantly comes to rest with respect to the block. The block is suspended from the ceiling by means of thin wires. Calculate the height to which the block rises. Also,estimate the amount of heat produced in the block.

Water falls from a height of $60 \, m$ at the rate of $15 \, kg/s$ to operate a turbine. The losses due to frictional force are $10 \%$ of the input energy. How much power is generated by the turbine? $(g=10 \, m/s^2)$ (In $kW$)