$A$ ball of mass $10 \text{ g}$ is allowed to fall from a height of $10 \text{ m}$. After the collision with the ground, if $50\%$ of its energy is lost, then the height reached by the ball is: (in $\text{ m}$)

$A$ 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 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 .............. $s$.

$A$ ball is dropped from a height of $5 \ m$ on a planet. It rebounds to a height of $1.8 \ m$. What is the fraction of velocity lost by the ball?

$A$ smooth sphere is moving on a horizontal surface with a velocity vector $(2 \hat{i} + 2 \hat{j}) \ m/s$ immediately before it hits a vertical wall. The wall is parallel to the vector $\hat{j}$ and the coefficient of restitution between the sphere and the wall is $e = 1/2$. The velocity of the sphere after it hits the wall is:

$A$ particle of mass $m$ strikes the ground inelastically with a coefficient of restitution $e$. Find the relation between the angles $\theta$ and $\alpha$ as shown in the figure.