Two bodies $A$ and $B$ of masses $2m$ and $m$ are projected vertically upwards from the ground with velocities $u$ and $2u$ respectively. The ratio of the kinetic energy of body $A$ and the potential energy of body $B$ at a height equal to half of the maximum height reached by body $A$ is (in $ : 1$)

In the figure shown,two identical balls of mass $M$ and radius $R$ each are placed in contact with each other on a frictionless horizontal surface. $A$ third ball of mass $M$ and radius $R$ is coming down vertically with a velocity $v_0$. It simultaneously hits the two balls. The smaller ball (top ball) does not stop after the collision but continues to move downwards with a speed $v_0/2$. Find the speed of each bigger ball after the collision.

$A$ particle of mass $m$ moving eastward with a speed $v$ collides with another particle of same mass moving northward with same speed $v$. The two particles coalesce after collision. The new particle of mass $2m$ will move in north-east direction with a speed (in $m/s$) of:

It is well known that a raindrop falls under the influence of the downward gravitational force and the opposing resistive force. The latter is known to be proportional to the speed of the drop but is otherwise undetermined. Consider a drop of mass $1.00 \; g$ falling from a height $1.00 \; km$. It hits the ground with a speed of $50.0 \; m s^{-1}$. $(a)$ What is the work done by the gravitational force? $(b)$ What is the work done by the unknown resistive force?

$A$ spherical bowl of radius $R = 2 \ m$ is placed on a horizontal plane. $A$ particle of mass $m = 1 \ g$ oscillates on its inner surface. If the particle starts from a point on the bowl at a height of $h = 1 \ cm$ from the horizontal plane and the coefficient of friction is $\mu = 0.01$,find the total distance $s$ traveled by the particle before it comes to rest at the bottom of the bowl. (in $m$)

$A$ shell at rest on a smooth horizontal surface explodes into two fragments of masses $m_1$ and $m_2$. If just after explosion $m_1$ moves with speed $u$,then the work done by internal forces during the explosion is: