$A$ smooth sphere of mass $m$ moving with velocity $u$ collides with another smooth sphere of mass $2m$ at rest. What is the range of the velocity $v$ of the second sphere after the collision?

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?

Two statements are given below. Select the option that correctly explains both statements.
Statement-$1$: In a perfectly elastic collision between two particles moving in the same direction,they do not lose all their energy.
Statement-$2$: The principle of conservation of momentum is valid for all types of collisions.

Two cars,both of mass $m$,collide and stick together. Prior to the collision,one car had been traveling north at speed $2v$,while the second was traveling at speed $v$ at an angle $\phi$ south of east (as indicated in the figure). The magnitude of the velocity of the two-car system immediately after the collision is:

An engine pumps up $100 \ kg$ of water through a height of $10 \ m$ in $5 \ s$. Given that the efficiency of the engine is $60\%$. If $g = 10 \ m \ s^{-2}$,the power of the engine is .............. $kW$.

$A$ particle $P$ is sliding down a frictionless hemispherical bowl. It passes the point $A$ at $t = 0$. At this instant of time, the horizontal component of its velocity is $v$. $A$ bead $Q$ of the same mass as $P$ is ejected from $A$ at $t = 0$ along the horizontal string $AB$ (see figure) with the speed $v$. Friction between the bead and the string may be neglected. Let ${t_P}$ and ${t_Q}$ be the respective time taken by $P$ and $Q$ to reach the point $B$. Then