In a one-dimensional collision between two identical particles $A$ and $B$, $B$ is stationary and $A$ has momentum $p$ before impact. During impact, $B$ gives impulse $J$ to $A$.

Equal force $F (> mg)$ is applied to the string in all the $3$ cases. Starting from rest,the point of application of force moves a distance of $2 \ m$ down in all cases. In which case does the block have maximum kinetic energy?

If the kinetic energy of a body is directly proportional to time $t,$ the magnitude of force acting on the body is
$(i)$ directly proportional to $\sqrt{t}$
$(ii)$ inversely proportional to $\sqrt{t}$
$(iii)$ directly proportional to the speed of the body
$(iv)$ inversely proportional to the speed of the body

The block starts from rest as shown in the figure. Find the work done by the force of $10 \,N$ and friction in the time $0$ to $4 \,s$. [Take, $g=10 \,m/s^2$]

$A$ small block of mass $200 \ g$ is placed on a horizontal slab at a height of $2 \ m$ above the floor. The block is pressed against a horizontal spring fixed at one end to compress the spring by $10.0 \ cm$. Upon releasing, the block moves horizontally until it leaves the spring. Calculate the horizontal distance covered by the block after leaving the slab and just before hitting the ground. The spring constant is $50 \ N \ m^{-1}$. (Assume $g = 10 \ m \ s^{-2}$) (in $m$)

As shown below, bob $A$ of a pendulum having a massless string of length $R$ is released from $60^{\circ}$ to the vertical. It hits another bob $B$ of half the mass that is at rest on a frictionless table in the center. Assuming an elastic collision, the magnitude of the velocity of bob $A$ after the collision will be (take $g$ as acceleration due to gravity):