Two identical blocks $A$ and $B$,each of mass $m$,resting on a smooth horizontal floor,are connected by a light spring of natural length $L$ and spring constant $K$. $A$ third block $C$ of mass $m$,moving with a speed $v$ along the line joining $A$ and $B$,collides with $A$. The maximum compression in the spring is:

$A$ man of mass $60$ $kg$ wants to lose $5$ $kg$ of mass by climbing up and down stairs. Assume that twice the amount of fat is burned while climbing up compared to climbing down. If burning $1$ $kg$ of fat provides $7000$ $kcal$ of energy, how many times must he climb up and down the stairs to lose $5$ $kg$ of mass (in $\times$)? (Assume the energy spent in climbing up is $20$ $kcal$ and climbing down is $10$ $kcal$ per trip).

$A$ bob $B$ of mass $m$ at rest is hanging vertically from the ceiling via a massless string of length $10 \text{ m}$. $A$ point mass $A$ of mass $m$ travelling horizontally with speed $10 \text{ ms}^{-1}$ hits bob $B$ elastically. The bob $B$ rises $h$ meters after the collision. Taking the acceleration due to gravity $g = 10 \text{ ms}^{-2}$ and neglecting the size of the bob, the value of $h$ is:

Two particles,of masses $M$ and $2M$,moving as shown,with speeds of $10\, m/s$ and $5\, m/s$,collide elastically at the origin. After the collision,they move along the indicated directions with speeds $v_1$ and $v_2$ respectively. The values of $v_1$ and $v_2$ are nearly:

$A$ body of mass $0.5 \ kg$ is supplied with a power $P$ (in watt) which varies with time $t$ (in second) as $P = 3t^2 + 3$. If the velocity of the body at time $t = 0$ is zero, then the velocity of the body at time $t = 3 \ s$ is (in $ms^{-1}$)

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