The velocity of the bullet becomes one-third after it penetrates $4 \ cm$ in a wooden block. Assuming that the bullet faces a constant resistance during its motion in the block,the bullet stops completely after traveling a total distance of $(4+x) \ cm$ inside the block. The value of $x$ is $.....$ (in $cm$)

$A$ ball of mass $m$ falls vertically to the ground from a height $h_1$ and rebounds to a height $h_2$. The change in momentum of the ball on striking the ground is

Calculate the work done to pull a block of mass $M$ by a constant force $F$ as shown in the figure. The coefficient of friction between the block and the ground is $\mu$.

$A$ 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).

Two particles of equal mass start moving in opposite directions from point $A$ in a horizontal circular path. Their tangential velocities are $v$ and $2v$ respectively,as shown in the figure. At the time of collision,the particles move with the same speed. How many elastic collisions must occur after the first one so that these two particles reach point $A$ again?

$A$ train is moving along the tracks at a constant speed $u$. $A$ girl on the train throws a ball of mass $m$ straight ahead along the direction of motion of the train with speed $v$ with respect to herself. Then