$A$ bullet loses $\left( \frac{1}{n} \right)^{th}$ of its velocity while passing through one plank. The number of such planks required to stop the bullet is:

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):

$A$ 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).

$A$ ball of mass $m_1$ falls from height $h_1$ from rest to strike a spring of force constant $K$,which forces another ball of mass $m_2$ to jump on a horizontal floor at a height $h_2$ below it. Find the horizontal distance at which the ball of mass $m_2$ strikes the floor from the position of the start. [Assume the spring is fixed and the collision is elastic].

$A$ block of mass $m$ slides from rest at a height $H$ on a frictionless inclined plane as shown in the figure. It travels a distance $d$ across a rough horizontal surface with coefficient of kinetic friction $\mu$ and compresses a spring of spring constant $k$ by a distance $x$ before coming to rest momentarily. Then the spring extends and the block travels back attaining a final height of $h$. Then,

An engine is attached to a wagon through a shock absorber of length $1.5\, \text{m}$. The system with a total mass of $40,000\, \text{kg}$ is moving with a speed of $72\, \text{km/h}$ when the brakes are applied to bring it to rest. In the process of the system being brought to rest,the spring of the shock absorber gets compressed by $1.0\, \text{m}$. If $90\, \%$ of the energy of the wagon is lost due to friction,the spring constant is $....\, \times 10^{5}\, \text{N/m}$.