$A$ body of mass $2.9 \, kg$ is suspended from a string of length $2.5 \, m$ and is at rest. $A$ bullet of mass $100 \, g$ strikes the block horizontally with velocity $150 \, m/s$ and sticks to it. What is the maximum angle made by the string with the vertical after the impact? (Given $g = 10 \, m/s^2$)

$A$ block of mass $m = 5 \text{ kg}$ is released from the top of an inclined plane as shown in the figure. The inclined plane has a length of $10 \text{ m}$ and an angle of $30^{\circ}$. The horizontal surface has a coefficient of kinetic friction $\mu = 0.5$ and a length of $2 \text{ m}$ before the spring of spring constant $k = 100 \text{ N/m}$. Calculate the maximum compression $x$ in the spring.

Two bodies $A$ and $B$ of masses $1.5 \ kg$ and $3 \ kg$ are moving with velocities $20 \ m \ s^{-1}$ and $15 \ m \ s^{-1}$ respectively. If the same retarding force is applied on the two bodies,then the ratio of the distances travelled by the bodies $A$ and $B$ before they come to rest is

An object of mass $m$ is sliding down a hill of arbitrary shape and after traveling a certain horizontal path stops because of friction. The friction coefficient is different for different segments of the entire path but it is independent of the velocity and direction of motion. The work done that a force must perform to return the object to its initial position along the same path would be:

Statement $-1$: Two particles moving in the same direction do not lose all their energy in a completely inelastic collision.
Statement $-2$: Principle of conservation of momentum holds true for all kinds of collisions.

$A$ simple pendulum of length $L = \frac{10}{3} \text{ m}$ with a bob of mass $M = 3m$ is hanging freely from a rigid support. $A$ bullet of mass $m$ is fired with a velocity $u = 50 \text{ ms}^{-1}$ from the ground at an angle $\theta$ with the horizontal. When the bullet is at its highest point of its trajectory,it collides head-on with the bob of the pendulum and gets embedded in the bob. After collision,if the pendulum moves through a maximum angle of $120^{\circ}$,then the value of $\theta$ is $(g = 10 \text{ ms}^{-2})$.