$A$ particle of mass $m$ moving horizontally with velocity $v_0$ strikes a smooth wedge of mass $M$,as shown in the figure. After the collision,the ball starts moving up the inclined face of the wedge and rises to a height $h$. Choose the correct statement related to the wedge $M$.

$A$ bullet of mass $0.01 \, kg$ travelling at a speed of $500 \, m/s$ strikes a block of mass $2 \, kg$ which is suspended by a string of length $5 \, m$. The center of gravity of the block is found to rise a vertical distance of $0.1 \, m$. What is the speed of the bullet after it emerges from the block?

State whether the following statements are true or false:
$(a)$ If $\overrightarrow P \cdot \overrightarrow Q = 0$,then the angle between $\overrightarrow P$ and $\overrightarrow Q$ is $0^o$.
$(b)$ If two bodies stick together after a collision,the collision is called an elastic collision.
$(c)$ When an equal force is applied to a heavy and a light object on the same surface,more work is done by the force on the heavy object.

$A$ student skates up a ramp that makes an angle $30^{\circ}$ with the horizontal. He/she starts (as shown in the figure) at the bottom of the ramp with speed $v_0$ and wants to turn around over a semicircular path $xyz$ of radius $R$ during which he/she reaches a maximum height $h$ (at point $y$) from the ground as shown in the figure. Assume that the energy loss is negligible and the force required for this turn at the highest point is provided by his/her weight only. Then ($g$ is the acceleration due to gravity):
$(A)$ $v_0^2 - 2gh = \frac{1}{2} gR$
$(B)$ $v_0^2 - 2gh = \frac{\sqrt{3}}{2} gR$
$(C)$ The centripetal force required at points $x$ and $z$ is zero.
$(D)$ The centripetal force required is maximum at points $x$ and $z$.

$A$ block $C$ of mass $m$ is moving with velocity $v_0$ and collides elastically with block $A$ of mass $m$ which is connected to another block $B$ of mass $2m$ through a spring of spring constant $k$. What is $k$ if $x_0$ is the compression of the spring when the velocity of $A$ and $B$ is the same?

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