$A$ particle of mass $m = 2 \ kg$ is initially at rest. The force $(F)$ versus displacement $(x)$ graph is shown in the figure.
$(1)$ The speed of the particle will be maximum at $x = ..... \ m$.
$(2)$ The maximum speed of the particle is ...... $ms^{-1}$.
$(3)$ The speed of the particle will be zero again at $x = .... \ m$.

$A$ 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 $g = 10 \, m/s^2$)

In the figure shown,two identical balls of mass $M$ and radius $R$ each are placed in contact with each other on a frictionless horizontal surface. $A$ third ball of mass $M$ and radius $R$ is coming down vertically and has a velocity $v_0$ when it simultaneously hits the two balls and comes to rest. Then,each of the two bigger balls will move after the collision with a speed equal to:

$A$ sphere of mass $m$ slides down a smooth inclined plane from a point $B$ at a height of $h$ starting from rest. The magnitude of the change in momentum of the particle between position $A$ (at the bottom of the incline) and $C$ (on the horizontal surface) is (assuming the angle of inclination of the plane is $\theta$ with respect to the horizontal):

$A$ ball of mass $10\, kg$ moving with a velocity $10 \sqrt{3} \, m/s$ along the $x$-axis,hits another ball of mass $20\, kg$ which is at rest. After the collision,the first ball comes to rest while the second ball disintegrates into two equal pieces. One piece starts moving along the $y$-axis with a speed of $10 \, m/s$. The second piece starts moving at an angle of $30^{\circ}$ with respect to the $x$-axis. The velocity of the ball moving at $30^{\circ}$ with the $x$-axis is $x \, m/s$. The configuration of pieces after the collision is shown in the figure. The value of $x$ to the nearest integer 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 $\mu$.