$A$ ball is held in the position shown with a string of length $L = 1 \, m$ just taut and then projected horizontally with a velocity of $u = 3 \, m/s$. If the string becomes taut again when it is vertical,the angle $\theta$ is given by ........ $^o$.

$A$ particle of mass $m$ moving horizontally with $v_0$ 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 $h$. Identify the correct statement$(s)$ related to the situation when the particle starts moving downward.

$A$ block of mass $1\,kg$ is pushed up a surface inclined to the horizontal at an angle of $30^{\circ}$ by a force of $10\,N$ parallel to the inclined surface (see figure). The coefficient of friction between the block and the incline is $0.1$. If the block is pushed up by $10\,m$ along the incline,calculate:
$(a)$ Work done against gravity
$(b)$ Work done against the force of friction
$(c)$ Increase in potential energy
$(d)$ Increase in kinetic energy
$(e)$ Work done by the applied force

$A$ 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:

Consider the following two statements: $[A]$ The linear momentum of a system is zero. $[B]$ The kinetic energy of the particles of the system is zero.

$A$ body $x$ with a momentum $p$ collides with another identical stationary body $y$ one-dimensionally. During the collision,$y$ gives an impulse $J$ to body $x$. Then,the coefficient of restitution is