$A$ bullet of mass $0.01 \,kg$ travelling at a speed of $500 \,ms^{-1}$ strikes a block of mass $2 \,kg$ which is suspended by a string of length $5 \,m$. The centre 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 (in $\,ms^{-1}$)?

Two masses $m_1$ and $m_2$ are connected by a string of length $l$. They are held in a horizontal plane at a height $H$ above two heavy plates $A$ and $B$ made of different materials placed on the floor. Initially,the distance between the two masses is $a < l$. When the masses are released under gravity,they collide with $A$ and $B$ with coefficients of restitution $e_1 = 0.8$ and $e_2 = 0.4$ respectively. Find the time after the collision when the string becomes tight. (Assume $H >> l$)

$A$ particle of mass $m$ starts moving from the origin along the $x$-axis and its velocity varies with position $x$ as $v = k \sqrt{x}$. The work done by the force acting on it during the first $t$ seconds is ...........

State whether the following statements are true or false:
$(a)$ If the magnitude of force and length are increased by $4$ times,the magnitude of energy increases by $16$ times.
$(b)$ In an inelastic collision,both momentum and energy are conserved.
$(c)$ If work is done on a system by non-conservative forces,the potential energy increases.

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:

$A$ ball is falling freely from a height. When it reaches $10 ~m$ height from the ground,its velocity is $v_0$. It collides with the ground,loses $50 \%$ of its energy,and rises back to a height of $10 ~m$. Then the velocity $v_0$ is (in $~m / s$)