$A$ body of mass $2 \, \text{kg}$ moving with a speed of $4 \, \text{m/s}$ makes an elastic collision with another body at rest and continues to move in the original direction but with one-fourth of its initial speed. The speed of the two-body centre of mass is $\frac{x}{10} \, \text{m/s}$. Then the value of $x$ is ..... .

$A$ bullet loses $\left( \frac{1}{n} \right)^{th}$ of its velocity while passing through one plank. The number of such planks required to stop the bullet is:

$A$ ball of mass $0.2 \,kg$ is thrown vertically down from a height of $10 \,m$. It collides with the floor and loses $50 \%$ of its energy and then rises back to the same height. The value of its initial velocity is

$A$ small block of mass $M$ moves on a frictionless surface of an inclined plane,as shown in the figure. The angle of the incline suddenly changes from $60^{\circ}$ to $30^{\circ}$ at point $B$. The block is initially at rest at $A$. Assume that collisions between the block and the incline are totally inelastic $\left(g=10 \ m/s^2\right)$.
$1.$ The speed of the block at point $B$ immediately after it strikes the second incline is
$(A) \sqrt{60} \ m/s$ $(B) \sqrt{45} \ m/s$ $(C) \sqrt{30} \ m/s$ $(D) \sqrt{15} \ m/s$
$2.$ The speed of the block at point $C$,immediately before it leaves the second incline is
$(A) \sqrt{120} \ m/s$ $(B) \sqrt{105} \ m/s$ $(C) \sqrt{90} \ m/s$ $(D) \sqrt{75} \ m/s$
$3.$ If the collision between the block and the incline is completely elastic,then the vertical (upward) component of the velocity of the block at point $B$,immediately after it strikes the second incline is
$(A) \sqrt{30} \ m/s$ $(B) \sqrt{15} \ m/s$ $(C) 0$ $(D) -\sqrt{15} \ m/s$
Give the answers for questions $1, 2,$ and $3.$

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 $H >> l$)

Three objects $A, B$,and $C$ are placed on a frictionless horizontal surface. Their masses are $m, 2m$,and $m$ respectively. Object $A$ moves towards $B$ with a speed of $9 \ m/s$ and undergoes an elastic collision with it. Subsequently,$B$ undergoes a perfectly inelastic collision with $C$. All motion occurs along the same straight line. What will be the final speed of object $C$ in $m/s$?