Two beads $A$ and $B$ of masses $m_1$ and $m_2$ respectively are placed on a smooth vertical circular wire of radius $R$ as shown in the figure. Bead $A$ is given a very gentle push so that it slides down and collides with bead $B$ and comes to rest. After the collision,bead $B$ reaches the height of the center of the circle. Then $m_1 : m_2$ is equal to:

$A$ bullet of mass $m_1$ is moving with speed $v_0$ and hits a sand bag of mass $m_2$. If the speed of the bullet after passing through the sand bag is $\frac{v_0}{3}$,then the height $h$ up to which the bag rises is (assume,$g=$ acceleration due to gravity).

$A$ particle of mass $m$ is initially at rest at the origin. It is subjected to a force and starts moving along the $x$-axis. Its kinetic energy $K$ changes with time as $dK/dt = \gamma t$,where $\gamma$ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?
$(A)$ The force applied on the particle is constant
$(B)$ The speed of the particle is proportional to time
$(C)$ The distance of the particle from the origin increases linearly with time
$(D)$ The force is conservative

$A$ body moving with velocity $v$ has momentum and kinetic energy numerically equal. What is the value of $v$ in $m/s$?

$A$ small disc of mass $m = 1 \,g$ slides down a smooth hill of height $h = 10 \,cm$ from rest and gets onto a plank of mass $M = 100 \,g$ as shown in the figure. Due to friction between the disc and the plank, the disc slows down and moves as one piece with the plank. The work done by the frictional force is approximately (Use $g = 10 \,m/s^2$): (in $\,J$)

Three blocks $A, B$ and $C$ are kept as shown in the figure. The coefficient of friction between $A$ and $B$ is $0.2$,$B$ and $C$ is $0.1$,and $C$ and the ground is $0.0$. The masses of $A, B$ and $C$ are $3\, kg, 2\, kg$ and $1\, kg$ respectively. $A$ is given a horizontal velocity of $10\, m/s$. Blocks $A, B$ and $C$ always remain in contact and move together as a single system. The total work done by friction will be ........ $J$.