$A$ moving body with a mass $m_1$ and velocity $u$ strikes a stationary body of mass $m_2$. The masses $m_1$ and $m_2$ should be in the ratio $\frac{m_1}{m_2}$,so as to decrease the velocity of the first body to $\frac{2u}{3}$ and give a velocity of $v$ to $m_2$,assuming a perfectly elastic impact. Then,the ratio $\frac{m_1}{m_2}$ is

The bob $A$ of a pendulum,released from a horizontal position to the vertical,hits another bob $B$ of the same mass at rest on a table as shown in the figure. If the length of the pendulum is $1\,m$,calculate:
$(a)$ The height to which bob $A$ will rise after the collision.
$(b)$ The speed with which bob $B$ starts moving.
Neglect the size of the bobs and assume the collision to be elastic.

$A$ particle of mass $m$ moving with a speed $v$ hits elastically another stationary particle of mass $2m$ in a fixed smooth horizontal circular tube of radius $R$. Find the time when the next collision will take place?

In a carrom-board game,the striker and the coins are identical and of mass $m$. In a particular hit,the coin is hit when it is placed close to the edge of the board as shown in the figure,such that the coin travels parallel to the edge after the collision. If the striker is moving with speed $v$ before the strike,then the net impulse on the striker during the collision,if it moves perpendicular to the edge after the collision,is (assume all collisions to be perfectly elastic):

Six steel balls of identical size are lined up along a straight frictionless groove. Two similar balls moving with a speed $V$ along the groove collide with this row on the extreme left end. Then:

Two identical balls $A$ and $B$ having velocities of $0.5 \, m s^{-1}$ and $-0.3 \, m s^{-1}$ respectively collide elastically in one dimension. The velocities of $B$ and $A$ after the collision respectively will be