In two separate collisions,the coefficients of restitution $e_1$ and $e_2$ are in the ratio $3:1$. In the first collision,the relative velocity of approach is twice the relative velocity of separation. Find the ratio between the relative velocity of approach and the relative velocity of separation in the second collision.

Two identical balls $A$ and $B$ are released from the positions shown in the figure. They collide elastically on the horizontal portion $MN$. All surfaces are smooth. The ratio of the maximum heights attained by $A$ and $B$ after the collision will be (Neglect energy loss at $M$ and $N$):

The kinetic energy of a light body and a heavy body is the same. Which one of the following statements is $CORRECT$?

$A$ particle of mass $3 \ kg$ is moved slowly along the path $ABCDE$ from $A$ to $E$. The heights of $B$,$C$,and $D$ are $5 \ m$,$4 \ m$,and $6 \ m$ respectively. The total path length is $20 \ m$,the horizontal distance $AE = 10 \ m$,and the coefficient of friction $\mu = 0.6$. Calculate the work done in $J$ to slowly move the mass from $A$ to $E$. (Take $g = 10 \ m/s^2$)

The figure below shows a small mass connected to a string,which is attached to a vertical post. If the ball is released when the string is horizontal as shown below,the magnitude of the total acceleration (including radial and tangential) of the mass as a function of the angle $\theta$ is

$A$ rubber ball is dropped from a height of $5 \ m$ on a planet,where the acceleration due to gravity is not known. On bouncing,it rises to $1.8 \ m$. The ball loses its velocity on bouncing by a factor of