$A$ sphere of mass $m$ moving with a constant velocity $u$ hits another stationary sphere of the same mass. If $e$ is the coefficient of restitution,then the ratio of the velocity of the two spheres after the collision will be

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. Then,the ratio between the relative velocity of approach and the relative velocity of separation in the second collision is:

$A$ tennis ball dropped from a height of $2 \,m$ rebounds only $1.5 \,m$ after hitting the ground. What fraction of its energy is lost in the impact?

$A$ body $A$ moving with momentum $P$ collides one-dimensionally with another stationary body $B$ of same mass. During impact,$A$ gives impulse $J$ to $B$. Then which of the following is/are correct?
$(a)$ The total momentum of $A$ and $B$ is $P$ before and after impact and $(P-J)$ during the impact.
$(b)$ During the impact,$B$ gives impulse of magnitude $J$ to $A$.
$(c)$ The coefficient of restitution is $\left[\frac{2 J}{P}-1\right]$.
$(d)$ The coefficient of restitution is $\left[\frac{2 J}{P}+1\right]$.

Starting from rest on her swing at initial height $h_0$ above the ground,Saina swings forward. At the lowest point of her motion,she grabs her bag that lies on the ground. Saina continues swinging forward to reach maximum height $h_1$. She then swings backward and when reaching the lowest point of motion again,she simply lets go of the bag,which falls freely. Saina's backward swing then reaches maximum height $h_2$. Neglecting air resistance,how are the three heights related?

$A$ large block of wood of mass $M = 5.99 \, kg$ is hanging from two long massless cords. $A$ bullet of mass $m = 10 \, g$ is fired into the block and gets embedded in it. The (block $+$ bullet) then swing upwards,their centre of mass rising a vertical distance $h = 9.8 \, cm$ before the (block $+$ bullet) pendulum comes momentarily to rest at the end of its arc. The speed of the bullet just before collision is: (Take $g = 9.8 \, ms^{-2}$) (in $m/s$)