Three objects $A, B$ and $C$ are kept in a straight line on a frictionless horizontal surface. The masses of ${A}, {B}$ and ${C}$ are ${m}, 2\, {m}$ and $2\, {m}$ respectively. $A$ moves towards ${B}$ with a speed of $9$ ${m} / {s}$ and makes an elastic collision with it. Thereafter $B$ makes a completely inelastic collision with $C.$ All motions occur along same straight line. The final speed of $C$ is $....\,{m} / {s}$

A point mass of $1 \mathrm{~kg}$ collides elastically with a stationary point mass of $5 \mathrm{~kg}$. After their collision, the $1 \mathrm{~kg}$ mass reverses its direction and moves with a speed of $2 \mathrm{~ms}^{-1}$. Which of the following statement(s) is (are) correct for the system of these two masses?

$(A)$ Total momentum of the system is $3 \mathrm{~kg} \mathrm{~ms}^{-1}$

$(B)$ Momentum of $5 \mathrm{~kg}$ mass after collision is $4 \mathrm{~kg} \mathrm{~ms}^{-1}$

$(C)$ Kinetic energy of the centre of mass is $0.75 \mathrm{~J}$

$(D)$ Total kinetic energy of the system is $4 \mathrm{~J}$

A ball of mass $m$, moving with a speed $2v_0$, collides inelastically $(e > 0)$ with an identical ball at rest. Show that

$(a)$ For head-on collision, both the balls move forward.

$(b)$ For a general collision, the angle between the two velocities of scattered balls is less than $90^o$.

In an elastic collision of two particles the following is conserved

A rubber ball is released from a height of $5\, m$ above the floor. It bounces back repeatedly, always rising to $\frac{81}{100}$ of the height through which it falls. Find the average speed of the ball. (Take $g =10 ms ^{-2}$ ) (in $ms ^{-1}$)

A sphere $P$ of mass $m$ and moving with velocity $v$ undergoes an oblique and perfectly elastic collision with an identical sphere $Q$ initially at rest. The angle $\theta $ between the velocities of the spheres after the collision shall be ............... $^o$