A ball is projected vertically down with an initial velocity from a height of $ 20 m$  onto a horizontal floor. During the impact it loses $ 50\% $ of its energy and rebounds to the same height. The initial velocity of its projection is ............ $\mathrm{m} / \mathrm{s}^{-1}$

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 heavy body moving with a velocity of $6\,ms^{-1}$ collides elastically with a light body (whose mass is half of mass of heavy body) at rest. The velocity of light body will be (in $ms^{-1}$ )

A ball is thrown with a velocity of $6\, m/s$ vertically downwards from a height $H = 3.2\, m$ above a horizontal floor. If it rebounds back to same height then coefficient of restitution $e$ is $[g = 10\, m/s^2]$

For inelastic collision between two spherical rigid bodies

A body $A,$ of mass $m=0.1\; kg$ has an initial velocity of $3 \hat{\mathrm{i}}\; \mathrm{ms}^{-1} .$ It collides elastically with another body, $\mathrm{B}$ of the same mass which has an initial velocity of $5 \hat{\mathrm{j}} \;\mathrm{ms}^{-1}$. After collision. A moves with a velocity $\overline{\mathrm{v}}=4(\hat{\mathrm{i}}+\hat{\mathrm{j}})$. The energy of $\mathrm{B}$ after collision is written as $\frac{\mathrm{x}}{10} \;\mathrm{J}$ The value of $x$ is