$A$ smooth sphere of mass $M$ moving with velocity $u$ directly collides elastically with another sphere of mass $m$ at rest. After collision,their final velocities are $V$ and $v$ respectively. The value of $v$ is

$A$ particle of mass $m$ is moving with a horizontal speed of $6 \, m/s$ as shown in the figure. If $m << M$,then for a one-dimensional elastic collision,the speed of the lighter particle after the collision will be:

Two small particles of equal masses start moving in opposite directions from a point $A$ in a horizontal circular orbit. Their tangential velocities are $v$ and $2v$,respectively,as shown in the figure. Between collisions,the particles move with constant speeds. After making how many elastic collisions,other than that at $A$,will these two particles again reach the point $A$?

Two masses $m_{a}$ and $m_{b}$ moving with velocities $v_{a}$ and $v_{b}$ in opposite directions collide elastically. After the collision,$m_{a}$ and $m_{b}$ move with velocities $v_{b}$ and $v_{a}$ respectively. The ratio $m_{a} : m_{b}$ is:

$A$ billiard ball moving with a speed of $5 \, m/s$ collides with an identical ball originally at rest. If the first ball stops after collision,then the second ball will move forward with a speed of ........... $m/s$.

An alpha-particle of mass $m$ undergoes a $1$-dimensional elastic collision with a nucleus of unknown mass $M$ at rest. It is scattered directly backwards,losing $64\%$ of its initial kinetic energy. The mass of the nucleus is .......... $m$.