Two identical balls $A$ and $B$ having velocities of $0.5 \, m s^{-1}$ and $-0.3 \, m s^{-1}$ respectively collide elastically in one dimension. The velocities of $B$ and $A$ after the collision respectively will be

$A$ mass $M$ moving with a certain speed $V$ collides elastically with another stationary mass $m$. After the collision,the masses $M$ and $m$ move with speeds $V^{\prime}$ and $v$,respectively. All motion is in one dimension. Then,

Consider the following statements $A$ and $B$ and identify the correct answer:
$A$. In an elastic collision,if a body suffers a head-on collision with another of the same mass at rest,then the first body comes to rest while the other starts moving with the velocity of the first one.
$B$. Two bodies of equal mass suffering a head-on elastic collision merely exchange their velocities.

$A$ body of mass $2 \ kg$ makes an elastic collision with another body at rest and continues to move in the original direction with one-fourth of its original speed. The mass of the second body which collides with the first body is ............... $kg$.

Two balls $A$ and $B$,of masses $M$ and $2M$ respectively,collide with each other. If ball $A$ moves with a speed of $150 \ m \ s^{-1}$ and collides with ball $B$ moving with speed $v$ in the opposite direction,and after the collision ball $A$ comes to rest with the coefficient of restitution being $1$,then the speed of ball $B$ before it collides with ball $A$ is: (in $m \ s^{-1}$)

Derive the expressions for the velocities of two bodies after a one-dimensional elastic collision.