$A$ steel sphere of radius $1.2 \ cm$ collides with a second steel sphere at rest. If the collision is elastic and after the collision the first sphere continues to move in its initial direction with a velocity of $\frac{7}{9}$ times its initial velocity,then the radius of the second sphere is (in $cm$)

Six identical balls are lined in a straight groove made on a horizontal frictionless surface as shown. Two similar balls each moving with a velocity $v$ collide elastically with the row of $6$ balls from the left. What will happen?

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.

$A$ uniform rod of mass $M$ and length $L$ is placed on a smooth horizontal surface. $A$ particle of mass $m$ moving with velocity $v$ strikes the rod at one end perpendicularly and comes to rest. The velocity of the centre of mass of the rod after the collision is:

$A$ particle of mass $m$ collides with a heavy mass $M$ (initially at rest) elastically. After the collision, the particle returns with $4/9$ of its initial kinetic energy. The mass of the heavy object $M$ is ............... $m$.

$A$ smooth sphere $A$ moves on a frictionless horizontal surface with an angular velocity $\omega$ and a linear velocity $v$ of its center of mass. It undergoes an elastic collision with another identical sphere $B$ at rest. Neglecting friction everywhere,if the angular speeds after the collision are $\omega_A$ and $\omega_B$ respectively,then: