Two small balls A and B , each having mass m , are rigidly attached...

According to the given hint, the light rod can exert an impulsive force only along its length.

During the collision, the impulsive force between particle $P$ and ball $A$ acts in a direction perpendicular to the rod. Since the rod cannot transmit any impulse perpendicular to its length, it exerts zero transverse impulse on ball $B$.

As a result, the velocity of ball $B$ remains unchanged immediately after the collision:

$v_B = v_0$

For the system comprising ball $A$ and particle $P$, there is no external impulsive force in the direction perpendicular to the rod. Applying the principle of conservation of linear momentum for this system in the transverse direction:

$m v_0 + m(0) = (m + m) v_A$

$m v_0 = 2m v_A \Rightarrow v_A = \dfrac{v_0}{2}$

The impulse along the rod must be zero because the relative velocity of $A$ and $B$ along the rod must remain zero, and there are no initial velocities or external impulses in that direction. Thus, the linear speeds immediately after the collision are $\dfrac{v_0}{2}$ and $v_0$.