A ball of mass m moving with velocity V, makes a head on elastic collision with a ball of the same mass moving with velocity 2V towards it. Taking direction of V as positive velocities of the two balls after collision are

The correct option is: – 2V and V

Let us analyze the system before and after the collision step-by-step using the principles of physics.

1. Identify the given values:
Let the first ball be ball 1 and the second ball be ball 2.
Mass of ball 1, $m_1=m$
Mass of ball 2, $m_2=m$
Let the direction of the initial velocity of ball 1 be the positive direction.
Initial velocity of ball 1, $u_1=V$
Since ball 2 is moving towards ball 1 (in the opposite direction) with speed 2V, its initial velocity in the chosen coordinate system is:
$u_2=-2V$

2. Understand the type of collision:
The collision is given to be perfectly elastic and head-on between two bodies of equal mass.
A fundamental property of a one-dimensional elastic collision between two bodies of equal mass ($m_1=m_2$) is that they exchange their velocities after the collision.

Let the final velocities of the two balls after the collision be $v_1$ and $v_2$ respectively.

Using the velocity exchange property:
$v_1=u_2=-2V$
$v_2=u_1=V$

Thus, the final velocities of the two balls after collision are – 2V and V respectively.