Derive final velocities `V_1 and V_2` after perfectly elastic collision.

Explanation

To derive the final velocities \( V_1 \) and \( V_2 \) after a perfectly elastic collision, we will use the principles of conservation of momentum and the coefficient of restitution.

### Step 1: Define the Variables
Let:
- \( m_1 \) = mass of the first object
- \( m_2 \) = mass of the second object
- \( u_1 \) = initial velocity of the first object before collision
- \( u_2 \) = initial velocity of the second object before collision
...