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 can follow these steps:

### Step 1: Define the System
Consider two objects with masses \( m_1 \) and \( m_2 \). Let their initial velocities be \( u_1 \) and \( u_2 \) respectively. After the collision, their final velocities will be \( V_1 \) and \( V_2 \).

### Step 2: Apply Conservation of Momentum
Since no external forces are acting on the system, the total linear momentum before the collision is equal to the total linear momentum after the collision. This gives us the equation:

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