A uniform smooth rod of mass \( M = 2.5 \text{ kg} \) and length \( l = 4 \text{ m} \) is lying still on a smooth horizontal table. Now I shoot a small ball of mass \( m = 2 \text{ kg}\) from point \(P\) such that it traces the path shown and hit the rod at a distance of \( l / 4 \) from the centre. The ball collides elastically with the rod. Let \( \omega \) represents the angular velocity of the rod just after collision and \( {v}_\text{rel} \) represents the relative velocity of ball just after collision with respect to centre of mass of the rod.

Find \( {v^2}_\text{rel} \times \omega \).

**Details and Assumptions**:

- The ball was given an impulse \( I = 20 \text{ kg m/s} \) at point \(P\).
- Consider the figure for geometry.
- Friction is absent everywhere.
- The rod is completely free (that is free to do translational as well as rotational motion).

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