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There is a vertical pole perpendicular to the horizontal plane. From point \(P\) on the plane, 2 projectiles are fired simultaneously at different velocities. The first projectile is fired at an angle of \(30^{\circ}\) and it hits the foot of the pole. The second projectile is fired at an angle of \(60^{\circ}\) and it hits the top of the pole.

It is further known that the projectiles hit the pole at the same time.

If the angle subtended by the pole from \(P\) is \(\alpha\), find \(\tan \alpha\) to 3 decimal places.

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Details and Assumptions:

• Air resistance is negligible.
• A gravitational pull is present.