The angle of elevation of an airplane from a point on the ground is $60^\circ$. After a flight of $15$ seconds, the angle of elevation changes to $30^\circ$. If the airplane is flying at a constant height of $1500\sqrt{3}$ m, find the speed of the plane in m/s.
Options:
- A: $150$ m/s
- B: $200$ m/s
- C: $100\sqrt{3}$ m/s
- D: $250$ m/s
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