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This is exactly what I thought would happen, when you detach from earth you are flying straight line (minus the starting speed from rotation of earth) while earth continue to spin!
So, the question is: If I'm flying from one place to another, how would I know what azimuth should i hold to arrive where i need?
Well, if your in an airplane they don't use launch azimuth, they use NDBs and VORs. Other than that, they use GPS.