The simple answer is that the rotation of the earth does not do anything to the trip. The reason is that the earth’s atmosphere is generally moving along with the earth below it. And airplanes fly through the atmosphere – thereby gaining nothing from the earth’s rotation.
The not so simple answer is that the rotation of the earth does cause winds. Now these so called trade winds tend to be from east to west close to the equator (+/- 30 degrees). Further north and south, the winds blow from west to east.
The even more complicated answer is that, commercial jets can be influenced by the jet stream. These winds are at high altitudes and generally blow from west to east – but these can curve and also go north south at times.
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