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.
TL;DR: it’s complicated.
Assuming the wind doesn’t blow, there’s no difference. You’re flying X miles either way.
However, the wind does blow, and IIRC it tends to blow from west to east near the poles and east to west near the equator. Since planes fly in air, flying *with* the wind moves you over the ground faster, while flying *into* the wind slows you down (relative to the ground). So flying from the UK, it’s easier to fly westward than to fly eastwards, until you hit the equatorial region where it gets harder. But once you’ve crossed over well south of the equator, it will get easier again.
This is ignoring that most planes fly great-circle routes, which often route “unpredictably” (to common intuition) on a map, because the Earth is a sphere and not a 2D grid.
Earth rotates at about 1,000 miles per hour at the equator, but they don’t experience 1,000 mile per hour winds because most of the air moves with the Earth. Therefore, Earth’s rotation isn’t a significant factor aside from how it drives wind currents.
Biggest factor will be the wind/ocean currents. This is why a flight between New York and LA can have different flight times based on what direction you’re flying. If you’re flying/sailing with these currents, you will move faster and therefore have a shorter trip than you would it you’re working against them.
Others have fully answered the question about rotation and winds if your question was about the speed of the airplane. But if you want a practical answer about the trip time, what’s faster will always be determined by the available flight routes. For London to Sydney this will always be East, since connecting through the Middle East or Asia is simply a shorter trip.
http://www.gcmap.com/mapui?P=lhr-sin-syd;lhr-yvr-syd;lhr-lax-syd;lhr-dxb-syd
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