Because of the difference in air speed pressure and lift above and below the wing. Earth rotation is irrelevant. Next.

The air is rotating along with the surface of the Earth. And so is the plane. The movement of a plane is relative to the moving surface.

Except for very near the poles, or a few supersonic aircraft, a plane traveling "west" is in fact still being carried eastward by the Earth's rotation. It's just traveling eastward less quickly, so its position relative to the surface moves westward.

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Planes fly due to two forces, lift which is generated by air traveling across the wing, and thrust which pushes the plan forward. Combine those two and you get flight.

The Earth's rotation doesn't really enter into it because in theory everything Earthy also rotates, including the air in the sky and the plane on the ground. In theory, since the plane and the air in the sky are part of the spinning Earth system it all cancels out and spinning or no spinning, the motion of the Earth doesn't interact with the physics of an air plane.

But of course the air in the sky isn't just spinning with the Earth, we have wind and storms and jetstreams and things like that mean the bulk motion of the air IS important to flight. For example, because of the Jetstream that blows roughly from NYC to Ireland, it takes around 45 minutes longer to fly back to the NYC from Ireland, because you're flying into the wind.

Think about when you’re riding in a car or train. If you’re not speeding up or braking suddenly, you can still throw a paper airplane as if you weren’t moving.

The air around earth is similar. It’s moving with the earth’s rotation and travel through space. That’s why it’s lucky space is virtually empty. If it weren’t, the air would blow away with anything else not tied down. Like riding in a convertible!