Question about spinning Earth The earth spins on its axis at 1674 km/h from west to east. Would it not imply that an airplane flying eastbound at 500 km/h would cover a distance of 500 kilometres in an hour while another plane flying westbound at the same speed would cover 500+1674 or 2174 kilometres in an hour? What am I missing?
 A: What you are missing is the fact that the atmosphere moves along with the rotating earth (plus or minus local wind).  You can see this because otherwise just standing still would subject you to enormous winds.
Note also that your figure of 1674 km/h only applies at the equator.
A: A scaled down example might help:


*

*Stand on a roundabout or turntable spinning clockwise at 10m / sec.

*Take a step clockwise. If your stride is 1m long, you are now 1m around the turntable from where you started. If the step took you a second, you were moving at 1m / sec.

*Take a step anti-clockwise. You are now back where you started. You've again moved 1m, at 1m / sec.


The fundamental insight is that speed is a relative measure. In the example above, we measured your steps relative to the turntable; in every day life, we measure our steps, or an aeroplane's flight, relative to the Earth. 
If I watch you standing on the turntable, you are moving relative to me at a constant speed of 10 m / sec, then briefly at 11 m / sec when you step clockwise, and 9 m / sec when you step anti-clockwise. Similarly, an observer on the sun would see the Earth spinning, and movement going "with" or "against" that spin; an observer in a neighbouring galaxy would see the entire solar system moving at an enormous speed around the centre of the Milky Way.
A: The speed of Earth's rotation is 1674 km/h on the equator. The surface, the airport, the atmosphere and the plane (before taking off) all move with 1674km/h relative to the centre of earth. They all move 0 km/h (well, more or less) relative to the airport. After the plane takes off and reaches 500 km/h, it will move with 500 km/h relative to the airport. Of cource, it keeps the tangential speed relative to the centre of earth. It's total speed relative to the centre of earth will be 2174 km/h if it moves east, 1164 km/h if it moves west.
For an airplane, surface speed is the most relevant one. If you start a rocket to reach orbit, on the other hand, you want to start east because that way you get the rotational speed for free.
A: As others have said, the important factor is that air rotates along with the earth. However, If you go east or west at the same speed relative to the air, on average, there will be a small effect due to dominant winds and high altitude air currents, which move in different directions relative to the Earth, depending on latitude. The wind will then help you go faster or slower relative to the ground. For example, if you look at the time taken to fly from New York to Paris, on average usong the same plane, to the time taken to do the opposite, there will be small difference.
This was not in your question, but there is a small difference that can be detected in westward and eastward trips around the world due to the addition of speeds of the plane and the Earth. Relativity does not care about ground movements, but about absolute movements. Clocks taken around the world in different directions will track time differently as they will be moving at different speeds, as mentioned in the question. You can look into the Hafele–Keating experiment for more details.
