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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?

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  • $\begingroup$ space.stackexchange.com/q/13866 and space.stackexchange.com/q/13815 don't actually answer your question, but may help. @olaf_b is correct: the plane itself starts off at the same speed as the ground it's on. $\endgroup$ – barrycarter Nov 9 '17 at 12:44
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    $\begingroup$ The question is: Would cover a distance relative to what? To the ground? To the atmosphere? To a static observer from outside the solar system? $\endgroup$ – Jörg W Mittag Nov 9 '17 at 13:16
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    $\begingroup$ What if you take the universe expansion into account !?! $\endgroup$ – J. Chomel Nov 9 '17 at 17:06
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    $\begingroup$ Airplane at 500 km/h (a rather low speed for commercial jets) relative to what rest frame? $\endgroup$ – Carl Witthoft Nov 9 '17 at 18:13
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    $\begingroup$ The thing that you are missing is that all of the reference points on the ground are also moving at roughly 1674km/h. It turns out that objects on the ground move really fast if you view them from an inertial frame that isn't rotating. $\endgroup$ – Cort Ammon Nov 9 '17 at 20:28
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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.

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    $\begingroup$ This is slightly wrong, because an airplane does not fly at zero altitude. At X meters altitude, the plane needs to "keep up" with the angular speed, not the linear speed. If you're at the equator and shoot a gun straight up, ignoring atmospheric effects, it won't fall down at the barrel but to the West. $\endgroup$ – Carl Witthoft Nov 9 '17 at 18:15
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    $\begingroup$ This is true, at 10000m the plane would need to fly with 1677 km/h relativev to the centre of earth to keep up. But this already taken into account, because the question stated that the plane flew with 500 km/h relative to the surface. Am I missing something? $\endgroup$ – olaf b Nov 9 '17 at 19:46
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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.

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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.

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  • $\begingroup$ This fails to explain why the airplane doesn't float independently of the Earth. The OP seems to understand the concept when on the ground, but does not understand how it translates to atmosphere. (Otherwise they could just ask why it doesn't take less time to drive east than west.) $\endgroup$ – AJ Henderson Nov 9 '17 at 19:44
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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.

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