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Suppose we have a helicopter that is able to stay stationary in flight for extended periods of time. If such a helicopter stayed at point A in the sky for 12 hours straight, would it reach the other side of the globe?

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No. A helicopter that "stays stationary" does so in relation to the atmosphere around it and the atmosphere pretty much follows the ground underneath it. The atmosphere does not stand still while the earth rotates. If it did, we would experience constant winds on the order of 1000 km/h. That would not be pleasant.

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