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Imagine a helicopter that took off from point A on the ground and goes vertically in the air to a certain fixed point B in the atmospheric area at a certain fixed altitude with enough fuel to stay there (at point B) during let's say the whole week. Point B is fixed according to an external system - Copernic for instance. By the end of the week and after several earth rotations, does the helicopter remains at point B? I mean does the point B remains vertically and exactly above point A?

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The rotors of the helicopter act against the air in the atmosphere. As a result, the helicopter will always vary its location or speed relative to the air around it.

At the equator, a point on the earth travels at a speed of around 1600 km/h (40,000 km in 24 hours). This reduces at higher latitudes, but becomes zero only at the poles. We do not notice this movement because the air circles around with the earth. The helicopter, acting against the air, will do exactly the same: it will travel with the earth.

Of course, cold and hot spots on the earth cause different pressures at different points, which in turn creates winds. These winds are also affected by the rotation of the earth; that is why cyclones are circular. The helicopter will again follow the air, unless the pilot consciously angles the rotor blades to counteract the movement of the wind.

In answer, if we ignore winds, point B will remain above point A.

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If the helicopter only exerts force to counteract gravity, it will be carried away by the winds.

If the helicopter also actively counteracts the effects of wind it will stay above the same point.

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