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If you were going fast enough. But if you want too fast I imagine you would see western sunrise again. However at that speed you would run out of fuel quite fast. It would depend upon your latitude and the time of year. For example if you were above the Arctic circle in the summer, any speed would keep you in the light, but in the winter you would always be in the darkness. The farther away you are from the equator the slower you could travel. (Assuming you are traveling parallel to the equator.) The necessary speed at the equator is approximately 1038 MPH.

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Q: If you were in an airplane with a maximum speed of 600 miles per hour and you were traveling in the direction opposite the earth's rotation could you travel fast enough to never see the sun set?
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