About a year (without stopping for a rest )
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It would need to be launched into a tangent plane parellel to that of the earth's orbit around the sun, with the same speed of rotation around the sun
The equator
Your at the equator if you see Polaris at the horizon
We can't feel the Earth's rotation because we're moving along with it. So is the atmosphere. Also . . . You NEVER feel "uniform" motion. You only feel CHANGES in motion, i.e., changes in speed or direction. On the rotating Earth, the speed is constant, and the direction changes by only about 1 degree in 4 minutes, which is too gradual to be noticeable. Imagine you went to the equator, where the Earth's rotational speed is greatest. If you stood on a balance to measure your weight there, you would notice a slight reduction in your weight because of the "centrifugal effect". However the difference would be too small to "feel". The difference, caused by the Earth's rotation, between your weight at the poles and at equator would much less than 1%.
it ways the same gomer (just google gomer)