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One rotation of the Earth, gives up 1 day. One day is split into night time and day time.
Earth's rotation speed doesn't affect the ability to escape Earth's gravity. Escaping Earth's gravity requires reaching a velocity of about 11.2 km/s regardless of Earth's rotation speed. Earth's rotation does provide a slight boost to the velocity required to escape in the direction of the rotation.
Days on Jupiter are shorter than those on Earth because Jupiter has a much faster rotation, completing a full rotation on its axis in just under 10 hours. Earth, in comparison, takes about 24 hours to complete one full rotation. This difference in rotation speeds is why Jupiter's days are shorter.
Yes, the rotation of the Earth affects flight time. Flights that go against the direction of the Earth's rotation take longer, while flights that go with the rotation of the Earth can be quicker due to the boost in speed provided by the Earth's rotation.
If you're talking about Earth's rotation, it is taking place all the time. It takes 24 hours for the Earth to complete a rotation, however.