Imagine that the Earth is spinning much faster than real situation and a man on the Earth's surface feels weightless. Suddenly the man decides to jump vertically with respect to the Earth. What will happen to him and the trajectory ignoring all the frictional forces?
Here's what I think. Since the man feels weightless he is just like a satellite in close circular orbit around the planet. So jumping process is just like giving a radially out impulse to a satellite in which the circular orbit becomes elliptical. More about this: https://physics.stackexchange.com/a/70418/192515
But how does the event happen in the man's frame?
At the moment he jumps he feels a sudden momentary weight, and he gets to the highest altitude and then comes down to the same spot he jumps (ignoring coriolis effect which might be too powerful this time?) and touches the ground with the same feeling of weight which will be momentary again.
I'm especially not sure about the second part of my answer. Please correct me if I have any mistakes.