Last night I couldn't sleep for some time because of thinking this problem. The starting point of this problem was actually "If we were to jump perfectly vertically on earth, would we land on the same spot as we jumped?". However, I would like to ask a simplified version:

Suppose that there is a spherical rotating body of mass, which I assume it to be a planet such as earth, and an object on the surface of planet. Now, at some point this object is thrown vertically outwards the planet with some initial velocity.

I would like to know the distance between the thrower and the thrown object when it lands on this planet after it is thrown. If there is a minimum velocity which this object does not land on the planet after thrown, it would be awesome as well!

Note: My major is not in physics and its been a while since I have worked on these types of problems, so it would be helpful if you can explain the path to equations, derivations in more detail.

Last night I couldn't sleep for some time because of thinking this problem. The starting point of this problem was actually "If we were to jump perfectly vertically on earth, would we land on the same spot as we jumped?". However, I would like to ask a simplified version:

Suppose that there is a spherical rotating body of mass, which I assume it to be a planet, and an object on the surface of planet. Now, at some point this object is thrown vertically outwards the planet (90° angle between velocity vector and the planet surface) with some initial velocity.

I would like to know the distance between the thrower and the thrown object when it lands on this planet after it is thrown. If there is a minimum velocity which this object does not land on the planet after thrown, it would be awesome as well!

Note: My major is not in physics and its been a while since I have worked on these types of problems, so it would be helpful if you can explain the path to equations, derivations in more detail.