I was trying to solve this problem myself but i dont know how to work out certain variables like gravity, centripetal force and the changing impact point due to the curve of a planet (any planet).

So i wondered if you guys could try and help me giving out some of the intuition to get the result and i also wondered if its possible to describe the motion of a projectile over a planet using polar coordinates.

So yeah, we have a initial launching point from the surface of a circle, and then a initial velocity and initial angle of launch, how do i know what will the distance over the surface up until the impact point be for that projectile accounting for the planetary body circular shape and the changing gravitational force?