Jump on the Moon
It's interesting fact that Moon has always the same orientation relative to the Earth. However, let's consider following situation:
If we are staying on the visible Moon side, then Earth attraction allow us to jump higher in comparison with the other side (back side). On the back side Earth attraction will prevent us from jumping higher.
Flying ball's trajectory
Consider folowing problem. Ball having a mass m and diameter d flies out from the starting point with a certain velocity v0 at an angle to the horizon α. Air's drag force is considered. Ball's rotation is not considered.
We will compose motion differential equations (in plane) and simulate ball's flying trajectory.