Suppose that you want to launch an object that travels the longest distance, given the starting speed (or the force applied). You have to determine the angle that al you to reach the longest distance, before it touches the ground.
For simplicity, the launch happens at the ground level, so no change in altitude happens (same gravitational potential energy at the start and at the end).
In the case of a flat surface and with no air resistance, I would say that the optimal angle is 45° (and it should be easy to determine). But how the earth curvature plays in the calculation?
Optional: how can you introduce the effect of air resistance, without making the problem too complex? Is it possible to model it in a simple way, such as an arbitrary force depending only on the speed, mantaining the problem solvable? It would be also sufficient to say if the angle is unaffected or goes up/down, with some justification.
For simplicity, assume that the object has to remain in the atmosphere, so putting it in the orbit is not an option :)