I am familiar with standard distance-time models for paths of projectiles in perfect conditions, ie, where the curvature doesn't play a role, and where gravity is constant no matter the height. My question is what if you launch a projectile so high that the curvature of earth plays a role, and gravity varies as you increase and decrease height, is there and way to model it's motion, say a distance to surface-time equation? It would probably be similar to basic ones like s(t)=.5at^2+v*t, but since the acceleration changes over time it becomes difficult. It seems the acceleration at any point would be (Fg-Fc)/m (force of gravity minus force centripital, divided my mass), but then you get distance in the equation twice... Also, since (mv^2)/r = Fc how would you know the tangential velocity? Do equations already exist for this?
Basically, at this point, you are dealing with orbital motion. Kepler's Laws will give you the basic trajectory. From there, you'll have to look into more detailed orbital mechanics for things like time of flight, velocities at each point, and so on. Of course, in real world, if velocity is high enough for these things to matter, your bigger concern will be drag. It is difficult enough to account for drag with level ground and constant gravity. There are methods, though. Orbital mechanics with drag pretty much have to be solved with numerical methods. If gravity changing with height is a factor, density changing with altitude will certainly be an even bigger factor. That means having a pretty good barometric model on top of which you'll be running your simulation. It gets tricky.
It's a field of study called exterior ballistics, and the work usually involves the numerical solution of complicated differential equations of motion. In WWI when the Germans were firing their Paris gun, they had to account for, among other variables, changes in gravity, density of the atmosphere, curvature of the earth, coriolis forces, the gravitational effect of the moon, the temperature, and the condition of the propellant.
It's my understanding that the accuracy of the Paris Gun was terrible. The few projectiles that were fired landed somewhere in a 10 kilometer radius around the center of Paris (so they still landed in the suburbs). For calculating the amount of propellent needed the crudest approximation would have been sufficient. Density of the atmosphere at different stages of the flight: probably yes, the projectiles climbed tens of kilometers high. My understanding is that all other effects were totally swamped. The gun could fire 10 projectiles or so, then it had to be shipped back to the factory to resurface the inside of the barrel. In terms of strategic value the Paris Gun was a waste of resources. The effect was psychological: the fact that the Germans were able to reach Paris with that Gun.