First, ignore air resistance. Always ignore air resistance.
Using kinematics for gravitational acceleration systems works within a specific scope, and when the system's scope widens too far, they become an oversimplification. Kinematics are great for systems subject to constant acceleration, and gravity fundamentally cannot be a constant acceleration (take a look at its formula).
I know how to accurately model velocity as a function of displacement in a gravitational system using potential energy, more specifically the changes in potential energy. Basically the difference between initial potential energy (using gravity calculated with the object's initial height) and final potential energy (with gravity calculated from the object's current position) tells us the change in the object's kinetic energy and thus velocity. I've checked these calculations against highly precise computer simulations and found them to be accurate.
This set up models velocity as a function of displacement. My question is how to derive a formula which models velocity as a function of time in a gravitational system. I've asked some colleagues about this and none can produce an accurate model. I tried to integrate the acceleration to yield velocity, but that approach can't convert acceleration vs displacement into velocity vs time.
Any insight would be much appreciated.
EDIT: To clarify, the model I'm looking for is not so much of an orbit as it is a simple (one dimensional) free fall straight down to the "earth"