The True Nature of Gravity

F/M = Gm/r^2 = a

This makes sense when M is the reference point, because the acceleration due to gravity is infinitesimally small at r=inf and huge near r=0. This prompts me to integrate for the total acceleration from r2 to r1. Thus, I get the equation:

a® = Gm (1/r1 - 1/r2)

Ok, you've derived the equation for a change in gravitational potential. Now what? Since you want to find out the *true* shape of the displacement time curve for an object moving under gravity, i'd think integrating from your start point with respect to time would be better. (Gm/r^2 = a)
Then you'd need to express a or r as function(s) of time. I'd go for r, since you can just go with int(a)dt= velocity, int(v)dt= displacement
You would need to use a equation of motion of some sort to express r as some function of t, however since all the equations of motion that spring to my mind at the moment will give you a bunch of other variables that are also functions of t, I'm not entirely sure how well this approach would work.