Subject: Instantaneous velocity in a gravitational acceleration sequence.

I am to begin my Physics & Astronomy major this fall, but have 
been theorizing to some extent about the mechanics of linear 
motion, and gravity's connection therein. As I examined the 
curves for 1/x^2, I noticed that it increases at an exponential 
rate. I thought then that perhaps that instantaneous velocity 
could be expressed in terms of higher rates of acceleration such 
as m/s^3 and m/s^4. I have gone so far as to develope the linear 
motion equations for these acceleration rates (which I have 
deemed triceleration and tetraceleration), and run numerical 
tests. My intent was to find a mathmatical relationship. However, 
I have not been able to do this. I do understand that these 
acceleration rates can be expressed as average acceleration, but 
that does not acurately describe the motion of the object in 
question. My question, in a nutshell, is this: Is there a 
noncalculus-based method of determining instantaneous velocity 
and distance traveled in a gravitational acceleration sequence? I 
should also like to know if I am at least on the right track, 
more importantly if this phenomenon occurs, and if the 
aformentioned acceleration rates have any real application. This 
all came from my interest in colliding galaxies, as my career 
intent is astrophysics. Keep in mind that I am using Newtonian 
gravitational mechanics, not relativity. I know this will take 
some time and effort, so I will await patiently. Thank You! 

Re: Instantaneous velocity in a gravitational acceleration sequence.