Integrating law of gravity into a parabola?

[Chuckstabler]

Homework Statement

Hey all, I've been learning some incredibly INCREDIBLY basic calculus on my own, so please take it easy on my stupidity.

So here's what I was wondering. In a 1 dimensional theoretical system, let the acceleration experienced by an object = A, with the signage +/- indicating the direction (forward or backwards, + or -) of the acceleration, seeing as this is a 1 dimensional system a +/- acceleration should suffice.

Integrating with respect to time, we find that v = (a*t) + c (constant of integration for velocity). Integrating again with respect to time we find that d (displacement) = (1/2at^2) + ct + di (initial displacement, or constant of integration). I will assume that the constant of integration for displacement will be 0, meaning that I am assuming that the initial starting distance can be assigned the coordinate of 0. We can further simplify the final displacement equation to be...

D = 1/2t(at + 2c)

When graphing this I put a (acceleration), c (initial velocity) as constants, and I get the typical parabola.

Here is my question: We all know that the force experienced by a body = G(m1*m2)/d^2, and that the acceleration experienced by a body = G*Mcentral/d^2. I was wondering, is it possible to somehow integrate this equation into the above equation so that the acceleration changes as a function of displacement? If so, (I am assuming it is possible), how can I go about doing so? Thanks for your time :).

I just realized that this may be considered a homework / schoolwork question , it's not for any homework or school work but it may fit into that category in any case, so I posted this here as well as general physics

Homework Equations

D = 1/2t(at + 2c)

The Attempt at a Solution

I honestly don't know where to start, I was just wondering if anybody could give some assistance. Thank you so much :)

 

[Orodruin]

So let me see if I can decipher your question. You are used to integrating to solve for the displacement as a function of time in the case when the acceleration is constant. You are wondering if you can do it also when it is not constant, but depends on the displacement? If so, the answer is yes, but it is a more complicated problem which requires some more knowledge about differential equations that you would typically not learn in basic calculus.

 

[jedishrfu]

We once had to do this problem in Classical Mechanics. In our case, we had to show that an object falling from some distance in space took 5/9 or the time to fall half the distance. We couldn't figure out how to change Newtons formula to one with a time in it.

It stumped everyone until a prof mentioned that you needed to use one of Kepler's laws to solve it.

http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion