# Integrating Gravitational Acceleration

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[help]Here's what I'm trying to do- Given: Xo: the initial distance from the source of the gravitation Vo: the initial radial velocity Is there any way to use the following equation (the universal law of gravitation)
M
A(X) = G ---
X2
to solve for A(t), V(t), X(t)? When I tried, I started from
M
A(t) = G ---
X(t)2
and then I got stuck on the next step. I started that with
V(t) = Vo plus the indefinite integral of A(t)dt
and
X(t) = Xo plus the indefinite integral of V(t)dt
Is there any way I can continue? When I plugged in A(t) into the V(t) equation and attempted the integration, I made no progress. Can this be done exactly or will I be forced to use some sort of approximation technique [sad] [crying] [bawling] ?

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A(t) = G M / X(t)^2
V(t) = Vo + A(t)*t
X(t) = Xo + V(t)*t + 1/2 A(t)*t^2

But, you likely need to do A(t-1) inorder to get answers from that, since it depends on X(t).
This is no different from any other acceleration. You just have a changing force.

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Your equation can be rewritten like this:

d2y
y2 --- = gM
dx2
This is a second-order non-linear differential equation. It can't be solved by simple integration, but there are standard methods of solving certain second-order non-linear differential equations. I'll leave it at that for now. Good luck.

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