A bit off topic here Mike but if you're a mathematician (I'm not) here you go:
mikeandgerry wrote:Gravity is a an accelerative force, meaning that for every unit of distance further away the two bodies are, the attractive force gets weaker by the square of the distance.
II. RELEVANT CLASSICAL MECHANICS
Let x, v, and a denote distance, velocity, and acceleration respectively of an object along a line. Then, from the chain rule we have:
(1) a = dv/dt = (dv/dx)*(dx/dt) = v*(dv/dx)
(2) a*dx = v*dv
For convenience let f(x) be the indefinite integral of a*dx.
Consider an object moving position x1 to position x2 with initial velocity v1 and final velocity v2. We integrate both sides of (2) to get
(3) v2**2 - v1**2 = 2 * (f(x2)-f(x1))
Now consider the case where the acceleration of the the object is due to gravitational forces from one or more masses M_i located at positions X_i. The acceleration due to mass M_i is given by
(4) a_i(x) = G*M_i/(x-X_i)**2
Full formulas here:
This link is broken, either the page no longer exists or there is some other issue like a typo.