Re: Solution for motion of two masses under gravity

Brock,

It is true that every mechanics problem is soluble from F=ma.  However, it
is usually worthwhile to consider other ways of looking at the problem, in
order to take advantage of symmetries and other simplifications.

The situation you present, two masses released from rest in a vacuum, is
relatively straightforward.  The only force acting (gravity) gives an
acceleration on the line connecting the masses, and they start from rest,
so there is really only one coordinate -- it is a 1-D problem.  What
happens?  Intuitively, the masses will start accelerating toward one
another, and eventually they will collide.

The details of x versus time?  As you correctly note, the equations of
motion for masses #1 and #2 would be:

    for #1:   (G)(m2) = (x1")(x2 - x1)^2
    for #2:   (G)(m1) = (x2")(x2 - x1)^2

which is a coupled set of second-order differential equations, and not a
lot of fun to solve directly.  However, one can certainly ask what it is
you are really wanting to know, and how much time you want to spend doing
this sort of problem for your own edification.  At this point, most
mechanics texts would suggest moving to the center-of-mass frame, which
changes the problem to two decoupled second-order differential equations
(where one of them, the equation for angular motion, vanishes by your
assumptions).  From there, the solution is fairly straightforward, and is
an exercise in every mechanics text I have on my bookshelf.  I suggest:

"Mechanics", third edition, by Keith R. Symon, chapter 3 but especially
pages 124 and 125.
"Classical Mechanics", second edition, by Herbert Goldstein, chapter 3 in
general but especially Section 3-8 "The Motion of Time in the Kepler
Problem" on page 98.

There are others, but these are the ones within reach.  I have no doubt
that your libraries (personal, departmental or college) will have these or
 equivalent texts.  I suggest cracking the books.

Good luck!
Aaron J. Redd