Infinite Distance In Finite Time

It has recently been found that there exist configurations of a
small number of point masses such that, using Newton's laws of
motion and Newtonian gravity, at least one mass will shoot off
to infinity in a finite time.  Most of the configurations are 
similar to one proposed by Joseph Gerver in 1984.  Since they 
involve point masses, it's debateable if they are in any sense 
realistic, but anyway...

Consider three heavy masses at the points of a triangle, with an 
obtuse angle at the heaviest one.  Give them initial velocities
so that the triangle expands but maintains its shape.

Now add a tiny particle that orbits around the outside of
the triangle, looping very close to each of the large masses
and using the "sling shot" effect to gain energy from each
near approach.  In effect, the small particle distributes the
relative kinetic energy of the three masses in such a way as
to pull them apart at ever-increasing speed.  But where does
all the kinetic energy come from?

There's a fifth object orbiting around the heaviest mass, and
it gives up energy as it spirals in to smaller and smaller radius.  
Remember these are *point* masses, so there is an infinite potential 
well of energy available.  So the overall idea is to let an object 
spiral in to an arbitrarily small radius, releasing limitless amounts
of kinetic energy, which is then distributed to the three masses
by the tiny mass that's sling-shoting around the outside perimeter,
pulling them apart.  

Supposedly if you set everything up just exactly right, there is a 
trajectory that result in all three corners of the triangle (with 
all five masses) being shot off to infinity in finite time.

I'm not sure how astounding this really is.  After all, if you're
looking for "singularities" in classical dynamics, and if you're
willing to allow point masses, then you need look no further.  A 
point mass is already a singularity, in the sense that it represents 
an infinite amount of potential energy, and the force of gravity is
infinite ar r=0.

Interestingly, Newton nicely avoided the problem by taking density
and volume as his fundamental qualities, and then defining mass
as the product

    "The quantity of matter is the measure of the same,
     arising from its density and bulk conjointly."