Like John Denker, I played a bit with the numerical solutions to
Carl's equation, a = k t / x, and developed some intuition about its
behavior. I have three comments to add to what John has already said:
1. It takes only a few moments' consideration of the DE to make it
clear that, for every solution, x1(t), that starts at (x_o, v_o),
there is a mirror image solution, x2(t) = -x1(t), that starts at (-
x_o,-v_o).
2. The "stiffness" of the DE near x = 0 suggests pathological
behavior. Clearly, any launch toward the origin will be repelled,
but the question arises: Is it possible to launch with a high enough
initial velocity toward the origin to "overcome" the repulsion and
pass to the other side?
If one considers the specific impulse (i.e., the integral of a dt),
which is equal to the change in v over any given time interval, one
may be led to think that it *is* possible. After all, it might seem
that the faster one launches the less time one will spend in the high
acceleration region thus reducing the change in velocity to
arbitrarily small values. Indeed, numerical solutions will show
precisely this behavior if the time step is chosen to be large enough
to enable skipping over the origin without sampling the arbitrarily
large repulsions in its vicinity. The real question is whether or
not the integral converges and that is a tough question to answer
without knowing x(t) in advance.
It's easier to consider the specific work (i.e., the integral of a
dx), which is equal to the change in the squared velocity over any
given spatial interval. Doing this integral is technically
complicated by the fact that the acceleration depends on time, but
for large enough velocities the time dependence is arbitrarily
small. It is easy to see that the integral from any given value of x
to the origin diverges logarithmically, suggesting pretty
conclusively that no finite initial velocity can overcome the barrier
at the origin. For large enough velocities, the trajectory
essentially bounces off the origin.
3. Like John Denker, I became interested in the assymptotic
behavior. My interest in the question was stimulated by noticing
that, at any given time the acceleration is inversely proportional to
x. Thus, any solution that falls behind the analytical solution that
Carl gave will be in the process of catching up and vice-versa. John
has given an elegant scaling analysis of this behavior. I took a far
more seat of the pants approach, simply looking at the difference
between the analytical solution and a numerical solution for x_o = 4,
v_o = -2 (which gives very different early behaviors) as a function
of time. The numerical solution obviously starts out "ahead of" the
analytical solution but is passed at about t = 1.3. It regains the
lead at about t = 13.5 and is passed again at about t = 1220. That
behavior appears to continue with increasingly large periods between
passings. Moreover, the magnitude of the maximum difference gets
bigger between each passing. Thus, it seems that the convergence
involves oscillations of *growing* absolute magnitude. Nevertheless,
the graphs appear almost identical at large time because the
*relative* magnitude of the oscillations (i.e., their size relative
to the current values of x) decreases with time.