Sliding Ladders in the Abstract

Consider a ladder leaning against a wall, and suppose the base is
pulled outward at a fixed rate.  Eventually, if we assume the top of
the ladder maintains sliding contact with the wall, it must achieve
infinite speeds, but of course this is unrealistic.  To simplify the
description of what actually happens, assume the ladder is massless 
and it supports a mass at the top-most point (in contact with the 
wall).  If we keep pulling the lower end of the ladder outward at a 
constant horizontal rate, the top end will lose contact with the 
wall precisely when the acceleration of the kinematic point of 
contact exceeds the acceleration of gravity.  From then on (until 
it hits the floor) the top end will be in free-fall.  

On the other hand, if we somehow constrain the top end to remain in 
sliding contact with the wall, then the force required to pull the 
lower end would approach infinity as the ladder approaches horizontal.
Of course, taking into account the actual distribution of mass along 
the ladder, this analysis would have to be adjusted for the ladder's 
moment of inertia, etc.

But suppose we disregard these physical details, and assume there are
no masses involved at all, i.e., we wish to interpret the problem in
purely kinematical terms, without regard to masses and forces, etc.,
and we assume the top of the ladder is constrained to remain in 
sliding (frictionless) contact with the wall.  On this basis, we might
ask, when does our premise fail, viz, at what point do the stated
conditions become logically untenable?

If L is the length of the ladder and x is the distance between the
wall and the bottom of the ladder, then it might appear that the 
model fails for all x > L, reasoning that for all such x it's 
impossible for the top of the ladder to be in contact with the wall.  

However, since the vertical height at which the ladder contacts 
the wall is  h = sqrt( L^2 - x^2 ),  the height of contact when 
x = 2L is simply h = iL*sqrt(3), and the ladder makes an angle of 
invtan(i*sqrt(3)/2) with the floor.  At this point the speed 
of the ladder top is -2iv/sqrt(3), where v is the constant 
horizontal speed of the bottom (i.e., v = dx/dt).  So the 
"model" is well-behaved for all values of x, with the possible 
exception of x=L, where the velocity exhibits a singularity.

This singularity is very reminiscent of the singularity at the 
event horizon of a black hole when viewed in terms of the usual 
Schwarzchild coordinates.  I haven't worked out the details, 
but I suspect this ladder singularity is likewise dependent on 
the choice of coordinates, and that the actual "surface" of the 
solution is well-behaved at x=L for some suitable choice of 
"proper time".

Another interesting question raised by this general approach (i.e.,
extending a problem beyond its "real" representability) is whether 
the "point of departure from reality" necessarily coincides with a
singularity, or with a transition to complex parameters.  I can 
think of at least one case where it does not.

Consider an elipse of the form  (x/a)^2 + (y/b)^2 = 1.  The curvature
of the perimeter at any given x value is

                               -ab
             C  =  ------------------------------
                   {a^2 - [1 - (b/a)^2]x^2}^(3/2)

Notice that this curvature is a well-behaved real function even for
x > a, meaning that the curvature of the perimeter of an elipse 
extends in a natural way beyond the bounds of the elipse itself.