Intuition behind the "infinite velocity" of a falling ladder

In Calculus there is a "classic" related rates problem involving a falling ladder. Say the ladder is $25$ ft tall and is leaning against a wall. The bottom edge of the ladder is pulled away from the wall at a constant rate of $2$ ft/sec; as it moves, the top of the ladder slides down the wall. The student is asked to express the downward velocity of the top of the ladder in terms of the position $x$ of the bottom of the ladder, and finds that $$\frac{dy}{dt}=-\frac{2x}{\sqrt{625-x^2}}$$ Of course it makes sense that the velocity should only be defined up to $x=25$, because beyond that point the ladder comes away from the wall. But it seems strange (even to me, who has taught this stuff) that the downward velocity approaches $\infty$ as $x \to 25$. Why is that a "reasonable" result? If I imagine a speedometer attached to the top of the ladder, it's hard for me to believe that in the moments before the ladder hits the ground the speedometer readout increases without bound.

Is there an intuitive explanation of why the downward velocity of the top of the ladder ought to diverge to infinity as the ladder hits the ground?

In terms of mechanical systems, this is what we would call a singular configuration which is a position where the system looses a degree of freedom. In this case when $x=25$ the system cannot move by pushing in the $x$ direction (note that I did not say that the system cannot move, only that it cannot move by pushing in the $x$ direction). The reason that $\frac{dy}{dt}=\infty$ is unreasonable is because that it is equally unreasonable that $\frac{dx}{dt}=2$ at $x=25$. The same thing happens in other systems. An example would be a piston in an engine when it is at top dead center... the system looses a degree of freedom.. i.e. no matter how hard you push on the piston, it cannot (in theory) move the crank shaft. Furthermore if we computed the crankshaft speed based on the piston speed, we would get the same infinite result. The case of Gimbal Lock is another example of a situation where this occurs... the system looses a degree of freedom, in this case from three to two.