1. ## Find out fallacy(2)

Consider a ladder of length L leaning against a frictionless wall which is at right angles to the ground. You pull the bottom of the ladder horizontally away from the wall, at constant speed v. The claim is that this causes the top of the ladder to fall infinitely fast.

Common sense tells us this can't possibly be true, but can you find the flaw in the following supposed "proof" of this claim?

The Fallacious Proof:

* Step 1: As shown, let x denote the horizontal distance from the bottom of the ladder to the wall, at time t.

* Step 2: As shown, let y denote the height of the top of the ladder from the ground, at time t.

* Step 3: Since the ladder, the ground, and the wall form a right triangle, $\displaystyle x^2+y^2=L^2$ .

* Step 4: Therefore, $\displaystyle y=\sqrt{L^2-x^2}$ .

* Step 5: Differentiating, and letting x' and y' (respectively) denote the derivatives of x and y with respect to t, we get that

$\displaystyle y'=\frac{xx'}{\sqrt{L^2-x^2}}$

* Step 6: Since the bottom of the ladder is being pulled with constant speed v, we have x' = v, and therefore

$\displaystyle y'=\frac{xv}{\sqrt{L^2-x^2}}$

* Step 7: As x approaches L, the numerator in this expression for y' approaches -Lv which is nonzero, while the denominator approaches zero.

* Step 8: Therefore, y' approaches $\displaystyle -\infty$ as x approaches L. In other words, the top of the ladder is falling infinitely fast by the time the bottom has been pulled a distance L away from the wall.

2. Spoiler:
The problem assumes that the top of the ladder is always in contact with the wall. It is not. When the bottom of the ladder is a certain distance away from the wall, the top of the ladder will not longer be touching the wall and the Pythagorean relationship will break down.

3. Suppose however that the top of the ladder were attached to the wall via a sliding pivot, say, so that it could move freely up and down but not away from the wall. What then?

I think the problem in this case would be the difficulty in trying to maintain a constant speed for the lower end of the ladder. In the limiting case when the ladder is horizontal and therefore fully extended from the wall, how is the constant speed to be maintained?

You will probably agree that if the ladder were to fall freely in this situation then the lower end of the ladder would slow down as the upper end approached the floor closely, coming to instantaneous rest when in the horizontal position. To overcome this natural motion, I suspect an ever increasing force would need to be applied at the lower end, a force which would "tend to infinity" as the ladder became horizontal. The effects of such a force on the ladder, on the wall and on the person pulling like crazy at the lower end might well be inconvenient for all concerned.

But in theory, I suppose, an infinite force would be consistent with the infinite speed suggested by the proposer.