Claim: A ladder will fall infinitely fast when pulled. An incorrect proof using
ID: 1948795 • Letter: C
Question
Claim: A ladder will fall infinitely fast when pulled. An incorrect proof using Calculus and Physics. Find which step is incorrect. Write down which one and why. It might be useful to experiment with a ladder or even with a pen or pencil! Step 1: As shown above, let x = x(t) denote the horizontal distance from the bottom of the ladder to the wall, at time t. Step 2: As shown, let y = y(t) 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, x2(t) + y2(t) = L2. Step 4: Therefore, y(t) = root L2 - x2(t). Step 5: Differentiating, and letting X' and y' (respectively) denote the derivatives of x and y with respect to t, we get that (look up the chain rule from Calculus 1) y' = xx'/root L2 - x2 Step 6: Since the bottom of the ladder is being pulled with constant speed v. we have x' = V, and therefore y' = - xy/root L2 - x2 Step 7: As x approaches L, the numerator in this expression for y' approaches -Lv which is nonzero, wrhile the denominator approaches zero. More precisely lim x rightarrow L y' = limx rightarrow L -xv/root L2 - x2 = - infinity Step 8: Therefore, y approaches - infinity 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.Explanation / Answer
step 3 is not correct There is the implicit assumption here that the top of the ladder remains resting against the wall. However, that is not always true. Once the ladder has reached a sufficiently small angle to the horizontal, your pulling of the bottom away from the wall will actually cause the top to pull away from the wall too. When this happens, there is no longer the relationship , because x, y, and L no longer form the sides of a closed right triangle If you want to verify this for yourself, just try it out on your favourite ladder. You will see that the top pulls away from the wall just before the ladder becomes horizontal. To understand mathematically (or rather, physically) why this is true, suppose for a moment that there was no gravity. Then, when you pull the ladder, you will pull the entire ladder as a unit, with both top and bottom moving away from the wall with the same speed. The only reason the top stays in contact with the wall for most of the time is because of gravity. If the wall were not there, gravity would cause the top of the ladder to fall to the ground Therefore, the effective force of gravity on the top of the ladder (actually, the combination of gravity and the rigidity of the ladder that prevents the ladder from shrinking in length and collapsing in a heap) is as shown in the diagonal purple arrow in the picture below. This effective force can be thought of as the sum of two forces (the horizontal and vertical blue arrows): one that pushes the top of the ladder against the wall, and one which slides the top of the ladder down the wall. As the ladder gets more and more horizontal, the effective force acting on the top of the ladder (the diagonal arrow) becomes closer to vertical, and the force pushing the top against the wall will become less and less Once the force pushing the top of the ladder against the wall becomes less than the force with which you are sliding the bottom of the ladder away from the wall, your force will prevail and you will pull the ladder away from the wall. Once this happens, the top will be in free-fall and no longer governed by the same set of equations.