ma010 - The Falling Ladder Paradox Paul Scholten Miami...

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The Falling Ladder Paradox Paul Scholten, Miami University, Oxford, OH 45056 Andrew Simoson, King College, Bristol, TN 37620 The College Mathematics Journal , January 1996, Volume 27, Number 1, pp. 49–54 A nyone who has studied calculus has probably solved the classic falling ladder problem of related rates fame: A ladder L feet long leans against a vertical wall. If the base of the ladder is moved outwards at the constant rate of k feet per second, how fast is the tip of the ladder moving downward? The standard solution model for this problem is to assume that the tip of the ladder slips downward, maintaining contact with the wall until impact at ground level, so that if the base and tip of the ladder at any time t have coordinates and Figure 1. The standard falling ladder model. respectively, the Pythagorean theorem gives see Figure 1. Differentiating with respect to time t yields the formula (1) The paradox in this solution is that as the ladder nears the ground, attains astronomical proportions. In fact, in [ 5 ] the student is lightheartedly asked to find (for a particular k and L ) at what height y the ladder’s tip is moving at light speed. Of course, the resolution of this paradox is that the ladder’s tip leaves the wall at some point in its descent. A few classroom experiments using a yardstick lend observational support for this explanation, for as the base of a stick or ladder is moved away from the wall at constant speed, at the moment of impact it appears as if the tip lands some small distance from the wall, although the action transpires so quickly and catastrophically that it is hard to be certain about what happens. A paper [ 2 ] in the physics literature points out the flaw of using (1) and demonstrates the correct model for the falling ladder. Our approach is somewhat simpler, making no use of the force exerted by the wall on the ladder’s tip; we furthermore show how to numerically plot the path of the ladder’s tip, from the time it leaves the wall until its crash landing. y ? y ? 5 2 kx y . x 2 1 y 2 5 L 2 ; s 0, y s t dd , s x s t d , 0 d
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Let’s determine the critical height at which the ladder leaves the wall and (1) ceases to be valid. We will do this by examining the differential equations governing these two different physical situations: the moving ladder supported by the wall and the
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ma010 - The Falling Ladder Paradox Paul Scholten Miami...

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