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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View Full DocumentLet’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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 Calculus, Pythagorean Theorem, Moment Of Inertia, ladder

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