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How Not to Land at Lake Tahoe!
Richard Barshinger
The American Mathematical Monthly,
May 1992, Volume 99, Number 5, pp. 453–455.
T
he following problem gives a simplified model of landing an airplane. It is
adapted and extended from Trim [
1
] and is regularly presented in first semester
calculus at my campus, where it is unanimously enjoyed and wins some converts
to the methods of calculus.
Problem.
An aircraft landing approach pattern is shaped generally as in Figure 1 below.
The following conditions are imposed:
a) The cruising altitude is
h
when descent begins at a horizontal distance
L
from
the airstrip.
b) A constant horizontal airspeed
U
must be maintained throughout descent
(somewhat unrealistic).
c) At no time must the vertical component of acceleration exceed (in absolute
value) some fixed constant
k
,
where
g
is the acceleration constant for
gravity; i.e.,
(English units).
Model the plane’s approach path by means of a cubic polynomial, using a coordinate
system with origin at the beginning of the runway, so that descent starts at the point
in units of your choice. Impose suitable conditions at the beginning of
descent and at touchdown. Discuss the implications of condition c) above, in the cases:
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 Winter '08
 STAFF
 Math, Calculus

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