Microworld: Calculus
in Action: Chapter 1, Section 3: Maximizing the Range and Chaos
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Author:James
E. White
In
Section 3, we return to the problem posed in Section 1: Ballistics,
finding the maximum range of the ballistic trajectory. Given that the projectile
was launched at altitude 10 meters with speed V
= 50 meters/sec, we asked which angle will cause our projectile to travel
the greatest horizontal distance. For that, we observed that the
correct angle would be that for which
You saw that such an angle will be a solution to the equation:
Newton's
method gives a solution that is slightly less than 45 degrees. We can check
the solution in the Ballistics Lab. We can also change the values of
V = 50 and h = 10 to find solutions in other cases. This raises
a delicate question. Will the solution eventually be less than 45 degrees
whenever V and h satisfy a certain condition: that h
is small compared to the height a vertical projectile would rise at speed
V ? We see that Newton's method alone cannot give the answer. Another
method of approximation is appropriate here.
So,
we approach this problem in a different way. The solution using Newton's Method
above required that the altitude h = 10 and the speed V = 50
be fixed in advance. Suppose we want an "engineering" solution to
the problem in which we seek a solution in terms of V and h.
Here, we want to leave V and h variable, but would like an approximate
angle for maximum range in terms of V and h . We also would
like conditions on V and h that guarantee that this approximation
is a good one. We see that differential approximation leads us to a quartic
equation that we can solve.
Finally,
will explore the fact that for certain functions, Newton's Method can lead
formally to chaos: chaotic sequences that do not converge to any point,
but wander chaotically in an interval, never settling down. In
other cases, it can lead to stable oscillations of period 2 or greater. This
behavior is essentially independent of the choice of start point. The function
pictured above whose graph has a vertical tangent at the zero is one of the
chaotic ones that we shall examine in the microworld. We will also explore
the well-known "logistic map" in the Recursive Sequences
lab.
The
following is the Table of Contents for this 7 page Microworld
- Section 2: Sequences and Iteration
- Section 2: Recursive Sequences
- Section 2: Newton's Recursive Method
- Symbolic Calculator
- Maximizing the Range with Newton's Method
- Newton's Method and Chaos
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| - James E. White, Ph.D. , Library Director, | ||
| author of this website, Mathwright 2000, MindScapes, | ||
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