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{\Large MAT 331, Fall 2002}\\[\baselineskip]
{\Large\bf Project 2: Gliders and Differential Equations}\\[.25\baselineskip]
{\large\it Due Mon, November 11}
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In class, we have been discussing a system of differential equations which
approximately models the flight of a glider. In this project, you are to
explore a similar system, which models such a glider with a propeller or
small engine attached. In this case, our model is
$$
\frac{dv}{dt} = -\sin\theta - 0.3\, v^2 +k, ~~~~~~
\frac{d\theta}{dt} = {{v^2 - \cos\theta}\over{v}}.
$$
Here, as in class, $v>0$ is the speed of the glider through the air ({\em
not} the horizontal speed), and $\theta$ is the angle the nose makes with
the horizontal direction. Note that we have fixed the drag coefficient to
be 0.3, but have added an additional term $k$ to account for acceleration
caused by the propeller.
\bigskip
You are to analyze and classify the solutions of this system for all
$k\ge 0$ (note that $k=0$ was covered in class). This means that you should
find the various ranges of $k$ where the behavior is qualitatively
different.
Such analysis should include a discussion of the existence of fixed points
(equilibrium solutions) and their linearizations (i.e., a discussion of
eigenvalues and eigenvectors and how this relates to the solutions), as well
as a discussion of the long-term behavior of the solutions.
In addition to the description of the trajectories in the
$(\theta,v)$-plane, you should also relate the solutions to properties
of glider flight. For example, discuss whether the glider eventually must
crash or can stay aloft indefinitely, whether the glider does loops, etc.
\bigskip
In your writeup, you should include a number of relevant pictures and
graphs, preferably (but not necessarily) produced by Maple. These pictures
should be used to illustrate your exposition, not merely included without
comment or reason (remember: ``a picture is worth a thousand words, but a
thousand pictures are worthless''). Please pay attention to clarity of
exposition; {\bf do not} merely hand in an annotated Maple worksheet. While
including relevant Maple commands is useful, your goal is to explain what
you are doing from a {\em mathematical} point of view, not to describe how
to use Maple to perform a certain task. You need not (although you can)
state or prove the relevant theorems, but you should explain how you are
using them to do your analysis.
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