\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\newcommand{\R}{{\mathbb R}}
\newcommand{\points}[1]{\phantom{.}\hfill\textbf{(#1~points)}}
\begin{document}
\begin{center}
\begin{huge}
MATH 609-602: Numerical Methods
\end{huge}
\end{center}
\begin{tabular}{ll}
Lecturer: & Prof. Wolfgang Bangerth \\
& Blocker Bldg., Room 507D \\
& (979) 845 6393 \\
& \texttt{bangerth@math.tamu.edu}\\[5pt]
Teaching Assistant: & Seungil Kim \\
& Blocker Bldg., Room 507A \\
& (979) 862 3259 \\
& \texttt{sgkim@math.tamu.edu}
\end{tabular}
\section*{Homework assignment 11 -- due Tuesday 11/29/2005}
\paragraph{Problem 1 (Numerical solution of a vector-valued ODE).}
Reconsider Problem 2 from last week's homework. You were asked to compute the
solution to an accuracy of 100 meter, a task that proved almost impossible
with the first order forward Euler method. Solve the same problem with the
fourth order Runge-Kutta method and determine the step size $h$ you need to
achieve above accuracy.
\points{4}
\paragraph{Problem 2 (A modeling challenge).}
\textit{This problem is meant as a bonus question. It doesn't give a whole lot
of points, but if you feel bored sitting around the table with your family
during Thanksgiving, do what mathematicians typically do: scribble the
solution to questions like this on napkins or the back of envelopes. The
problem has two parts: a theoretical part that you can do on the napkin
(worth two points), and a practical part where you have to implement your
model on a computer (worth only 1 point, so that if you can't do it at home
it doesn't matter too much).}
Here's the theoretical problem: Thanksgiving turkeys aren't particularly good
at flying. They may try, but they don't really get into the air very gracefully
and for extended periods of time. Derive an ODE model for turkey flight that
takes into account the following rules (all quantities have units meter, meter
per second, etc, as appropriate):
\begin{itemize}
\item[a)] the turkey is initially at rest;
\item[b)] it then runs horizontally, accelerating at a modest pace of
$1.5$;
\item[c)] when it reaches the lift-off speed of $v=8$ it gets airborne;
from thereon, its vertical (upward) acceleration is $-4+v$ (in
other words, the initial upward acceleration after getting airborne is
$4$); at the same time, air friction reduces the horizontal
velocity by a decelleration of $-v^2/10$;
\item[d)] at some point, the turkey's speed will become too slow to sustain
flight, and it will fall back to earth.
\end{itemize}
To write an ODE model for this, you will have to use the following variables:
$x(t)$--horizontal distance from the starting point; $v(t)$--horizontal
velocity; $h(t)$--height above ground; $u(t)$--vertical velocity.
Practical part: Solve these equations from the turkey's start until where it
falls back down to earth. Plot $x(t)$ and $h(t)$ in a single plot to show the
turkey's trajectory. If you feel challenged, compute the length of the
flight in both seconds and meters.
Hint: A plot of $x(t)$ vs. $h(t)$ (i.e. the turkey's trajectory) would look
like the figure below.
\points{3}
\begin{center}
\includegraphics[width=.6\textwidth]{q2-1}
\end{center}
\vspace*{1cm}
\begin{center}
\bf Happy Thanksgiving!
\end{center}
\end{document}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End: