%% This LaTeX-file was created by <jl> Tue Feb 11 15:53:53 1997
%% LyX 0.10 (C) 1995 1996 by Matthias Ettrich and the LyX Team

%% Don't edit this file unless you are sure what you are doing.
\documentclass[12pt,letterpaper,oneside,onecolumn]{article}
\usepackage[]{fontenc}
\usepackage[dvips]{epsfig}

%%
%% BEGIN The lyx specific LaTeX commands.
%%

\makeatletter
\def\LyX{L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\spacefactor1000}%
\newcommand{\lyxtitle}[1] {\thispagestyle{empty}
\global\@topnum\z@
\section*{\LARGE \centering \sffamily \bfseries \protect#1 }
}
\newcommand{\lyxline}[1]{
{#1 \vspace{1ex} \hrule width \columnwidth \vspace{1ex}}
}
\newenvironment{lyxbibliography}
{
\begin{thebibliography}{99}}
{\end{thebibliography}}
\def\lxq{"}
\newenvironment{lyxcode}
{\list{}{
\rightmargin\leftmargin
\raggedright
\itemsep 0pt
\parsep 0pt
\ttfamily
}%
\item[]
}
{\endlist}
\newcommand{\lyxlabel}[1]{#1 \hfill}
\newenvironment{lyxlist}[1]
{\begin{list}{}
{\settowidth{\labelwidth}{#1}
\setlength{\leftmargin}{\labelwidth}
\addtolength{\leftmargin}{\labelsep}
\renewcommand{\makelabel}{\lyxlabel}}}
{\end{list}}
\newcommand{\lyxletterstyle}{
\setlength\parskip{0.7em}
\setlength\parindent{0pt}
}
\newcommand{\lyxaddress}[1]{
\par {\raggedright #1 
\vspace{1.4em}
\noindent\par}
}
\newcommand{\lyxrightaddress}[1]{
\par {\raggedleft \begin{tabular}{l}\ignorespaces
#1
\end{tabular}
\vspace{1.4em}
\par}
}
\newcommand{\lyxformula}[1]{
\begin{eqnarray*}
#1
\end{eqnarray*}
}
\newcommand{\lyxnumberedformula}[1]{
\begin{eqnarray}
#1
\end{eqnarray}
}
\makeatother

%%
%% END The lyx specific LaTeX commands.
%%

\pagestyle{plain}
\setcounter{secnumdepth}{3}
\setcounter{tocdepth}{3}
\begin{document}

lift=body weight. \( mg=C_{l}\rho \frac{v^{2}}{2}A_{wing} \) with \( C_{L}=2\pi \alpha  \)with \( \alpha \sim 6 \)degrees. so \( \frac{C_{L}}{2}\sim .3 \). 

Then cruise velocity set by \( \frac{mg}{A}=.3\rho v^{2} \). So \( v=5m/s*(\frac{\frac{m}{A}}{kg/m^{2}})^{\frac{1}{2}}(\frac{\rho _{sealevel}}{\rho })^{\frac{1}{2}} \).

\begin{itemize}

\item for seagulls, m=.37kg, A=.115m*m

\item 747, \( m/A=600kg/m^{2} \) so v=\( 120m/s*2(densityratio) \)

\end{itemize}

Scaling: \( A\sim L^{2} \) , and \( m\sim L^{3} \), so \( v\propto m^{\frac{1}{6}} \)

737 has short wings to increase speed.

Power required to fly:

drag: \( P_{I} \) induced drag + \( P_{p} \) parasitic drag...friction.

\( lift/area=\frac{mg}{Ll} \) mass of air deflected down, something like \( L^{2}\rho v \). Then \( P_{I}\sim \frac{1}{2}v_{z}^{2}(L^{2}\rho v)\propto \frac{1}{v} \)

\( P_{p}=\frac{1}{2}C_{D}\rho v^{3}A \). There is a minimum power needed to fly.

\( v^{4}_{min}=\frac{1}{3}\frac{(mg)^{2}}{\rho ^{2}L^{2}AC_{D}} \)

Velocity for maximum range: \( v^{4}_{min}=\frac{(mg)^{2}}{\rho ^{2}L^{2}AC_{D}} \)

This means minimum power to fly is: \( mg(\frac{mg}{\rho L^{2}})^{\frac{1}{2}}(\frac{C_{D}A}{L^{2}})^{\frac{1}{4}}\sim mgv(\frac{l}{L})^{\frac{1}{2}} \). Maximum distance is \( E_{fuel}\frac{v}{P}=\frac{E_{fuel}}{mg}C_{D}^{\frac{-1}{2}}(\frac{A}{L^{2}})^{-\frac{1}{2}} \). Assume
\( \frac{E_{fuel}}{m}=4*10^{11}erg*.2 \), you get: \( 10^{3}km*C_{D}^{-\frac{1}{2}}(\frac{A}{L^{2}})^{-\frac{1}{2}} \)

\end{document}
