\documentclass[12pt]{article} \pagestyle{empty} \renewcommand{\today}{February 10, 1997} \special{src: 5 LAB2A.TEX} %Inserted by TeXtelmExtel % \ilf is inline formula in displaystyle \newcommand{\ilf}[1]{$\displaystyle #1 $} \special{src: 10 LAB2A.TEX} %Inserted by TeXtelmExtel \special{src: 13 LAB2A.TEX} %Inserted by TeXtelmExtel \begin{document} \noindent{\bf Math 3226 \hfill Laboratory 2A}\\ \begin{center} {\bf Ordinary Differential Equations and World-class Sprints\footnote{This laboratory is based on the article {\em ``The ODE of World-Class Sprints''} by Steven R. Dunbar, which appeared in C $\cdot$ ODE $\cdot$ E, Spring 1994.}} \bigskip\end{center} \special{src: 24 LAB2A.TEX} %Inserted by TeXtelmExtel \noindent{\em According to a theory put forward by J.B. Keller\footnote{J.B. Keller,{\em ``A Theory of Competitive Running''}, Physics Today, 9-1973, p. 43.}, track sprints of up to 300~m can be described by the following differential equation:% \renewcommand{\thefootnote}{}\footnote{\bf Keywords: first order linear differential equation, modeling.} \[\frac{dv}{dt}=A-\frac{v(t)}{B},\] where $v(t)$ denotes the speed of the sprinter. Keller estimated the constants $A$ and $B$ for a (male) world-class sprinter in 1973 as follows:} \[A=12.2 \mbox{ m}/\mbox{sec}^2,\ B=0.892 \mbox{ sec.}\] \begin{enumerate} \renewcommand{\theenumi}{{\bf \arabic{enumi}}} \item What is an appropriate initial value condition for the problem? \special{src: 35 LAB2A.TEX} %Inserted by TeXtelmExtel \item In your own words, explain each of the terms in the differential equation. What is an interpretation of the parameter $B$? Does this differential equation seem reasonable in your experience? \special{src: 39 LAB2A.TEX} %Inserted by TeXtelmExtel \item Find the sprinter's acceleration function $a(t)$. \special{src: 43 LAB2A.TEX} %Inserted by TeXtelmExtel \item Solve the initial value problem for $v(t)$ symbolically for general parameters $A$ and $B$. \special{src: 47 LAB2A.TEX} %Inserted by TeXtelmExtel \item Find the distance function $s(t)$ travelled in a sprint symbolically for general parameters $A$ and $B$. \special{src: 51 LAB2A.TEX} %Inserted by TeXtelmExtel \item Draw a graph of the distance, velocity and acceleration functions over a reasonable interval, using the 1973 parameters. \special{src: 55 LAB2A.TEX} %Inserted by TeXtelmExtel \item When does the maximum acceleration occur, and what is it? \special{src: 59 LAB2A.TEX} %Inserted by TeXtelmExtel \item Using the 1973 parameters, how long does it take for the acceleration to drop to 10\% of its maximum value? \special{src: 63 LAB2A.TEX} %Inserted by TeXtelmExtel \item What is the 1973 runner's maximal speed? In races of 100~m and 200~m, is his final speed the same? Explain! \end{enumerate} \special{src: 68 LAB2A.TEX} %Inserted by TeXtelmExtel \noindent{\em How did Keller find the parameters $A$ and $B$? In the second part of this laboratory you will address this question. \special{src: 72 LAB2A.TEX} %Inserted by TeXtelmExtel Race officials often record the split-times of runners in addition to their final times. The following table contains split-times for some sprinters during the 1993 World Championships in Stuttgart, Germany:}\medskip \begin{center} \begin{tabular}{|l|r|r|r|r|} \hline {\bf Name}&{\bf 30 m}&{\bf 60 m}&{\bf 80 m}&{\bf 100 m}\\ \hline \multicolumn{5}{|l|}{\bf MEN}\\ \hline Linford Christie&3.85&6.45&8.15&9.87\\ \hline Andre Cason&3.83&6.43&8.15&9.92\\ \hline Carl Lewis&3.95&6.59&8.30&10.02\\ \hline \multicolumn{5}{|l|}{\bf WOMEN}\\ \hline Gail Devers&4.09&6.95&8.86&10.82\\ \hline Gwen Torrance&4.14&7.00&8.92&10.89\\ \hline Irina Privalova&4.09&7.00&8.96&10.96\\ \hline \end{tabular} \end{center} \begin{enumerate} \renewcommand{\theenumi}{{\bf \arabic{enumi}}} \addtocounter{enumi}{9} \item Using the symbolic solution you found earlier, give estimates for the parameters $A$ and $B$ for each of the sprinters listed. {\em One way to do this is to first eliminate $A$ by taking the ratios of the $s(t)$ values, then solving for $B$ in the resulting equation using one of the split times.} \special{src: 103 LAB2A.TEX} %Inserted by TeXtelmExtel \item Graph the distance function (with the parameters $A$ and $B$ you obtain) for each of the sprinters, and compare it to the given split times. How good is the fit? Do you think that Keller's model describes sprints adequately? \special{src: 108 LAB2A.TEX} %Inserted by TeXtelmExtel \item Has athletic performance in 100 m sprints improved since 1973? Explain your observations! \special{src: 113 LAB2A.TEX} %Inserted by TeXtelmExtel \item Compare the parameter values $A$ and $B$ for men and women. Some track experts speculate that men's and women's abilities are becoming identical. Do your observations support this? \special{src: 117 LAB2A.TEX} %Inserted by TeXtelmExtel \end{enumerate} \vspace*{\fill} \flushright\tiny\copyright\ H. Knaust. \today . \end{document}