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\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.}}
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\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}
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\item What is an appropriate initial value condition for the problem?

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\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?

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\item Find the sprinter's acceleration function $a(t)$.

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\item Solve the initial value problem for $v(t)$ symbolically for general parameters $A$ and $B$. 

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\item Find the distance function $s(t)$ travelled in a sprint symbolically for general parameters $A$ and $B$.

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\item Draw a graph of the distance, velocity and acceleration functions over a reasonable interval, using the 1973 parameters.

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\item When does the maximum acceleration occur, and what is it? 

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\item Using the 1973 parameters, how long does it take for the acceleration to drop to 10\% of its maximum value?

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\item What is the  1973 runner's maximal speed? In races of 100~m and 200~m, is his final speed the same? Explain!
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\noindent{\em How did Keller find the parameters $A$ and $B$? In the second part of this laboratory you will address this question.

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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}
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\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.}

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\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?

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\item Has athletic performance in 100 m sprints improved since 1973?
Explain your observations!

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\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?

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\end{enumerate}
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\flushright\tiny\copyright\  H. Knaust. \today .
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