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\begin{document}

\newpage
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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\DESIGNING A FAIR HOME RUN STADIUM}





\markboth{Designing a Fair Home Run Stadium}{Designing a Fair Home Run Stadium}






\author{Brian Winkel, Director 
SIMIODE\\
Chardon OH 44024 USA\\
BrioanWinkel@simiode.org}

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\begin{abstract}
For a given playing field outline how high must the outfield fence be at each
point in order to make a homerun equally likely in all fair directions?
\end{abstract}

\section*{SCENARIO DESCRIPTION}

Consider a  baseball field with prescribed irregular outfield and fence configuration.

Suppose a player can hit a home run which just clears the fence at one given point if the ball leaving the bat (2 feet off the ground) has initial velocity v0 and launch angle at 45 degrees.  Assume no wind resistance.  Redesign the fences so that this player (and hence any player) has the same chance of hitting a home run to anywhere in the outfield (i.e.  in the outfield and fair!) 

We consider Hypothetical Stadium (HS) with the following outfield wall configuration. See Figure~(\ref{fig:3-35-Stadium-Layout}).   Home plate is at $(0, 0)$; left field foul pole is straight up the positive y-axis; right field foul pole is straight down the positive $x$-axis.

The left field outfield fence runs parallel to first base line for 135 ft at a distance of 360 ft from home plate.

The right field outfield fence is a straight line from right field foul line at 
$(409, 0)$ to the point $(300,252)$. 

Center field outfield fence is a circular wall going through deepest center field point out at $(200 \sqrt{2}, 200 \sqrt{2})$, point ${\rm LC} = (135,360)$ in left center, and point ${\rm RC} = (300,252)$ in right center.


\begin{figure}[!h]
\centering

\includegraphics[width=9cm]{3-35-Stadium-Layout.eps}


   
\parbox{7.5 true cm}{\caption{\label{fig:3-35-Stadium-Layout} Outfield wall configuration in our Hypothetical Stadium. Home plate is at the origin $(0,0)$.}}

\end{figure}

\begin{itemize}
  \item [a)] Write equations which completely describe the outfield fences and plot them with home plate at $(0, 0)$. 
\item [b)] Determine the farthest point of the outfield fence from home plate, measured along the ground to the base of the fence. 
\item [c)] Place a 10 ft high fence at this farthest point found in (b)  and determine the initial velocity a hitter must impart to a ball hit 2 ft high over home plate to get it to just clear the 10 ft high fence here.  We shall assume there is no air resistance.  Hint:  It is known (and you should be able to prove it is true!) that a hitter should always try to launch the ball from the bat at a $45\,^{\circ}$ angle with the horizontal for maximum range.
\item [d)] Now determine for every other point at the base of the outfield fence just how high the fence must be so that this player can hit a home run by launching a ball from his bat 2 feet above home plate with exactly the same initial velocity at a 45 degree angle with the horizontal. 
\item [e)] Discuss the reasonableness of your answers. Would baseball ``buy'' your idea?
\end{itemize} 

How would resistance alter (if at all) the shape of your wall?


\teacher{
\section*{NOTES FOR TEACHERS}

In Figure~\ref{fig:3-35-Stadium-HomeRunBall} we see a home run ball traveling to center right field (actually at an angle of $30\,^{\circ}$ off the right field foul line. Notice how hight the fence is in short left field to give fairness when the distance to the fence from home plate is small and how shallow the fence is at straight away center field and down the long right field foul line. 
\begin{figure}[h]
\centering

\includegraphics[width=13.0cm]{3-35-Stadium-HomeRunBall.eps}


   
\parbox{9.7 true cm}{\caption{\label{fig:3-35-Stadium-HomeRunBall} Home run ball headed to right center field, at angle of $30\,^{\circ}$ off the right field foul line in a totally fair stadium for home runs.}}

\end{figure}

We provide a complete analysis with detailed commentary in a Mathematics notebook, 3-35-Mma-T-StadiumDesign-TeacherVersion.nb and a pdf form of the same file for those who cannot read Mathematica.

The basic analysis is to construct an outfield fence (give height at each position along the fence) so that the same initial velocity and angle of flight will produce a home run (just clear the fence) in every direction in this ball park with outfield fence outline given.

First, we must determine a two dimensional projectile motion equation for each given angle from the right field foul line to ``hit'' an exact spot at the top of the fence along a prescribes outfield fence of given distance from home plate. We use the two differential equations (\ref{eq:two}) obtained from Newton's Second Law of Motion which says that the mass times the acceleration of the ball is equal to the sum of external forces acting on the ball. 

\begin{eqnarray}\label{eq:two}
\label{eq:vec}
m\cdot x''(t) &=& 0 \\
m \cdot y''(t) &=& - m\cdot g \, .
\end{eqnarray}

\noindent Here the assumption is that the launch angle for maximum drive is $\pi/4$. 

From (\ref{eq:two}) we can integrate twice and find that 
\begin{eqnarray}\label{eq:twosol}
\label{eq:vec}
x(t) &=& v_0 \sin(\pi/4) t \\
y(t,\theta) &=&   -\frac{1}{2}\cdot 32 t^2 + v_0 \sin(\pi/4) t + 2\, .
\end{eqnarray}


We use this information to determine the time it takes to get to the correct height at the time it takes to get the ball to the point on the top of the outfield fence which is at a given distance $x$. From here we ascertain the necessary initial velocity $v_0$ necessary to have the ball go over the fence at 10 ft high to the farthest point in the outfield, namely 409 feet down the right field line.

Now using this initial velocity, $v_0$, the $45\,^{\circ}$ launch angle, and the fact that our flight path for the ball stays in a plane (an assumption) we now wish to find out how high the fence must be at each point on the fence in all fair directions  so that a ball hit with initial velocity of $v_0 =115.538$ ft/sec will just clear the fence. The analysis is carried out, along with graphics, in the Mathematica notebook and accompanying pdf file of the same name, 3-35-Mma-T-StadiumDesign-TeacherVersion.nb.

\subsection*{ISSUES RELATED TO THE PROBLEM}





 
\subsubsection*{Prerequisites}

Students need to know how to build the equations of motion using Netwon's Second Law of Motion for a two dimensional projectile motion without resistance  in parametric or rectangular form and use this in context.

\subsubsection*{Time allotment - time management}

This is a big project and could be set up in one class hour period with several visits to the problem over a few day (or weeks) project period.
 

\subsubsection*{Extensions}
 

Do the same problem with resistance.  This would require a great deal of numerical work in the solving of differential equation model for two dimensional projectile motion as a system of such differential equations.
}

\section*{ACKNOWLEDGEMENT}


The production of this material was supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849 from 1993-1996 entitled Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. This material was produced by Dave Horn a high school mathematics teacher from Michigan City IN.


\end{document}




