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\begin{document}

\newpage
% You only need to change the NAME OF SCENARIO in the next line and delete the footnote. See above for instructions on toggling from Teacher Version to Student Version.
\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\RELAY THE BALL HOME}





\markboth{Relay the Ball Home}{Relay the Ball Home}






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

\makeStitlePDFLaTex


\begin{abstract}
We use a differential equations of one dimensional projectile motion and an integration of velocity for total distance to model the relay between an outfielder and an infielder in throwing the ball to home plate. We determine the location of the infielder so as to minimize the time of the relay.
\end{abstract}


\section*{SCENARIO DESCRIPTION}

A baseball player (outfielder) throws a ball with initial velocity  135 ft/s  (about 92 mi/hr -  less than a good pitcher's high speed fastball of 100 mi/hr) toward home plate some 400 feet away.  Another player  (an  infielder who cuts the ball off - catches it  -  from the outfielder's throw) places himself in direct line between the outfielder and home plate and catches the ball $x$ feet from home plate and then throws (relays) the ball home with initial velocity 110 ft/s.   

It  takes  2  seconds to accomplish the relay.  Assume  that  due  to  air  resistance the velocity $v = v(t)$ obeys the differential equation (\ref{eq:relay}).
 \begin{equation}\label{eq:relay}
v'(t)  = -0.1 v(t)\, ,
 \end{equation}
and ignore vertical motion.  

\begin{enumerate}
\item Find the time, $T$,  it takes for the ball to reach home  plate as  a function of $x$, the distance from the plate of the infield cut-off player. 
\item Find x which minimizes $T$.
\item For a range of infielder capabilities, i.e. initial  throwing velocities  from  60  ft/s to 140 ft/s, prescribe the cutoff distance $x$ and the total time for the relay to the plate. 
\item Consider the infielder who can throw 110 ft/s.  Would you rather improve the infielder's  relay  time, say down to 1.8 sec, or increase his initial velocity on the ball  by 10 ft/s?  Explain.  
\end{enumerate}

\teacher{
\section*{NOTES FOR TEACHERS}

This activity is an exercise in solving a differential equation and integrating the result wherein an objective function of time  in terms of the location of the infielder from home plate at distance $x$ ft from home plate is minimized and further analyzed.

This activity can be done in class in a 20 minute session as students work in groups and help each other to realize the necessary steps for solution. We presume they have access to technology, preferably a computer algebra system. We do our work in a Mathematica notebook 3-54-T-Mma-TeacherVersion which we offer int he Supplemental Docs for this modeling scenario and an accompanying pdf version of this file for those who do not read Mathematica files.

We first need to solve the differential equation for $v(t)$, the velocity  of the ball given the governing equation prescribed, i.e. $v'(t) = - 0.1 v(t),\quad v(0) = v_0$  and then for the distance the ball will travel with that varying velocity.  We obtain

\begin{equation}\label{eq:vel}
v(t) = v_0 e^{-0.1 t}\, .
\end{equation}

The total distance traversed for this leg of the relay is then

\begin{equation}\label{eq:dist}
s(t) = \int_{T = 0}^t  v(T)\, dT = \int_{T = 0}^t  v_0 e^{-0.1 T}\, dT = 10 v_0(1 - e^{-0.1 t})\, .
\end{equation}

We then need to find the time of the throw of the outfielder, tOut,  from distance 400 ft to distance $400 - x$, i.e to where the infielder is standing at distance $x$ ft from home plate. Using Mathematica's Solve command we find tOut:

\begin{equation}\label{eq:tOut}
{\rm  tOut} = -10 \log \left(\frac{0.1 (10 
   v_1+x-400)}{v_1}\right)\, .
\end{equation}

Next, we determine the time of the throw of the infielder, tIn, from distance $x$ ft from home plate to home plate. Again we use Mathematica's Solve commend to determine tIn.

\begin{equation}\label{eq:tIn}
{\rm tIn} = -10 \log \left(\frac{0.1 (10  v_2-
   x)}{v_2}\right)
  \end{equation} 
  
  Finally, we determine the total time of the relay, Tot, in terms of the outfielder's throwing velocity $v_1$, the infielder's throwing velocity $v_2$, and the time of the relay, Ex, in which the infielder transfers the ball from glove to hand and releases it in a throw to the plate.
  
  \begin{equation}\label{eq:Tot}
{\rm Tot} =  {\rm Ex} -10  \log \left(\frac{0.1 (10 
   v_1+x-400)}{v_1}\right) -10  \log \left(\frac{0.1 (10  v_2-
   x)}{v_2}\right)
  \end{equation} 
  
We determine the distance, xOpt, at which  Tot as a function of $x$ is minimum by taking the first derivative of Tot with respect to $x$ and solve for $x$ to obtain the optimal location of the infielder given these speeds:
  
  \begin{equation}\label{eq:opt}
 {\rm xOpt} = 5 (40 - v_1 + v_2) \, .
  \end{equation}
  
Using the numbers $v_1 = 135$ ft/s and $v_2 = 110$ ft/s we determine the distance $x$  the infielder should stand from home plate is  75 ft for total minimum time for the relay to the plate, which is well into the infield part of the field. So we have the minimum total time for the throw  when the outfielder can throw the ball at 135 ft/s, the relay exchange is 2 s, and the infielder can throw the ball at 110 ft/s is  5.4603 s with the infielder standing $x = 75$ ft from home plate.
  
We refer to the files for this modeling scenario,  3-o54-T-Mma-TeacherVersion, for the rest of the solution.
  
\section*{ACKNOWLEDGMENT}


The production of this material was supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849 from 1993-1996 which was  entitled Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. This, and other problems, are located at {\tt http://www.rose-hulman.edu/Class/CalculusProbs/}.


The initial problem  concept  for this scenario and   wording  came from correspondence with  Roland Minton,  Department  of  Mathematics,  Roanoke  College, Salem VA 24153 USA, back in the early 1990's
.}


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