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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Whiffle Ball Fall}






\markboth{Whiffle Ball Fall}{Whiffle Ball Fall}

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

\makeStitlePDFLaTex


\begin{abstract}
We are given data on the time and position of a whiffle ball as it falls to the ground. We attempt to model the falling ball and we confront the different resistance terms and models. Finally, we introduce a new way of comparing models,  the Akaike Information Criterion, and apply it to our models.
\end{abstract}



\section*{SCENARIO DESCRIPTION}

We are given data on time and position of a free falling whiffle ball (see Figure~\ref{fig:3-13-whiffleball}) from a university laboratory in Table~\ref{table:data}. We will make a mathematical model of this phenomenon and estimate the parameters in the model. We shall also compare the model's predictive power with the actual data. 



\begin{figure}[h]
\centering
\includegraphics[width=11cm]{3-13-whiffleball.eps}

\vspace*{-8 true cm}
\parbox{8 true cm}{\caption{\label{fig:3-13-whiffleball}  Whiffle ball lying on the ground next to a bat used to propel the ball in competition.}}

\end{figure}
The teacher who wrote up  this experiment \cite{BrandQuintanilla2013}  his students conducted, says,

\begin{quote}

To begin, teams of students are given a few softballs, a few whiffle balls,
a strip of paper (at least 20 feet long) marked at every foot or half-meter, a pad of
large adhesive paper, foam pads to minimize bouncing (optional), and a long tape
measure. Each student is also asked to bring, if possible, a digital camera. On the day
of the activity, the class meets at a place (or places) suitable for dropping balls from
various heights. Open stairwells or sets of balconies do well. Naturally, safety is a
consideration, since softballs falling large distances can be dangerous.

Each team hangs the long strip of paper so that it touches the ground, and someone
goes to a spot where a softball and a whiffle ball can be dropped simultaneously in
front of the lined paper. The distance from this point to one of the lines on the paper is
measured, recorded on adhesive paper large enough to be readable from a photograph,
and placed on the chosen line. As the balls are dropped, the other members of the team
photograph them against the backdrop of the lined paper before the softball lands. With a little practice, students seem to be able to get useful data from about
half their attempts, though capturing both balls in the same photograph is difficult
from large heights. The team repeats this procedure from at least four different heights,
perhaps capturing more than one photograph from each.\cite[p.~58]{BrandQuintanilla2013}
\end{quote}

\begin{table}[h]\label{table:data}
\begin{center}
\begin{tabular}{|l|r|} \hline \hline
Time (s) & Distance (ft)\\ \hline \hline
\multicolumn{1}{|c|}{0}& \multicolumn{1}{|c|}{0}\\ \hline
0.32601& 1.6154\\ \hline
0.72587& 8.0730\\ \hline
0.86624& 11.3219\\ \hline
0.97305& 14.2320\\ \hline
1.18558& 20.0783\\ \hline
1.28771& 23.4167\\ \hline
1.41354& 27.3056\\ \hline
1.61894& 34.1013\\ \hline
1.76001& 39.3958\\ \hline \hline
\end{tabular}

\parbox{7.1 true cm}{\caption{\label{table:data} Time and position data on a free falling whiffle ball which is dropped from rest. }}
\end{center}

\end{table}

The mass of the whiffle ball is not given, nor are its dimensions.

\subsubsection*{Activity 1}

Create a Free Body Diagram of all the forces acting on the whiffle ball. Be sure you make assumptions clearly and use the assumptions.


\subsubsection*{Activity 2}

Use Newton's Second Law of Motion which states that the sum of the forces acting on a body are equal to the product of the body's mass and acceleration to produce an equation in which one side is $m y''(t)$ where $y(t)$ is the distance fallen (in ft) at time $t$ (in sec) and $m$ is the mass of the whiffle ball. Comment on the reasonableness of your model.


\subsubsection*{Activity 3}

Attempt to solve your model in Activity 2.  If you cannot solve it analytically then put in some numbers. Reasonable numbers for mass of a 20 g and its diameter is  7.2 cm.\cite{RossmannRau2007}  Be aware that the units in the data given are in feet and seconds! This will be significant in any analysis you do beyond this point.  you may rightfully assume the ball is at rest and has no velocity when the experiment start as it is merely dropped from a given height.



\subsubsection*{Activity 4}

Estimate the parameters in your model using the data to help you and then to verify the reasonableness of your estimates.

 \subsubsection*{Activity 5}

Offer a plot of your model with parameters estimated with the data to give a visual as to how good your modeling really is.



\teacher{


\section*{NOTES FOR TEACHERS}

The data in Table~\ref{table:data} was sent by John A.~Quintanilla, co-author of \cite{BrandQuintanilla2013},  to this author via email. In  \cite{BrandQuintanilla2013} the authors had their class collect the data on the falling whiffle ball in a very clever manner by using simultaneous falling softball as a stopwatch. 

The classical models for $y(t)$, the distance fallen in time $t$ are all given by (\ref{eq:model}):

\begin{equation}\label{eq:model}
m y''(t) = m g - k y'(t)^r \, .
\end{equation}

Quite often $r$ is presumed to be 1 or 2, but we address a more general analysis in our accompanying Mathematica notebook 3-013-Mma-T-WhiffleBallFall-TeacherVersion.nb along with pdf version of the file for those who do not have access to Mathematica.

This model comes directly from Newton's Second Law of Motion which states that the sum of the forces acting on a body are equal to the product of the body's mass and acceleration to produce an equation in which one side is $m y''(t)$ where $y(t)$ is the distance fallen in time $t$. One force (downward)  is that due to gravity $g$ (ft/s$^2$) acting on the mass $m$ and an opposite force (upward) is a resistive force proportional to ($k$) the velocity $y'(t)$ raised to some power. 


\subsection*{Summary of Analyses and Comparisons of Models}

We attempt to approximate $K = k/m$ and $r$ in the differential equation with  power resistance. 

 $m y''(t) = m g - k y'(t)^r$   or dividing through by $m$ to obtain  $y''(t) = g - K y'(t)^r$.
 

For $r = 1$ we have $K =  0.381793$ and ${\rm SSE} = 2.80858$.


For $r = 2$ we have $K =  0.0179628$ and ${\rm SSE} = 0.232112$.


For $r = 2.3755$ we have $K =  0.00545$ and ${\rm SSE} = 0.109867$.


Thus it would appear that our best fitting model is for $r = 2.3755$, where we have $K =  0.00545$ and $SSE = 0.109867$. Of course, there is no physics which could produce the value of $r$ we obtain through this analysis, but this model will fit the data and be predictive. We demonstrate that all three models do a very good job in modeling the data in Figure~\ref{fig:3-13-FinalPlots}.



\begin{figure}[h]
\centering
\includegraphics[width=11cm]{3-13-FinalPlots.eps}


\parbox{8 true cm}{\caption{\label{fig:3-13-FinalPlots}  Final plots of data with the three models for $r=1$, $r=2$, and $r = 2.3755$.}}

\end{figure}


This is where the complexities of physical phenomena and attempts to scientifically explain these phenomena come face to face with data and numerical accuracy. At times there is not exactness, not clarity of one answer. However, there is no denying that with regard to  the Whiffle ball data the general model with $r = 2.3755$ cuts down the error between theory and observations (1) with $r=1$ from $2.80858$ to $0.109867$ - that is a reduction of over 96\% in error - and (2) with $r=2$ from $0.232112$ to $0.109867$ - that is a reduction of over 52\% in error. We rest in our accuracy and algorithms for determining the least sum of square errors and the attendant estimates of the parameters, $K$ and $r$ and let our students (and the reader!) feel the ambiguities of the situation. Just settling for one approach without considering options is unsettling. However, in this milieu there is much to think about and we are in the business of providing thoughtful experiences for our students and our readers. Thus we are not that unsettled!

This is an example of what the Australian philosopher, Robert E.~Goodin, offers up in his treatise {\it On Settling} \cite{Goodin2012}. Just what will it take to satisfy us?  What can we live with? What will we settle for?  What offers the best model explanation of an observed phenomenon? 


\section*{Comparing Models}

Thus we see that we  have several models for the falling whiffle ball.  The first (Model (1)), with one parameter $K$, using $r=1$; the second (Model (2)) with one parameter $K$, using $r=2$; and the third  (Model (3)) with two parameters $r$ and $K$.  The first has a sum of square errors of 2.80858. The second has a sum of square errors of 0.232112. The third has a sum of square errors of 0.109867. 

Some texts refer to the sum of square errors (SEE) as the {\it residual sum of errors\/} (RSE). Now certainly if we increase the number of parameters we can get a better fitting model in terms of reducing the SSE. Indeed, if we took a 10$^{\rm th}$ degree polynomial we could fit it to go through all the 10 data points and our SSE would be 0. However, this model would not be valuable in any predictive sense. Moreover, it would not explain anything as to the actions of the underlying model. So there is a trade-off, namely,  if we increase the number of parameters in our model we can decrease our SSE. We can ask which model  is best, in general. Namely which model keeps both the SSE and the number of parameters low.

In 1974 a formula  was published  \cite{Akaike1974} which would allow us to use both SSE and number of parameters at the same time to compare models. The statistic is called the {\it Akaike Information Criterion\/} or AIC.  A reasonable formulation (\ref{eq:AIC}) for our purpose is given in  \cite[p.~138]{Ledder2013}:


\begin{equation}\label{eq:AIC}
{\rm AIC}(n, k, {\rm SSE}) = n \ln\left(\frac{{\rm SSE}}{n}\right) + 2 (k+1) \,,
\end{equation}
where $n$ is the number of observations, $k$ is the number of parameters to be estimated, and SSE is the least sum of square errors.

AIC is used to compare models and the model with the smallest AIC is considered best of the set if the difference between that model's AIC and all other AIC's is more than 2. Otherwise the AIC cannot tell us which model is best.  AIC does not tell us if the model is good, only if it is the best of the ones being considered.

Smaller (even negative) values of AIC represent a higher level of statistical support for the corresponding model. The smallest AIC indicates the best model - in terms of balancing ACCURACY (minimum sum of square errors) and COMPLEXITY (number of parameters).  A difference of 2 in the AIC's is needed to suggest a significant difference. If all AIC's are close to each other, i.e. smallest one is less than 2 from the next smallest one, then we cannot tell which is the best model.

We apply  this AIC analysis to our situation.


\section*{SUMMARY OF ANALYSES}

We offer a summary (see Table~\ref{table:Summary}) of three models Model (1) - Model (3). Model (3) offers the lowest SSE.    Moreover,  Model (3)  possesses an AIC value  sufficiently below (more than 2) the other two models. This means   Model (3) is the best of the three models.  



\begin{table}[h]


\begin{center}\small 
\begin{tabular}{|c|c|c|c|c||} \hline
\multicolumn{1}{|c|}{Model} & \multicolumn{1}{|c|}{Parameters} & \multicolumn{1}{|c|}{$k$}& \multicolumn{1}{|c|}{SSE}&\multicolumn{1}{|c|}{AIC }\\ 
\multicolumn{1}{|c|}{Number} &  \multicolumn{1}{|c|}{Constant(s)}& \multicolumn{1}{|c|}{Values}&\multicolumn{1}{|c|}{Value}&\multicolumn{1}{|c|}{Value}\\ \hline \hline 
1 ($r = 1$)& $K$ &\multicolumn{1}{|c|}{0.381793}&2.80858&-8.69906\\ \hline 
2 ($r=2$)  & $K$ &\multicolumn{1}{|c|}{0.0179628}&0.232112&-33.6312\\ \hline  
3& $r,\,\,  K$&\multicolumn{1}{|c|}{2.3755, \,\,  0.00545}&0.109867&-39.1107\\ \hline \hline
\end{tabular}




\parbox{13.0 true cm}{\caption{\label{table:Summary} Summary table of data for three models for whiffle ball fall, in which we display Model Number, parameters, rate constant(s),  Sum of Square Errors (SSE), and AIC Value.}}
\end{center}
\end{table}

Now, from our computed AIC values for the three models  as given in Table~\ref{table:Summary} it is clear that Model (3) is the best model. Note, the AIC for Model (3) is at least 2 lower than that for Models (1) and (2).  This means our best model, according to the AIC,  is the more complex model of Model (3) with the two parameters $r$ and $K$.

This suggests that the power to which we raise the velocity term, $y'(t)$ in our model (\ref{eq:model}), is neither $r = 1$ nor $r = 2$, but higher, namely $r = 2.3755$. This means that our resistance for the whiffle ball is more than actually offered in the literature with the $r=1$ and $r=2$ models. Obviously, this might be disconcerting, but it merits further investigations with other objects, perhaps in other media.




\subsection*{Alternate experiments}

Students can collect their own data using objects such as whiffle balls, badminton, shuttlecocks, coffee filters, etc. and perform the same analysis. The approach in \cite{BrandQuintanilla2013} was intended to be low tech. However, students can use sophisticated equipment such as  a Vernier motion detector with a sophisticated timer to collect very precise position data at very small time increments.


A rich area for student efforts in mathematical modeling is in the study of terminal velocity and the general modeling of falling objects subject to resistance.  Many studies have been published in this area \cite{Brauer2001, LongWeiss1999, LongWeiss2006} and a number of pedagogically motivated papers have also appeared  \cite{PeastrelEtAl1980, Barrye1991, BrandQuintanilla2013}.   In \cite{BrandQuintanilla2013} the authors describe a class project in which a class  collects data to model a falling Whiffle ball to determine terminal velocity. Falling badminton shuttlecocks form the basis of a comparable study in \cite{PeastrelEtAl1980}. While in \cite{Barrye1991} the author uses balloons  in short free fall to collect data and estimate parameters using the same basic models as in \cite{BrandQuintanilla2013}.

}
 
\begin{thebibliography}{99}

\teacher{
\bibitem{Akaike1974} Akaike, H. 1974.  A new look at the statistical model identification. {\it  IEEE Transactions on Automatic Control\/}.  19(6): 716-723.

\bibitem{Barrye1991} Battye, A. 1991. Modelling air resistance in the classroom. {\it Teaching Mathematics and Its Applications\/}.  10(1): 32-34. 
}
\bibitem{BrandQuintanilla2013} Brand, N. and J.~A.~Quintanilla. 2013. Modeling Terminal Velocity.  {\it The College Mathematics Journal\/}. 44(1): 57-61.


\teacher{
\bibitem{Akaike1974} Akaike, H. 1974.  A new look at the statistical model identification. {\it  IEEE Transactions on Automatic Control\/}.  19(6): 716-723.

\bibitem{Brauer2001} Brauer, F. 2001. What Goes Up Must Come Down, Eventually. {\it The American Mathematical Monthly\/}. 108(5): 437-440.

\bibitem{Goodin2012} Goodin, R.~E. 2012. {\it On Settling\/}. Princeton:  Princeton University Press.



\bibitem{LongWeiss1999} Long, L.~N. and H.~Weiss. 1999. The  velocity dependence of aerodynamic drag:  a primer for mathematicians.  {\it The American Mathematical Monthly\/}. 106(2): 127-135.

\bibitem{LongWeiss2006} Long, L.~N. and H.~Weiss. 2006. How terminal is terminal velocity?  {\it The  American Mathematical Monthly\/}. 113: 127-135.


\bibitem{PeastrelEtAl1980} Peastrel, M., R.~Lynch, and A.~Armenti, Jr. 1980. Terminal velocity of a shuttlecock in vertical fall. {\it American Journal of Physics\/}. 48(7): 511-513.
}

\bibitem{RossmannRau2007} Rossmann, J. and A.~Rau. 2007. An experimental study of Wiffle ball aerodynamics. {\it American Journal of Physics\/}. 75(12): 1099-1105.
\end{thebibliography}









\end{document}
 
