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\usepackage{color}
\usepackage{graphicx}
\usepackage{hyperref}

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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\\FALLING IN WATER}






\markboth{Canister Falling in Water}{Canister Falling in Water}






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


\makeStitlePDFLaTex



\begin{abstract}
We model a small canister with ballast to keep it vertical as it falls through water in a graduated cylinder filled with water. We suggest several models in terms of resistance due to the media and estimate appropriate parameters. We then ask students to compare   models using the Akaike Information Criterion which we define and reference.
\end{abstract}


\section*{SCENARIO DESCRIPTION}

In the iconic differential equation text \cite[pp.~43-48]{Braun1978},  there is a section on modeling  a falling canister of nuclear waste in the ocean to determine the velocity that the canister might have upon striking the bottom of the ocean. This is due to concern that the canister might break open and the lead containment surrounding the nuclear material might fail.  


\begin{figure}[h]
\begin{center}
\includegraphics[width=3cm]{3-70-SmallShotcanister.jpg}

\vspace*{-.3 true cm}
{\caption{\label{fig:3-70-SmallShotcanister} Canister used in experiment.}}
\end{center}
\end{figure}



\begin{figure}[h]
\begin{center}
\hspace*{2 true cm}\includegraphics[width=7cm]{3-70-SetUp.jpg}


\parbox{9.5 true cm}{\caption{\label{fig:3-70-SetUp} Experiment set up with  cylinder of water and meter stick with canister released and falling in the foreground and laptop computer with digital clock for timing in the background. This image was taken with an iPad as was the YouTube video \cite{SIMIODE2016}.}}
\end{center}
\vspace*{-.6 true cm}
\end{figure}



\subsection*{Activities}

\begin{enumerate}
\item Model the fall of our own canister in a  cylinder of water as shown in  Figure~\ref{fig:3-70-SmallShotcanister}  in our own small little ``ocean'' as we place it in the water. Figure~\ref{fig:3-70-SetUp} shows you our setup for making the video posted on YouTube \cite{SIMIODE2016} from which we took data in Table~\ref{table:dataFirst}.


You will need to capture your own data  using the set up depicted in (see Figure~\ref{fig:3-70-SetUp}) from the YouTube video \cite{SIMIODE2016}. This video was captured by an iPad 5 and then the app SlowMo was used to slow the video down to 25\% of actual speed for ease in collection of data by stopping and starting the video.

You should produce two models, one with resistance proportional to velocity and one in which resistance is proportional to velocity squared. Be sure, in both cases to consider all forces acting on the canister as it falls.

\item You might consider a more general situation in which resistance is proportional to velocity to some unknown power?

\item Compare your models in some manner. We offer one method called the  Akaike Information Criterion (AIC) \cite[p.~138]{Ledder2013}. 
\end{enumerate} 


\subsection*{Akaike Information Criterion}



The Akaike Information Criterion (AIC) offers a way to compare different models of the same phenomenon. Variables used for the number or score which is the AIC are: $p$, number of parameters to fit; $n$, the number of observations; and RSS, the minimum residual sum of square errors (or just sum of square errors, SSE). The actual formula for AIC is given in (\ref{eq:AIC}):


\begin{equation}\label{eq:AIC}
 AIC = 2 (1+p)+n \log\left[\frac{\rm RSS}{n}\right] \, .   
\end{equation}
NB: when RSS or SSE is small then the log function is negative. 

Smaller (even negative) values of AIC represent a higher level of statistical support for the corresponding model. A difference of 2 in the AIC's is needed to suggest a significant difference between models, with the smallest  AIC signifying the best model - in terms of balancing ACCURACY (RSS) and COMPLEXITY ($p$).  However, if all AIC's are close (within 2) to each other then one cannot distinguish which model is better.     




\section*{APPENDIX}

Table~\ref{table:dataFirst} shows the formatted data taken from an earlier video \cite{SIMIODE2016First}. For this data set the mass of the canister with the steel balls was 15.0 g as it had two more steel balls than the canister used in \cite{SIMIODE2016}, but the dimensions of the canister were the same, height of 3.6 cm and radius of 1.1 cm.


\begin{table}[h]
\begin{center}
\small
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
$t$&0.00 & 0.26 &0.4,&  0.56&  0.70&  0.90& 1.00&  1.19& 1.26&  1.40&  1.52& 
1.66&  1.80&  2.06\\ \hline
$d(t)$& 0.0 &1.1 & 2.3 & 4.1 & 6.2& 8.5 &  10.4&  13. & 14.1 & 16.2 & 18.7
 & 20.4 & 22.5 & 26.8\\ \hline
\end{tabular}
\normalsize
\parbox{7.8 true cm}{\caption{\label{table:dataFirst} Data on falling canister from video. $t$ is time in seconds while $d(t)$ is depth of fall in cm at  time $t$.}}
\end{center}
\end{table}

\begin{thebibliography}{98}



\bibitem{Braun1978} Braun, M. 1978. {\it Differential Equations and Their Applications, Second Edition\/}. New York:  Springer Verlag.

\bibitem{Ledder2013} Ledder, G. 2013. {\it Mathematics for the Life Sciences\/}. New York: Springer. 

\bibitem{SIMIODE2016First} SIMIODE. 2016. Canister Falling in Water, Early Version.  \url{https://www.youtube.com/watch?v=40MCsVJD1FI}. Accessed on 17 April 2016.

 

\bibitem{SIMIODE2016} SIMIODE. 2016. Canister Falling in Water.  \url{https://www.youtube.com/watch?v=oNoPw9GFf2g} or at SIMIODE Website \url{https://www.simiode.org/resources/1703/download/CanisterFallInWater.mov}. Accessed on 17 April 2016.


\end{thebibliography}


\teacher{
\section*{NOTES FOR TEACHERS}

We offer complete analysis of the data from Table~\ref{table:dataFirst} in a Mathematica notebook, 3-70-T-Mma-FallingInWater-TeacherVersion.nb and an accompanying pdf version of the file. We also offer complete analysis of data taken from the video \cite{SIMIODE2016} in  a Mathematica notebook, 3-70-T-Mma-FallingInWater2-TeacherVersion.nb and an accompanying pdf version of the file

We work with the model offered by differential equation (\ref{eq:model}).

\begin{equation}\label{eq:model}
m*y'' (t) = m* g - {\rm BuoyantForce} - k*y'(t)^r
\end{equation}
where the BuoyantForce is the weight of the water displaced by the canister, in this case $\pi R^2 H g \rho$ where $R$ is the radius of the cylindrical canister in cm, $H$ is its height in cm, $g$ is the acceleration due to gravity in cm/s$^2$, and $\rho$ is the mass density in g/cm$^3$. Students can obtain this differential equation using a Free Body Diagram to keep track of forces applied to the canister and applying  Newton's Second Law of Motion which says that $m y''(t)$ is equal to the sum of the external forces applied to the canister. $m*g$ is the force due to weight downward (hence positive in the differential equation) as we will presume that downward distance and velocity are positive. BuoyantForce $ = \pi R^2 H g \rho$ is a force upward (hence negative in the differential equation) due to the displacement of water by the canister, and the resistance force is proportional to some power of velocity, $k*y'(t)^r$ and is also upward (hence negative in the differential equation).   

\begin{figure}[h]
\begin{center}
\includegraphics[width=11cm]{3-70-FirstDataModels.eps}


\parbox{12 true cm}{\caption{\label{fig:3-70-FirstDataModel} Plots original data from Table~\ref{table:dataFirst} with the three models, i.e., $r = 1$, $r = 2$, and $r = 2.3$. The first two models are almost identical in shape (blue and red) overlaid) while the general $r$ model tracks the data considerably better (green).}}
\end{center}

\vspace*{-.4 true cm}
\end{figure}

In Figure~\ref{fig:3-70-FirstDataModel} we see the three main models one might consider here with  resistance terms of $k y'(t)^r$ for $r = 1$, $r = 2$, and $r = 2.3$. The latter value of $r$ is obtained through a best fitting process to the data.
 Here, for the best fitting model in each case, the respective sum of square errors are SSE1 =11.2701, SSE2 = 17.9636, and SSEr = 6.73187, while the respective AIC values are AIC1 = 0.963365, AIC2 = 7.49007, and AICr = -4.25087. This analysis then says that the best model occurs when $r = 2.3$ since, in that case, the AIC is more (much more) than 2 less than the other two AIC values.




}





\end{document}
