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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\\STACKED COFFEE FILTERS FALLING}




\markboth{Stacked Coffee Filters Falling}{Stacked Coffee Filters Falling}






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


\makeStitlePDFLaTex



\begin{abstract}
 Data on free falling 2, 4, 6, and 8 stacked coffee filters is offered for analysis. Students first form a model using a resistance term proportional to velocity, velocity squared, or velocity to some general power. Parameters need to be estimated and models compared.
\end{abstract}

\section*{SCENARIO DESCRIPTION}

In \cite{Doane2009b} data on several sets of stacked coffee filters falling is presented. The stacks contain 2, 4, 6, and 8 coffee filters. This is part of a larger set of data and videos offered by the Doane College (Crete NB USA), Department of Physics \cite{Doane2009a}. We enclose the data with additional information in the Excel file 3-17-S-Excel-StackedCoffeeFiltersFalling-StudentVersion.xls.
farthest.     

\begin{enumerate}
  \item Draw a free body diagram of the stack of coffee filters in free fall.
  \item From your diagram using Newton's Second Law of Motion, which says that the mass times the acceleration of that mass is equal to the sum of the external forces acting on that mass, to construct several differential equations of the form $m y''(t) = \rule{3 true cm}{.1 pt} $. Hint:  Consider various resistance terms due to the medium (air).
  \item Solve the differential equations.
  \item Using the data for the situations (one at a time) offered in 3-17-S-Excel-StackedCoffeeFiltersFalling-StudentVersion.xls estimate the parameters in your model. Consider the mass of one coffee filter as $m$ and absorb it into the constant for resistance, but note how many filters for each data set, e.g., in first data set the mass of the stack of coffee filters is $2m$ while in the second data set the mass of the stacked coffee filters is $4m$.
  \item Compare your models to the data and to each other.
\end{enumerate}

\teacher{
\section*{NOTES FOR TEACHERS}


In a Mathematics notebook 3-17-T-Mma-StackedCoffeeFiltersFalling-TeacherVersion.nb and accompanying pdf format for those who do not have access to Mathematica we offer discussion and responses to the questions.  Basically we consider three cases with two approaches to the third  case. Here $m$ is the mass of the falling body, $g$ is the acceleration due to gravity, and $k$ is the resistance parameter.

\begin{enumerate}
  \item (Stokes' model of air resistance) $m y''(t) = m g - k y'(t)$.
  \item (Newton's  model of air resistance) $m y''(t) = m g - k y'(t)^2$.
  \item (General power model of air resistance) $m y''(t) = m g - k y'(t)^r$.
\end{enumerate}

Two approaches are presented for  case (3)  (General power model of air resistance). One is to form a grid in the parameter space $(k, r)$, say, 400 by 400, i.e. solving the differential equation numerically 400 $\times$ 400 = 160,000 and computing the sum of square errors obtaining 160,000 points of the form $(k,r, {\rm SSE}(k,r))$ in the parameter space, and searching for the lowest value of ${\rm SSE}(k,r)$.  We offer a contour grid to depict this minimum.  In the second approach we use a gradient search (highly numerical) in the paramter space to move from an initial estimate to a lower and lower value of ${\rm SSE}(k,r)$ and thus converge on the minimum value of ${\rm SSE}(k,r)$ and the best estimates of the parameters $k$ and $r$.

Further we seek to determine the true resistance parameter $k$, for when we divide both sides of the differentiao equations found in Cases (1) - (3) we actually are estimating the parameter $k/m$ and if we multiply by the appropriate mass, assuming $m=1$, without loss of generality for the cases of 2, 4, 6, and 8, stacked coffee filters we can get some idea of the true value of $m$. Indeed, the case where the $k$ values are close in all four mass situations is likely to be a good model.

We summarize our studies  in 3-17-T-Excel-StackedCoffeeFiltersFalling-TeacherVersion.xls.

Finally, we use the Akaike Information Criterion to compare models.


\subsection*{Comparing models using the Akaike Information Criterion (AIC)}
 
The Akaike Information Criterion (AIC) \cite{Akaike1974} is a way to compare models in terms of their accuracy  using sum of square errors and number of parameters. We know that if we have $m$ data points, ${\rm data} = \{(t_i,y_i)\vert i=1,2,3, \ldots , m\}$, then we can put a polynomial of degree $m -1$ exactly through the data with a sum of square errors of 0. However, this is costly with $m$ unknowns, the coefficients of the fitting polynomial of degree $m - 1$, and even though it gives a perfect fit ($SSE = 0$) it uses a large number of parameters. Moreover, this polynomial explains nothing about the underlying phenomenon we are modeling.  The AIC compensates for each of these notions - SSE and number of parameters. 

The Akaike Information Criterion, where $k$ is number of parameters to fit, $n$ is the number of observations, and RSS is the minimum residual sum of square errors (or just sum of square errors, SSE) is given in (\ref{eq:AIC}): 

\begin{equation}\label{eq:AIC}
AIC =2(1+k)+n \log\left(\frac{\rm RSS}{n}\right) \,.
\end{equation}
NB: When 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 for two models is needed to suggest a significant difference between these two models.
 
\begin{table}
\centering
\begin{tabular}{|l|c|c|c|c|c|} \hline
Model & $r$ & $k = a$ & SSE & AIC   \\ \hline
Source & 2 && 0.00920356 & {\bf -123.811}  \\ \hline 
No Resistance && 0 & 128.369& 36.3689 \\ \hline  
Case &$r = 1$ &  1.06174 & 0.523775  &-55.1584  \\ \hline 
Case &$r = 2$ &  0.211834 & 0.00920356 & {\bf -123.863}  \\ \hline 
General $r$ & 2.019 & .0205 & 0.00919924 &  {\bf -121.871} \\ \hline  
\end{tabular}

\parbox{6.8 true cm}{\caption{\label{table:AICSummary} 
Information we need about our models to decide which model is best is presented.}}
\vspace*{-.7 true cm}
\end{table} 

 
Thus if AIC's are close to each other then the models are comparable. The smallest AIC means best model - in terms of balancing ACCURACY(RSS) and COMPLEXITY(k).
 
So after we complete all model analyses we will see if one(or several) of the models is best. We offer the parameters, sum of square errors, and AIC values for the best parameters in our models in Table~\ref{table:AICSummary}, where the boldface AIC values are close to each other and far below the other AIC's so these three models have about the same value. Indeed, they are all of the quadratic, $r =2$ or $r \approx 2$, model with respect to the power of velocity in the model for resistance. 

}

\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{Doane2009a} Doane College Physics Department. 2009. {\it Gravity and Air Resistance}.  \href{http://physics.doane.edu/physicsvideolibrary/default.html}{http://physics.doane.edu/physicsvideolibrary/default.html}. 

\bibitem{Doane2009b} Doane College Physics Department. 2009. {\it Filter Data}.  \href{http://physics.doane.edu/physicsvideolibrary/spreadsheets/html/filters.htm}{http://physics.doane.edu/physicsvideolibrary/spreadsheets/html/filters.htm}. Accessed  31 March 2016.


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

\end{thebibliography}






\end{document}
