% SIMIODE-TeacherStudent-Template.tex and simiode.cls for LaTeX2e
% Version 22 November 2016.
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% The latest version of this class and template files 
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%       <http://www.SIMIODE.org> in the Author section.
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% Appreciation to Mark Frantz, IUPUI; John Thoo, Yuba College; and Eric Sullivan, Carroll College.
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\documentclass{simiode}
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% With this version of the tex file and the associated SIMIODE.cls file you can produce BOTH a Student Version and a Teacher Version of your modeling scenario or technique narrative.
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% Student Version contains the title STUDENT VERSION along with the NAME OF SCENARIO, with no abstract, no keywords, no tags, no author identifiers, and no COMMENTS section. The Student Version contains only the STATEMENT of the modeling scenario only with appropriate figures, data, instructions, and perhaps a bibliography appropriate for student resourcing, but nothing in the bibliography which would "give away" the nature of the modeling scenario, etc.
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%The Teacher Version contains the material in the Student Version, namely the title NAME OF SCENARIO, now headed with   TEACHER VERSION, but also full author  identification, abstract, key words, tag words, and, following the STATEMENT, a section of COMMENTS which can be discussion of presentation  ideas, "solutions," techniques, etc.
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% Authors can prepare a full Teacher Version using the notions immediately below  to create a pdf for review, but in review please eliminate all information about the author as we use a double-blind referee system. Such information will be restored when accepted and published in SIMIODE at www.simiode.org.
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%***********   Toggle the following line to turn the teacher version on (1) or off (0). ***********
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\def\TeacherEdition{0} % Toggle this line's value.
\newcommand{\teacher}[1]{\ifnum\TeacherEdition=1 #1 \fi}
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% The \teacher{ } command wraps any text that will appear only in the teacher version of
% the document. See this file for examples. This has been done for you in the document below.   
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\usepackage{amsmath,amssymb}
\usepackage{color}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{url}
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\renewcommand{\baselinestretch}{1.3}
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% NOTE regarding macros %%
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% Any macros defined for your paper should be contained in 
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% Top matter is above and the beginning of the document below.
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\begin{document}
\newpage
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% 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.
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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \footnote{DELETE THIS FOOTNOTE BEFORE SUBMISSION.  In the \TeX \, source code there is a toggle/option to produce a Teacher Version or a Student Version by ``carving'' out with  $\backslash$teacher\{\} that material which should only appear in the Teacher version. See the source code.} \else STUDENT VERSION \fi\\Name of Scenario}


\title{TEACHER VERSION\\HANG TIME }


\markboth{Hang Time}{Hang Time}






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

\newpage
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% 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.
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\makeStitlePDFLaTex

\begin{abstract}
We present a modeling opportunity for students in which they  model the vertical position of a basketball player taking a jump shot.  The goal of the scenario is to get students to consider the meaning of the notion ``hang time.'' 
\end{abstract}


\section*{SCENARIO DESCRIPTION}



Let us consider the phenomena known as ``hang time" in sports. Often the announcer in a televised football game will refer to the hang time for a punter's kick or a basketball announcer will say a player appears to ``hang in the air." What is that all about? Why would something appear to be so? We consider a one dimensional representation of an object going up and coming down and ask what the notion of hang time might suggest in this context.

Suppose we consider a basketball player about to take a jump shot. 
\begin{itemize}
\item [a)] Estimate how high his hip rises from his standing position to the top of his jump.
\end{itemize}

Suppose we try to model the height of his hip and let us say $y(t)$ is the height of his hip at time $t$ seconds, where $y(0) = 0$ is the starting height shifted to compensate for his real hip height. 
\begin{itemize}
\item [b)]  Newton's Second Law of Motion   says that the sum of the forces acting on a body are equal to the product of the body's mass and it acceleration,  $m*y''(t)$) it experiences can be computed by summing all the external forces acting on the body. Use that Law and the fact that the only force acting on the player's hip is due to gravity to build a differential equation for $y(t)$. What would be the initial conditions?
\item [c)] Solve the differential equation with one of your initial conditions being unknown. Which condition is unknown?
\item [d)] Based on your observations for (a) and your solution in (c) determine numerical values for your unknown initial condition.
\item [e)] Now use your model with your initial condition to determine the times at which the player's hip is at 75\% of its maximal height.
\item [f)] Determine the entire length time of this player's jump.
\item [g)] What percentage of time does the player's hip spend higher than 75\% of its maximum height?
\item [h)] Relate the finding in (g) to the appearance of hang time alluded to in the opening of this activity.
\end{itemize}

\teacher{
\section*{NOTES FOR TEACHERS}


We include a Mathematica notebook both in Mathematica notebook format (1-008-T-HangTime-Mma-TeacherVersion.nb)  and universally readable pdf (1-008-T-HangTime-Mma-TeacherVersion.pdf) format in which we offer up a typical write-up that students produce after about 10-15 minutes of class discussion and time at home to do the mathematics and write-up.  We have given students about 2-3 days to complete this assignment. 

We have used the assignment in many classes and students have written up their work in a word processor and computer algebra format.  We have used it in calculus courses, a differential equations course, and a modeling with differential equations course.

The computer algebra system (CAS) is the one we tend to recommend to  our students. Indeed, we expect them to communicate in Mathematica, the CAS we have used for years. The blending of the mathematics with the exposition is very important for student communication development.



Students are intrigued with this problem for sports jargon is a part of many of their lives. I encourage them to cite  resources and justify their estimates, say in (a) of the statement of the situation, but each student has to construct a set of assumptions, build a mathematical model, offer up a solution with analysis, and a write a complete summary.



Some students will try to make things very complicated, e.g., point out that the center of mass of the ``object'' changes as the jumper will pull up her feet, thus changing the geometry of the situation. The suggestion  is to keep the modeling simple.

This problem grabs the attention of the students and  gets them thinking about and doing assumptions, modeling, analysis, and communication. 
}
\end{document}
