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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\\UP DOWN PROJECTILE PASSING}





\markboth{ Up Down Projectile Passing}{Up Down Projectile Passing}






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

\makeStitlePDFLaTex



\begin{abstract}
Shoot a projectile straight up into the air.  Determine the  maximum height, $H$, the given projectile will go.  Then consider
the time $T(a)$ ($0 < a < 1$) it takes between when the projectile passes
distance $a \cdot H$ going up and then coming down. Develop $T(a)$ as a function of $a$.
\end{abstract}


\section*{SCENARIO DESCRIPTION}


Shoot a projectile straight up into the air.  Determine the  maximum height, $H$, the given projectile will go.  Then consider
the time $T(a)$ ($0 < a < 1$) it takes between when the projectile passes
distance $a \cdot H$ going up and then coming down. Develop $T(a)$ as a function of $a$. Finally, see if there are any relationships involving these two times at which the projectile passes through the same distance, going up and coming down.


\subsection*{Articulation}

Consider a projectile, projected upwards in a uniform gravitational field with no resistance. 

\begin{itemize}
\item [a)] Say the initial velocity is 100 m/sec and the gravitational field is due to earth's gravity, i.e. $g = - 9.81 {\rm m}/{\rm sec}^2$ downward.  

We fire a projectile directly up from ground zero.

\begin{itemize}
  \item [(i)] Determine the maximum height, $H$,  the projectile attains.  
  \item [(ii)] For heights $h = 0, 0.1 H, 0.2H, 0.3 H, \ldots , 0.9 H, H$ determine the time (from $t = 0$) it takes to reach height $h$ (on the UPward path) and the time (from $t = 0$) it takes to reach height $h$ again (on the DOWNward path). Remember, what goes up must come down!
  \item [(iii)] Look at the numbers you get and make a conjecture about the relationship between these times and the height $h$. 
\end{itemize}
\item [b)] Test your conjecture with another initial velocity.  Do you get the same results?  the same relationship?  Does the initial velocity seem to enter your relationship? 
\item [c)] Verify your conjecture you made in (a) (iii)  for the general equation for a projectile fired vertically with constant acceleration $a$, initial velocity $v_0$, and initial vertical displacement $x_0$.
\end{itemize}

\teacher{
\section*{NOTES FOR TEACHERS}


The difficulty in assigning this problem appears to be in how much of a hint to give the students in manipulating the data to obtain possible relationships.  Students who are familiar with fooling with data to obtain relationships and competent data manipulators in Mathematica, other CAS's, or a spreadsheet will be able to massage the data more easily. 


We offer analyses in a Mathematica file 3-41-T-Mma-UpDown-TeacherVersion.nb and a comparable pdf version of that file for those readers who cannot read Mathematica files. 

\subsection*{ISSUES RELATED TO THE PROBLEM}


 
\subsubsection*{Prerequisites}

Students need to know how to build the differential equation model describing projectile motion in a gravitational field in one dimension without resistance.


\subsubsection*{Extensions}
 


The students might consider the same problem with resistance.  

As is stated in the Acknowledgement section this product of  times is  independent of initial velocity in the case there is no resistance, but what if there is resistance?

One might also consider the problem for a bouncing object with 100\% coefficient of restitution or less than 100\% coefficient of restitution and ask for relationships between {\it later passages} through the given height.


\section*{ACKNOWLEDGEMENT}



This scenario comes from ``A Problem Characterization of Uniformly Accelerated Motion'' in the Problems and Solutions section of  {\it SIAM Review\/}. June 1995, Volume 37, Number 2, pp. 249-250.  Author of the problem M. S. Klamkin.  The problem posed in the journal was:

\begin{quote}
``It is a known result that if a projectile is projected upwards in a uniform gravitational field with no resistance, then the product of the two times it takes to pass through any point of its path is independent of the initial velocity of the projection. 

Prove or disprove that the result cannot hold if additionally the particle was subject to a resistance as some function of the velocity."

NB:  The solution by W. B. Jordan presented proves that the product of the two times is NOT independent of the initial velocity when resistance is present.''
\end{quote}


The production of this material was supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849 from 1993-1996 entitled Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN.  
}


\end{document}

