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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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\begin{document}

\newpage
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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\Falling Meteorite}




\markboth{Falling Meteorite}{Falling Meteorite}

\author{Lyle Smith III\\
Bryan College\\
Dayton TN  37321  USA\\
lylesmith333@gmail.com 
}

\makeStitlePDFLaTex


\begin{abstract}
After introducing the solution to the ordinary differential equation which models a falling object with drag (first-order, non-linear, separable), students will consider generalizing the model to a falling and disintegrating meteorite.  The focus is on creative identification of factors.  Students will also rank factors by perceived effect on the flight path and build a model incorporating one or more additional factors.  Students should also identify trends in the flight and boundary conditions.  Solutions may be attempted, but are not required in this scenario.
\end{abstract}



\section*{SCENARIO DESCRIPTION}
We have considered the situation of an object falling with air resistance or drag.  We found the ordinary differential equation (ODE) for the motion to be
\begin{align}
v'&=g-\frac{\rho}{m}v^2 \label{drag}
\end{align}
where $v$ is the velocity as a function of time, $g$ is the acceleration due to gravity, $\rho$ is the drag coefficient, and $m$ is the mass of the object.  We assume the initial velocity is zero and call the constant solution the {\it terminal velocity} $v_\infty$, which is given by 
\[v_\infty^2=\frac{mg}{\rho}.\]
After separating variables and using a partial fraction decomposition, the solution to (\ref{drag}) is found to be 
\[v(t)=v_\infty\tanh\bigg(\frac{\rho}{m}v_\infty t\bigg).\]
\teacher{For more information, see \cite[pp. 46-47]{Conrad2010}.}

In arriving at this solution, we have made a variety of assumptions, i.e. we have ignored factors that affect the motion of the object to a smaller extent.  Now consider a falling meteorite.  We will simplify the situation mathematically by assuming that the meteorite falls straight downward and consider only the vertical motion, i.e. we will look at equations in only one position variable.  However, consider all of the other factors that affect the flight of the object.

\section*{PROBLEM}
\begin{compactenum}
\item Make a list of every factor that affects the motion of a meteorite as it falls from above the atmosphere to the ground.  Be creative.  List everything you can think of, even if you are unsure.  Watch this amazing video of meteorites caught on camera \teacher{\cite{MeteorVideo2018}}.  There are three parameters listed in the original ODE (\ref{drag}).  How could each of them change?  Consider researching ``drag coefficient.''
\item Rank these factors from largest to smallest effect. It is fine to have a tie.
\item Modify (\ref{drag}) to incorporate the top one or two of the factors in your ranking.  Record your model here.
\item Choose one member of your group to present your model to the class.
\end{compactenum}
 


\teacher{
\section*{NOTES FOR TEACHERS}

\section*{Implementation Guide}
It is recommended to attempt this scenario in groups of two or three students.  Be sure to show the video, if possible, to drive home the fact that the meteors are burning up.  Also, emphasize the three parameters from (\ref{drag}), as noted in the first question.  We previously assumed that they are constant.  Is that still true?  That should help to begin the discussion.

The length of this scenario can vary greatly.  In 20 minutes, students should be able to tackle most of the questions above, plus additional time for group presentations.  This could also form the beginning of a project that could take a couple weeks to an entire semester.  One example of what a completed project might look like can be found here \cite{MeteorProject}.

The scenario directs students to web search for drag coefficient.  It is best if they first have some time to think through some of the issues before reaching for their phones.  However, most students will need some help in formulating some of the factors, such as drag.  

\subsection*{Example Solutions}
Students should be able to address the three parameters that are in (\ref{drag}).  Listed here are some factors you might find.

\subsubsection*{Mass:}
If the meteorite is burning up upon hitting the atmosphere, the mass should decrease over time.  This rate of decrease could be modeled in a variety of ways.  Here $m_0$ is the initial mass and $k$ is a parameter.
\begin{compactenum}
\item Linear: $m(t)=m_0(1-kt)$.  This model is simple, but has the obvious problem of yielding possibly negative mass.  Also note that zero mass is problematic in the ODE.
\item Exponential: $m(t)=m_0 e^{-kt}$.  This model is more accurate and avoids the problems of negative and zero mass.  However, intuitively, we expect the meteor to slow as it burns until it reaches a velocity where it is no longer burning.
\item ODE: The most natural way to consider the mass loss is to use a model of the mass as a function of velocity.  We want the mass to decrease quickly when the velocity is high and decrease very little when the velocity is slower.  The simplest model may be $m^\prime(t)=-kv^2$, which must be solved concurrently with $v(t)$. 
\end{compactenum}
 The issue of decreasing mass can quickly lead to complexity.
  
\subsubsection*{Gravity:}
This is definitely a factor, since the gravitational force will increase as the object falls, according to Newton's Universal Law of Gravitation.  The earth's atmosphere is about 300 miles deep, but the radius of the earth is about 4,000 miles, so this is less than 10\% difference in radius for the entire flight.  Encourage your students to search for these numbers themselves.  Thus, this effect should be one of the smaller effects and be ranked near the bottom of the lists.

\subsubsection*{Drag:}
Drag is dependent on a number of factors and the models here could become quite complex.  Drag is friction between the air and the object.  Empirically, drag is a function of $v^2$, but the coefficient $\rho$ can vary with size, air density, and other factors.
\begin{compactenum}
\item Cross-Sectional Area: This is known to be a major factor in the drag coefficient.  For example, a big, square van has much more drag than a small, short Corvette.  If the object is disintegrating, the cross-sectional area will decrease over time.  Again, linear models are the simplest, but this area should be a function of mass.  If we assume a uniform density for the meteorite, we can model it as a sphere.  Then the mass is $m=\frac{4}{3} \pi r^3 \cdot d$, where $d$ is the mass density.  The cross-sectional area is just a circle $\pi r^2$.  So we get an expression for the area as a function of mass,
\[A=\pi r^2=\pi\bigg(\sqrt[3]{\frac{3m}{4d}}\bigg)^2.\]
This could be combined with the models for mass.
\item Drag also varies with air density.  The atmosphere is much less dense at the furthest heights, so that the object would feel much less drag, increasing to some maximum amount as the object nears the ground.
\item Heat.  As the object burns, an “explosion” on the bottom side will greatly increase the drag, since no similar explosive force would be felt on the top side.  As the meteor slows, it will cool to a point where it no longer feels this force.
\end{compactenum}

\subsubsection*{Others:}
\begin{compactenum}
\item Relativity: As with all motion, the velocity we perceive is slightly different from the velocity felt by the object.  However, this effect is tiny and can be assumed to be negligible.  
\item Fracturing:  Students may want to consider the possibility of the object breaking apart.  This would create multiple ODEs to solve separately (as a first approximation).
\item Shape:  Students may want to consider a variety of shapes for the object.  This would modify the area model above, but can be safely ignored since the burning process would burn off the rough edges first, quickly creating an object with approximately circular cross-section.
\item Composition: Students may question our assumption of uniform density, and rightly so.  Many meteorites are composed of several materials, which burn at vastly different rates.  Without knowing the exact distribution of materials in the object, this can't be modeled properly.
\item Impact: It is possible that a large enough meteorite could survive and hit the ground.  Others might burn up in the atmosphere.  A robust model should allow for these possibilities, but the simpler models that students are constructing in this scenario probably will not.  It would be best if students consider these possibilities.
\end{compactenum}

We list some possible issues.  There are most likely others.  The point of this exercise is to encourage creativity and trial-and-error.  It may even pique their interest into doing a research project.

\subsection*{Prerequisites}
We have referenced the solution of the problem where an object falls with air resistance.  This could be removed to solely focus on the creative modeling aspects.  However, it would be more difficult without a familiarity of this problem.

This scenario is designed to further develop modeling skills.  As such, it is appropriate for most levels of experience by adjusting the level of teacher support during the scenario.

\subsection*{Objectives}
The goal in this scenario is to get students thinking about all of the different factors that might go into a model.  We want to drive home the fact that every model is a balancing act between achieving accuracy with reality and solubility.  A second, but also as important goal, is to provide more practice converting ideas into mathematical language.  For example, taking ``the mass should decrease over time'' and including a term in the right-hand side, such as $m_0(1-kt)$.  Lastly, we would like for students to become comfortable with considering a variety of models, instead of focusing solely on what they consider to be the ``correct'' model.  I find that this is often the biggest hurdle at the beginning of the course.  Students will be afraid to provide any answer with which they are unsure.  This activity fosters an environment where students can overcome this tendency in attempting specific models and making model assumptions.


%\subsection{Rubric}
%This rubric has been helpful for grading group/individual work during student presentations.
%\begin{itemize}
%\item Completion of activity components (0 1 2 3 4 5)
%\item Mastery of objectives (0 1 2 3 4 5)
%\item Depth of mathematical analysis/insight (0 1 2 3 4 5)
%\item Individual contribution to presentation/work (0 1 2 3 4 5)
%\end{itemize}

\begin{thebibliography}{98}
\bibitem{Conrad2010}
Conrad, Bruce. 2010. {\it Ordinary Differential Equations: A Systems Approach}. Available at \href{https://math.temple.edu/~conrad/}{https://math.temple.edu/~conrad/}. Accessed on 24 August 2019.

\bibitem{MeteorVideo2018}
Crazy Universe.  2018. ``10 Unbelievable Meteorite Falling Caught on Camera.''    \emph{YouTube}, 10 June 2018, \href{https://www.youtube.com/watch?v=AF9DfnTU-vg}{https://www.youtube.com/watch?v=AF9DfnTU-vg}.  Accessed on 24 August 2019.

\bibitem{MeteorProject}
Australian Space Academy. 2019. ``Meteor Flight Through the Atmosphere.''   \href{https://www.spaceacademy.net.au/watch/debris/metflite.htm}{https://www.spaceacademy.net.au/watch/debris/metflite.htm}.   1 August 2019. Accessed on 24 August 2019.
\end{thebibliography}


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\end{document}
