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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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\begin{document}

\newpage
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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Ball Drop in Water}





\markboth{Ball Drop in Water}{Ball Drop in Water}

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


\makeStitlePDFLaTex


\begin{abstract}
We conduct an analysis of a falling ball in liquid to determine its terminal velocity and to ascertain just what radius ball for a  given mass density is necessary to attain a designated terminal velocity.
\end{abstract}


\section*{SCENARIO DESCRIPTION}
A ball of mass  $m$ (gm) and radius $r$ (cm) is released from rest in a liquid of resistance to motion of  $\mu$ (g/(cm$^2$ s)). The ball falls under the influence of gravity, but the liquid exerts a drag force proportional to its velocity$v$ (cm/s) and there is the force due to buoyancy which is the weight of the liquid displace by the ball $\frac{4}{3} \pi r^4 \rho\cdot g$. If $y$ (cm) measures the vertical distance the ball has fallen, then one form of its equation of motion can be given by:

\begin{equation}\label{eq:FBD}
m \cdot y''(t) = - \pi r^2 \mu\cdot  y'(t) + m\cdot g - \frac{4}{3} \pi r^4 \rho\cdot g\, .
\end{equation}
                              
\begin{enumerate}
\item Derive and defend this mathematical model using a Free Body Diagram and Newton's Second Law of Motion. The latter says that for a body of mass m its mass times its acceleration is equal to the sum of all external forces acting on the body. 

Identify all the forces in this situation and then build the differential equation which models the position beneath the surface of the ball.


\item  Solve the differential equation for vertical distance ($y$).

\item Determine the velocity  $v = y'(t)$ and speak to its attributes.

\item Suppose a copper ball of density $\mu = 8.96$ gm/cm$^3$ and radius$ r = 0.60$ cm is dropped in water ($\rho  = 1$ gm/cm$^3$). Determine velocity  at which the ball will fall steadily after a period of time and roughly how long it takes the ball to reach this steady velocity.

\item Suppose a steel ball of density $\mu = 8.00$ gm/cm$^3$ is dropped in water ($\rho  = 1$ gm/cm$^3$). Determine the radius of the ball that will fall steadily at 5.50 mph (the fastest speed of a human swimmer).

\end{enumerate}


\teacher{
\section*{NOTES FOR TEACHERS}

We got the source idea for this Modeling Scenario from reading in \cite{Baker2016}. We changed things around a bit and included buoyancy force.


This is a situation in which students construct the second order, linear,  nonhomogeneous differential equation (\ref{eq:FBD1}), 

\begin{equation}\label{eq:FBD1}
m\cdot  y''(t) = - \pi r^2 \mu\cdot  y'(t) + m\cdot g - \frac{4}{3} \pi r^4 \rho\cdot g\, ,
\end{equation}
from applying Newton's Second Law to a Free Body Diagram.

Units are very important in this analysis and terminal velocity is an important concept.

We supply a Mathematica notebook 3-009-T-Mma-BallDropInWater and a pdf version for those who do not read Mathematica in the Supporting Docs in which we provide complete analyses.





\begin{thebibliography}{99}




\bibitem{Baker2016} Baker, Gregory. 2016. {\it Differential Equations as Models in Science and Engineering\/}. Works Scientific: Hackensack NJ USA.  pp.~109-110.


\end{thebibliography}
}





\end{document}
