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\newpage
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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\\How High?! Modeling Free Fall with Air Drag}



\begin{figure}
    \centering
    \includegraphics[width=130mm]{ff.jpg}
    %\caption{} \label{}
\end{figure}



\markboth{How High?! Modeling Free Fall with Air Drag}{How High?! Modeling Free Fall with Air Drag}

\author{Jacob P. Duncan\\
Winona State University\\
Winona MN  55987 USA \\
jacob.duncan@winona.edu
}

\makeStitlePDFLaTex


\begin{abstract}
Most projectile motion and free fall models are based on the assumption that gravity is the only force acting on the object.  Here we develop, solve, and analyze a second order nonhomogeneous differential equation model for free fall which incorporates air resistance.  Students will solve the model using two different methods -- reduction of order and separation of variables, and method of undetermined coefficients.  Using the solution, students derive an expression for the terminal velocity of the object as well as a prediction of the maximum height of the object (assuming is is fired directly upwards).  The model is then parameterized for a Nerf dart by an experiment performed by students.  The terminal velocity and muzzle velocity of the dart is measured using video frames of the dart's motion.  Finally, the model is validated by an experiment wherein students fire their darts upward and measure the assent time for comparisons with their predictions.
\end{abstract}


%\begin{figure}
%    \centering
%    \includegraphics[width=150mm]{mpb1.png}
%    %\caption{} \label{}
%\end{figure}

%\section*{STATEMENT}

%\section*{Motivation}


%\newpage
\section{INTRODUCTION AND SCENARIO DESCRIPTION}
%\begin{center}
%{\centering Jacob Duncan}\\
%{\centering Department of Mathematics and Statistics}\\
%{\centering Winona State University}\\
%\end{center}


Free fall refers to any motion of a body where gravity is the only force acting upon it.  Galileo’s law of free fall states that, in the absence of air resistance, all bodies fall with the same acceleration, independent of their mass.  See the following videos illustrating this concept:
 \begin{itemize}
   \item[--] Hammer vs Feather: Physics on the Moon\\ \verb"https://www.youtube.com/watch?v=KDp1tiUsZw8"\\
   \item[--] Brian Cox visits the world's biggest vacuum\\ \verb"https://www.youtube.com/watch?v=E43-CfukEgs"
 \end{itemize}
Objects falling in a gas medium will experience what is called a \textit{drag force}.  Air drag, a type of friction, is a force acting opposite to the relative motion of any object moving with respect to the surrounding gas.  Furthermore, the magnitude of the drag force increases in proportion (approximately) to the object's speed.  Hence, for an object in free fall in Earth's atmosphere, for example, there are two forces acting on the object -- gravity and air drag.  If the motion is essentially straight down, then these two forces act in opposite directions (Figure 1).

For an object dropped from rest, gravity accelerates the objects causing it to pick up speed.  However, as its speed increases, so does the drag force.  Eventually, the magnitude of the drag force approaches that of the force of gravity.  Given a sufficient amount of time, the two forces are essentially equal but in opposite directions resulting in zero net force.  Hence, the object will continue to fall at a constant speed -- its \textit{terminal velocity}.

\begin{figure}
    \centering
    \includegraphics[width=130mm]{Terminal-Velocity.jpg}
    \caption{Force diagram of an object in free fall with air drag.} \label{forceDiag}
\end{figure}

In this project we will predict the maximum height of a projectile (Nerf dart) subject to air resistance fired directly upwards. We first develop a differential equation governing the motion of such a projectile.  We then solve the differential equation which contains parameters for air drag and muzzle velocity.  Muzzle velocity of the firing apparatus (dart gun) is estimated experimentally using video frames of the dart being fired with a distance measure in the background.  Similarly, we estimate the terminal velocity of the dart by firing it directly upwards and measuring its speed on its descent.  The terminal velocity is then used to estimate the drag parameter.  Using the parameterized model, we predict the time required for the dart to reach its maximum height.  Finally, we test our prediction experimentally using a stopwatch to time the ascent of the dart.


\section{Materials and Setup}

 \begin{itemize}
   \item Several dart guns (one for each student group) with at least 3 darts each
   \item Tape measure
   \item Video camera
   \item VLC video editing software (free): \\
            \verb"https://www.videolan.org/vlc/"

 \end{itemize}

To parameterize the free fall model students must estimate the muzzle velocity of their dart gun and the terminal velocity of a dart using video footage of the dart fired horizontally (for muzzle velocity) and vertically (for terminal velocity).  Videos should be shot with some way to measure distance travelled by the dart in the background.  This can be accomplished with a tape measure affixed to a wall (horizontally for muzzle velocity and vertically for terminal velocity) or with a brick wall in the background.  Video files can be analyzed with VLC video editing software wherein each frame is advanced by pausing the video and pressing the `E' key.  If using a brick wall for measuring distance, students will need to measure the width (for muzzle velocity) and thickness/height (for terminal velocity) of a brick.

Each group will need to determine the frame rate, i.e., frames per second (fps) for their video recording device.  This can be found in the manual for the devise.  A common frame rate is 30fps.    





\section{Differential Equation Model}

Using Figure \ref{forceDiag} we see that the net force on an object in free fall in the atmosphere is the sum of the force due to gravity and the force due to air resistance,
\[
F_{\textrm{net}} = F_{\textrm{gravity}} + F_{\textrm{drag}}.
\]
Using Newton's Second Law of motion and the assumption that a drag force is proportional to velocity (and in the opposite direction of gravity), we have
\[
F_{\textrm{gravity}} = -mg
\]
(assuming that up is the positive direction) and
\[
F_{\textrm{drag}} = -\rho v
\]
where $m$ is the mass of the object, $g>0$ is the acceleration due to gravity, $v$ is the object's velocity, and $\rho$ is the drag coefficient.  Hence, we have
\[
F_{\textrm{net}} = -mg - \rho v
\]
Using Newton's Second Law on the left side of the equation, together with $v(t)=\frac{dy}{dt}=y'$ and $a(t)=\frac{d^2y}{dt^2}=y''$, where $y(t)$ represents the height of the object at time $t$, we have
\[
my'' = -mg - \rho y'.
\]
With $k=\frac{\rho}{m}$, we have
\begin{equation} \label{model}
y'' = -g - ky'.
\end{equation}




\section{Practice Problem}

Before proceeding to solve (\ref{model}), we will solve a similar problem without parameters:
    \[
       y''=-5-2y'
    \]

\begin{itemize}
  \item \textbf{Method 1}:
      \begin{enumerate}

          \item Let $v=y'$ and rewrite the equation in terms of $v$ and $v'$.

          \item Find the general solution of the resulting differential equation for $v(t)$.  Note that $v'=\frac{dv}{dt}$.

          \item Use $v(0)=10$ to find the particular solution.

          \item Find $\lim_{t\to\infty}v(t)$.

          \item Graph the solution and verify your answer from part 4.

          \item Using $y'=v$ and your solution for $v(t)$ (from part 3), find the general solution $y(t)$.

          \item Find the particular solution for $y(t)$ if $y(0)=0$.

          \item Graph the particular solution $y(t)$.

          \item Determine the maximum value of $y(t)$ on $[0,\infty)$ and verify your answer graphically. \\

      \end{enumerate}


  \item \textbf{Method 2}:
    \begin{enumerate}
        \item Find the general solution of $y''=-5-2y'$ using the method of undetermined coefficients (UC).  (Hint: Use a linear function for the particular solution of the UC method.)

        \item Using $y(0)=0$ and $y'(0)=10$, find the particular solution of $y''=-5-2y'$.

        \item Determine the maximum value of $y(t)$ on $[0,\infty)$ and verify your answer graphically.

        \item Let $v=y'$ and find $\lim_{t\to\infty}v(t)$.

        \item Graph $v(t)$ and verify your answer from part 4. \\



    \end{enumerate}

\end{itemize}




\section{Solving the Free-fall with Air Drag Problem}

We now use the technique(s) of the previous section to solve Equation (\ref{model}).

\begin{itemize}
  \item \textbf{Method 1}:
      \begin{enumerate}

          \item Let $v=y'$ and rewrite (\ref{model}) in terms of $v$ and $v'$.

          \item Find the general solution of the resulting differential equation for $v(t)$.  Note that $v'=\frac{dv}{dt}$.

          \item Use $v(0)=v_0$ to find the particular solution.

          \item Find $\lim_{t\to\infty}v(t)$ (in terms of model parameters).  Denote this value by $v_T$.  What does this value represent in the context of free-fall?

          \item Graph the solution (use $g=9.8\rm{\frac{m}{s^2}}$).  Verify your answer from part 4.

          \item Using $y'=v$ and your solution for $v(t)$ (from Part 3), find the general solution $y(t)$.

          \item Find the particular solution for $y(t)$ if $y(0)=0$.

          \item Graph the particular solution $y(t)$. \\

          %\item Determine the maximum value of $y(t)$ on $[0,\infty)$ and verify your answer graphically. \\

      \end{enumerate}


  \item \textbf{Method 2}:
    \begin{enumerate}
        \item Find the general solution of (\ref{model}) using the method of undetermined coefficients (UC).  (Hint: Use a linear function for the particular solution of the UC method.)

        \item Using $y(0)=0$ and $y'(0)=v_0$, find the particular solution of Equation (\ref{model}).

        \item Let $v=y'$.  Graph $v(t)$.

        \item Find $\lim_{t\to\infty}v(t)$ (in terms of model parameters).  Denote this value by $v_T$.  What does this value represent in the context of free-fall?  Verify your prediction using the graph of $v(t)$.



    \end{enumerate}

\end{itemize}







\section{Estimating Terminal and Muzzle Velocity (Model Parameterization)}

\begin{itemize}

 \item \textbf{Terminal Velocity}: We experimentally estimate the terminal velocity of the dart using video frames of the dart being fired vertically with a distance measure in the background.  We measure the dart's velocity on its descent and assume this is its terminal velocity.  The example and images below illustrate this process.

    \begin{figure}[H]
        \centering
        \includegraphics[width=75mm]{Snapshot1.png}
        \caption{Video frame of Nerf dart on its descent.  We note its position relative to the brick wall in the background.  }

        \centering
        \includegraphics[width=75mm]{Snapshot2.png}
        \caption{Next video frame of Nerf dart on its descent.  We note its position relative to the brick wall in the background.  We see the dart has travelled approximately 7 bricks since the last frame.  Since the frame rate for the video was 30fps, we note that the time between frames is 1/30 seconds. }

        \centering
        \includegraphics[width=75mm]{Snapshot3.png}
        \caption{Next video frame of Nerf dart on its descent.  We note its position relative to the brick wall in the background.  We see the dart has travelled approximately 7 more bricks since the last frame indicating that the dart has reached its terminal velocity.  }

      \end{figure}
      %\begin{figure}[H]
      %  \centering
      %  \includegraphics[width=75mm]{Snapshot4.png}
      %\end{figure}

      \noindent In each frame, the dart fell approximately 7 brick heights (75 mm including mortar).  The frame rate of the video was 30fps.  Hence, the dart fell $7 \times 75\rm{mm} = 525\rm{mm}$ in $\frac{1}{30}$ second.  This results in a terminal velocity estimate of $v_T\approx \frac{525}{\frac{1}{30}} = 15,750 \frac{\rm{mm}}{\rm{sec}} = 15.75 \frac{\rm{m}}{\rm{sec}}$.  See the following video for an illustration of this process:

        \begin{itemize}

            \item[--] \verb"https://www.youtube.com/watch?v=H2A2I_UbvKA"

          %\item[--] \verb"https://www.dropbox.com/sh/186bfb63m7vjpvq/AACc8wUIPqBps5XUi7esVRCVa?dl=0"
        \end{itemize}

    \begin{enumerate}

          \item Position a video camera (with vertical frame) approximately 6 to 8 feet from a wall with a tape measure affixed to it vertically.  \\
          *Alternatively, use a brick wall with known brick height for measuring distance.

          \item While recording video, fire the dart vertically upwards.  Make sure the flight of the dart is fairly smooth and that its descent is captured in the frame of the video.\\
          *Alternatively, drop the dart from a 3rd or fourth story window so that it falls along the brick wall and into the frame of the video at the bottom.

          \item Play the video back to determine how far the dart fell between each frame (for the first 2-3 frames of the video).  This distance should be roughly constant.  Average the distances.  Using the frame rate of the video camera and the average distance, calculate the velocity of the dart -- this is the terminal velocity, i.e., $v_T$.

          \item Use your experimental estimate of $v_T$ (from the previous part) in your prediction of $v_T$ (from part 4 of section 4) to approximate the value of $k$.  %Then use this value of $k$ to find an estimate of the drag parameter $\rho$ (using $k=\frac{\rho}{m}$ and $m=1$g).

    \end{enumerate}


%\newpage

  \item \textbf{Muzzle Velocity}: We experimentally estimate the muzzle velocity of the firing apparatus (dart gun) using video frames of the dart being fired horizontally with a distance measure in the background. The example and images below illustrate this process.

          \begin{figure}[H]
            \centering
            \includegraphics[width=75mm]{1.png}
          \end{figure}
          \begin{figure}[H]
            \centering
            \includegraphics[width=75mm]{2.png}
          \end{figure}
          \begin{figure}[H]
            \centering
            \includegraphics[width=75mm]{3.png}
          \end{figure}
          %\begin{figure}[H]
          %  \centering
          %  \includegraphics[width=75mm]{Snapshot4.png}
          %\end{figure}

      \noindent In each frame, the dart travelled approximately 7.6 cm.  The frame rate of the slow motion video was 240fps.  Hence, the muzzle velocity of the dart is approximately 18.25$\frac{\textrm{m}}{\textrm{sec}}$.   See the following video for an illustration of this process:

       \begin{itemize}

            \item[--] \verb"https://www.dropbox.com/s/ufd1nb4i6io1vqs/Muzzle2.mp4?dl=0" \\

          %\item[--] \verb"https://www.dropbox.com/sh/186bfb63m7vjpvq/AACc8wUIPqBps5XUi7esVRCVa?dl=0"
        \end{itemize}

%\newpage

      \begin{enumerate}

          \item Position a video camera (with horizontal frame) approximately 6 to 8 feet from a wall with a tape measure affixed to it horizontally.  Alternatively, use a brick wall with known brick length for measuring distance.

          \item While recording video, fire the dart horizontally from the zero mark on the tape measure.  Make sure the gun is on the left side of the frame and is fired to the right.

          \item Play the video back to determine how far the dart travelled in the first 2-3 frames of the video (this should be roughly constant).  Average the distances.  Using the frame rate of the video camera and the average distance, calculate the velocity of the dart -- this is the muzzle velocity, i.e., $v_0$.

      \end{enumerate}


\end{itemize}





\section{Predicting Ascent Time (Model Validation)}

\begin{enumerate}

  \item Use your solution from section 4 to determine the ascent time of the dart (time required for the dart to reach maximum height) in terms of the model parameters $g$, $k$, and $v_0$.

  \item Use your formula for $v_T$ (from section 4) to write the ascent time prediction in terms of $g$, $v_0$, and $v_T$.

  \item Verify your ascent time prediction graphically using the determined values for $v_0$ and $v_T$ \\(use $g=9.8\rm{\frac{m}{s^2}}$).

  \item Test your prediction of ascent time by timing (with a stopwatch) the actual ascent of the dart.  This should be done outside, in a stairwell, or a room with a very high ceiling.

  \item Calculate the relative error of your prediction:
  \[
  \textrm{relative error} = \frac{\textrm{observed value} - \textrm{predicted value}}{\textrm{observed value}} \times 100\%
  \]

\end{enumerate}















\section{Equivalent Model with no Air Resistance}

If we neglect air drag in our model, Equation 1 simplifies to
\begin{equation}
y'' = -g
\end{equation}
Here we show that if the effect of air drag is less and less, i.e., $k\rightarrow0$, the solution to Equation (1) becomes the solution to Equation (2).


\begin{enumerate}
	\item Solve Equation (2) using $v(0)=v_0$ and $y(0)=0$.
	\item Graph the solutions ($v$ and $y$) of both Equations (1) and (2) on the same coordinate axis.  Let $k$ approach $0$ using a slider.  What is the interpretation of this limit?  What do you observe in the graphs?
	\item To show that the solution of Equation (1) is equivalent to the solution of Equation (2) if we neglect air resistance, we can let $k$ approach $0$ in the solution of Equation (1).  What do you observe upon attempting this limit?
	\item Use the Taylor series expansion for $e^x$ to expand any exonential terms in your solution of Equation (1) for velocity $v(t)$ from section 4.
	\item Substitute this expansion into the velocity solution of Equation (1) and simplify.
	\item Now let $k\rightarrow0$.
	\item Compare the result to your velocity solution of Equation (2).
	\item Graph

\end{enumerate}

















%\newpage
%\begin{thebibliography}{98}

%\bibitem{Amman1988} Amman, Gene D., Schmitz, Richard F. 1988.  Mountain pine beetle: lodgepole pine interactions and strategies for reducing tree losses. \textit{Ambio} 17:62-68.



%\end{thebibliography}

















\teacher{

 \section{NOTES FOR TEACHERS}

We recommend this project be completed by students in a group setting to strengthen collaboration skills.  The data collection experiments (measuring muzzle and terminal velocities) are also more easily accomplished with a team of students working together.      

Each group will need to determine the frame rate, i.e., frames per second (fps) for their video recording device.  Note that video resolution can be diminished for slow-motion videos which may make seeing tape measure markings difficult.  In this case, we recommend using a brick wall in the background with known brick size to measure distance travelled by darts.  




}





\end{document}
