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\begin{document}

\maketitle

\hypertarget{why-walk-when-you-can-zipline}{%

\section{Why Walk When You Can

Zipline?}\label{why-walk-when-you-can-zipline}}

\hypertarget{sam-daitzman-and-jocelyn-jimenez}{%

\subsection{Sam Daitzman and Jocelyn

Jimenez}\label{sam-daitzman-and-jocelyn-jimenez}}

December, 2018

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\hypertarget{introduction}{%

\section{Introduction}\label{introduction}}

Although Olin is a very small campus, it is sometimes very tedious

having to walk through various curved paths and flights of stairs. We

noticed that there was great potential of a zip line from Olin's

Academic Center to West Hall. Initially there were multiple issues that

arose. For instance, where will the landing spot be? In the case of our

model, we believe it is best if the zip line ends on the corner room in

West Hall on the second floor because it gives us the fastest access to

most dorms in West Hall. If we were to ask ``Which floor in the AC

allows us to get faster to our destination?'' the answer would be

obvious, the 4th floor; but if you zipline at too steep an angle, you

are likely to get injured. As a result, we decided to take into

consideration the safety of the individual in order to make it a more

useful model. Our model asks what the fastest safe zip line starting

mount-point would be to get from the AC to West Hall as fast as (safely)

possible. Stick around to find out where to anchor a zip-line in the

Academic Center.

\hypertarget{code-setup}{%

\section{Code Setup}\label{code-setup}}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

To prove the point that is is a good idea for Olin to have a zip line,

we modeled how the change in position will affect the velocity of the

individual. We considered important initial/constant variables like the

starting position, where (0,0) is at the tallest part of the Academic

Center. For the following model, the starting velocity of the person in

the x and y dimensions is 0 m/s, which can be seen below.

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

Other constants include the mass, density, and area of the individual,

gravity, drag force, the height of the West Hall window, the maximum

landing speed and the end time. These constants are defined as the

parameters of the model.

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\hypertarget{force-functions}{%

\section{Force Functions}\label{force-functions}}

We decided to calculate the different forces that might affect the

zip-liner by including forces like gravity, drag, net and effective

force. The cells below demonstrate functions and equations that

contribute to the final velocity of the individual. We calculate the

force of gravity pulling the zipliner downward and the drag force

resisting their motion, and their sum is the net force. Then we

calculate the effective force, which is the component of the force in

line with the zip-line.

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\texttt{\color{outcolor}Out[{\color{outcolor}0}]:}

$\begin{pmatrix}0.0 & -981.0\end{pmatrix}\ \frac{\mathrm{kilogram} \cdot \mathrm{meter}}{\mathrm{second}^{2}}$

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\texttt{\color{outcolor}Out[{\color{outcolor}0}]:}

$\begin{pmatrix}-1200.0 & -1600.0\end{pmatrix}\ \frac{\mathrm{kilogram} \cdot \mathrm{meter}}{\mathrm{second}^{2}}$

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\hypertarget{simulation-setup}{%

\section{Simulation Setup}\label{simulation-setup}}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\texttt{\color{outcolor}Out[{\color{outcolor}0}]:}

$-103.10193984596022\ \mathrm{meter}$

\hypertarget{simulation}{%

\section{Simulation}\label{simulation}}

This simulation shows the zip-liner descending from the AC to West Hall

from a particular starting height.

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{center}

\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_23_0.png}

\end{center}

{ \hspace*{\fill} \\}

This plot shows the position over time. Each point along the line is a

different moment in time.

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{center}

\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_25_0.png}

\end{center}

{ \hspace*{\fill} \\}

These lines represent the change in X and Y position over time.

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

The cell above shows how we obtain the arrival velocity.

\hypertarget{sweeping-start-height}{%

\section{Sweeping Start Height}\label{sweeping-start-height}}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

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\begin{center}

\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_31_10.png}

\end{center}

{ \hspace*{\fill} \\}

The plot above shows the process of zip-lining from various starting

heights. We obtained the range of height differences using Olin

College's blueprints, by comparing the absolute heights of the top of

the AC, bottom of the AC, and our destination room in West Hall.

\hypertarget{finding-end-velocities}{%

\section{Finding End Velocities}\label{finding-end-velocities}}

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\texttt{\color{outcolor}Out[{\color{outcolor}0}]:}

$2.651921049001805\ \frac{\mathrm{meter}}{\mathrm{second}}$

\begin{Verbatim}[commandchars=\\\{\}]

\end{Verbatim}

\begin{center}

\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_37_0.png}

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{ \hspace*{\fill} \\}

This plot shows the landing speed depending on starting height. By

looking at this plot, we can choose a particular start height depending

on our desired landing speed.

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\hypertarget{ideal-starting-height}{%

\section{Ideal Starting Height}\label{ideal-starting-height}}

We wanted to calculate our ideal starting height. We wrote an error

function that approaches zero as the ideal starting height is

approached. The maximum landing speed is determined based on the maximum

safe landing speed of a hang-glider, which exerts analagous forces on a

human.

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\texttt{\color{outcolor}Out[{\color{outcolor}0}]:}

$9.348078950998195\ \frac{\mathrm{meter}}{\mathrm{second}}$

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\texttt{\color{outcolor}Out[{\color{outcolor}0}]:}

$2.1085355683680973e-12\ \frac{\mathrm{meter}}{\mathrm{second}}$

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\begin{center}

\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_46_0.png}

\end{center}

{ \hspace*{\fill} \\}

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\begin{center}

\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_47_0.png}

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{ \hspace*{\fill} \\}

This plot represents the ideal descent of a zip-liner from the AC to

West Hall.

\begin{Verbatim}[commandchars=\\\{\}]

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\texttt{\color{outcolor}Out[{\color{outcolor}0}]:}

$11.999999999997891\ dimensionless$

The zip-liner arrives incredibly close to, but slightly below, the

maximum safe velocity.

\hypertarget{conclusions}{%

\section{Conclusions}\label{conclusions}}

In this model, we find that the ideal starting height to arrive as

quickly as possible (but at a safe speed) when zip-lining to West Hall

would be around the third floor (about 15m above the first floor of the

AC). The zip-liner would arrive safely at a speed slightly below 12 m/s.

Before attempting this, we would want to conduct more precise modeling

and account for the forces we've abstracted out of our model, like the

tension in the rope and the changing normal force of the rope against

the zip-liner through the handle.

\hypertarget{future-steps-questioning-assumptions.-straight-zipline}{%

\section{Future Steps: Questioning Assumptions. Straight

Zipline?}\label{future-steps-questioning-assumptions.-straight-zipline}}

We didn't have time to finish this modeling work, but with more work we

might be able to simulate a more accurate zip-line curvature. To

simplify our modeling, we assumed the line would be under infinite

tension (in other words, perfectly straight) and the wheel would roll

perfectly. For the wheel to behave efficiently, it's more likely that

the line would have to maintain some slack. To get a more accurate

estimate, we would continue this modeling work.

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\begin{center}

\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_56_0.png}

\end{center}

{ \hspace*{\fill} \\}

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\begin{center}

\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_57_0.png}

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{ \hspace*{\fill} \\}

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