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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\ Bungee Jumping}




\markboth{Bungee Jumping}{Bungee Jumping}

\author{Gregory Tanner\\
Mathematics Department\\ 
Concordia College\\
Moorhead MN 56560 USA\\
gtanner2@cord.edu
}

\makeStitlePDFLaTex


\begin{abstract}
In this project, students design a Bungee jumping cord (cross-sectional area and length) that will keep jumpers safe. This open-ended project can be tackled with second-order differential equations, energy balance, or numerical simulation. Students communicate their recommendations as a letter to a client interested in starting a Bungee jumping business.
\end{abstract}



\section*{REQUEST FOR HELP}

Greetings Differential Equations Students,

I am participating in a competition for startups, and I need your help! I’m proposing a new Bungee jumping business marketed specifically to newlyweds on their honeymoons, to be named “Take the Plunge!” LLC and am soliciting opinions from several international experts on the technical side of things to strengthen my proposal.

My proposal involves sunset Bungee jumps from the Thomas Rukavina Memorial Bridge (the tallest bridge in Minnesota), which is 204 feet above a river, providing a spectacular view. I’m planning to use rubber Bungee cords. You can find more information on the type of cord and cord specifications below, as provided by my supplier and translated by your instructor. I need to select the length and diameter of the cords so as to maximize fun while avoiding potential negative outcomes (I would like happy and surviving customers at the end of this adventure!).

From my own experiences as a Bungee jumper and a student in introductory physics classes over a decade ago, I know that Bungee jumping is sort of like the classic spring-mass problem, but I’ve long forgotten the math involved to figure out a good recommendation for my business proposal. I’m hoping you can help me achieve this goal. I’ve arranged with your instructor to provide some extra details to help you (enclosed with this letter). I look forward to hearing from you.

\vspace{0.5cm}
\noindent Sincerely, \\
\noindent Harris Tottle

\teacher{
\vspace{0.5cm}
\textbf{Note:} Feel free to omit or modify the letter. I recommend swapping out the Thomas Rukavina Memorial Bridge for another such tall structure in your area that would be suitable for Bungee jumping.
}

\section*{SCENARIO DESCRIPTION}

You have been contracted by Mr.~Tottle to help jump-start his Bungee jump business proposal. Some additional instructions to model the relevant scenario are provided below.
\begin{enumerate}
\item Create a mathematical model to respond to the client's requests. To simplify the process, you will need to make appropriate assumptions. Explain your assumptions to Mr.~Tottle.
\item The stiffness coefficient $k=EA/L$ in the mass-spring problem depends on the elastic modulus $E$, cross-sectional area $A$, and length $L$ of the Bungee cord. Based on the specs provided by the client, he is able to get rubber cords with elastic modulus $E=0.70$ megapascals (MPa) in various diameters and lengths. The rubber cords are rated for a maximum strain (percent elongation) of 400\% to prevent excessive wear or potential failure. Make sure your response includes a recommendation as to what lengths and diameters are safe to use.
\item Mr. Tottle would like to avoid the following potential negative outcomes for his customers:
\begin{itemize}
	\item Strain beyond the manufacturer's recommendation of 400\%. That is, the maximum length of the cord is at most 5 times the initial length.
	\item Excessive accelerations. People tend to start feeling ill-effects when they  experience more than three time the acceleration of gravity (commonly phrased as $3 g$ of acceleration, where $g\approx 9.8 {\rm m}/{/rm s}^2$ is the acceleration of gravity at the Earth's surface) in a feet-to-head direction.
	\item Getting wet in the river 204 feet below.
\end{itemize}
\item Here is some guidance as you approach the scenario:
\begin{itemize}
\item We can separate this differential equation into two parts: when the jumper will be in freefall (Bungee cord has slack), and when the mass-spring system begins (Bungee cord is stretched). You will need to consider how to deal with the transition between differential equations.
\item After solving your model, find the maximum displacement and acceleration. Is the strain within the manufacturer's recommendation? Does the jumper hit the river? Does the jumper experience more than $3g$ of acceleration? 
\item Determine which cords can be safely used, and make a recommendation to the client. For the chosen cord, what is the maximum and minimum mass of jumpers that could use it? Can the client reasonably use a single cord for all of his customers or should he have different cords for different people?
\item Engineers often incorporate safety margins into their designs. What is the margin of safety for the largest/smallest jumpers with respect to the three potential negative outcomes? That is, how close to the maximum strain, maximum acceleration, or the ground, does the largest/smallest jumper get?
\item Extra Challenge: What effect would including air resistance have on your model?
\end{itemize}
\item Write your client a letter explaining your findings. Keep in mind the fact that your client may know very little mathematics. Include all relevant information, assumptions, calculations, and recommendations in your response. 
\end{enumerate}


\teacher{
\section*{NOTES FOR TEACHERS}

This scenario was designed to be given as a project. I gave students 2 weeks, one in-class (70 minutes) work day, and required a 15-20 minute check-in meeting outside of class. I also abbreviated some of the homework during the project to give students time to work on the project. 

The following are some hints and tips. You may want to give some of these hints right away or reserve them until students have thought about the problem a bit.

\begin{enumerate}
\item There are several approaches students could use to solve this: analytical, energy balance, or numerical simulation. I left it open to the students and had groups use each of the three approaches.

\item If students are using the analytical approach, make sure that they know how to transform a linear combination of sine and cosine into a single sine/cosine with amplitude and phase shift: 
\[
A \sin (\omega t) + B\cos(\omega t) = \sqrt{A^2+B^2} \cos(\omega t - \arctan(B/A))
\]

\item Students (especially those with a physics background) could use an energy balance approach to find the maximum displacement by equating the change in gravitational potential energy with the change in spring potential energy (since there is no kinetic energy at the very top and bottom). I had one student comment that this felt like ``cheating'' since it didn't require any differential equations. This was a good opportunity to talk about why the total energy is constant for conservative systems and the limitations if you wanted to include the effects of air resistance.

\item For numerical simulation, a useful trick for dealing with the spring force is to set:
\[
\text{stretch}=\max\left(0,x-L\right)
\]
so that the stretch is zero if the displacement ($x$) is less than the length of the cord ($L$).
\end{enumerate}

\section*{BRIEF SOLUTION}

The following is a brief analytical solution to the scenario. 
\begin{enumerate}
\item Let $x$ be the distance fallen measured from the top of the bridge with down as the positive direction. Some students needed help thinking through how they should help set up the coordinate system, especially where they should put the reference point (top of the bridge, length of the cord below the bridge, or at the equilibrium point). Each of these reference points can be used as long as the correct corresponding form for the differential equation is used.
 
\item During the free-fall portion, the differential equation is:
\[
mx''=mg, \quad x(0)=0\, , \quad x'(0)=0\, m/s\, , 
\]
with solution $x(t)=\frac{1}{2}gt^2$ where $g\approx 9.8\,  {\rm m}/{\rm s}^2$ is the acceleration due to gravity. At the end of the free fall portion ($x(t_L)=L, t_L=\sqrt{2L/g}$), the velocity is $x'(t_L)=\sqrt{2Lg}$. These will serve as the initial condition for the spring-mass portion.

\item During the spring-mass portion, the differential equation is:
\[
mx''=mg-k(x-L), \quad x(t_L)=L, \quad x'(t_L)=\sqrt{2Lg}
\]
with solution:
\[
x(t) = L + \frac{mg}{k} - \frac{mg}{k}\cos\left(\omega(t-t_L)\right) + \sqrt{\frac{2Lmg}{k}}\sin\left(\omega(t-t_L)\right)
\]
where $\omega=\sqrt{k/m}$. Be aware that this solution is only valid while $x(t)\geq L$.
\item This solution can be cleaned up significantly by introducing the dimensionless term $\theta = \frac{mg}{kL} = \frac{mg}{EA}$ and combining the linear combination of sine and cosine into a single cosine term to get:
\begin{align*}
\frac{x(t)}{L}&=1+\theta-\theta\cos\left(\omega(t-t_L)\right)+\sqrt{2\theta}\sin\left(\omega(t-t_L)\right) \\
&= 1+\theta +\sqrt{\theta^2+2\theta}\cos\left(\omega(t-t_L)-\arctan\left(\sqrt{\theta/2}\right)\right)
\end{align*}

\item The condition that the maximum strain is less than 400\% becomes:
\[
\theta +\sqrt{\theta^2+2\theta}\leq 4
\]
which is satisfied when $\theta \leq 8/5$.

\item The condition the maximum acceleration is less than $3g$ becomes:
\[
\frac{L\omega^2}{g}\sqrt{\theta^2+2\theta} =\sqrt{1+\frac{2}{\theta}} \leq 3
\]
which is satisfied when $\theta \geq 1/4$.

\item These two conditions $1/4\leq \theta \leq 8/5$ allow the cross-sectional area of the cord to be related to the mass of the jumper. For a given mass of the jumper $m$, we get a range of safe values for the cross-sectional area $A$:
\[
\frac{5 mg}{8 E} \leq A \leq \frac{4 mg}{E}
\]
Conversely, for a given cross-sectional area $A$, we get a range of acceptable masses $m$:
\[
\frac{EA}{4g} \leq m \leq \frac{8EA}{5g}
\]

\item To select an appropriate length $L$, we need that $5L$ is less than the maximum allowable distance that the jumper can fall (height of the bridge minus the height of the jumper and a reasonable safety margin). For the Thomas Rukavina Memorial Bridge (204 feet above the river), a 39 foot cord would keep jumpers above the water (with just a few feet to spare). 

There is some give and take between the length and cross-sectional area of the cord. For example, if a thicker cross-sectional area is chosen such that the maximum strain is less than 300\% ($\theta \leq 9/8$), then a longer cord could be used. This combination of a thicker, longer cord would result in a longer free-fall portion and greater max speed (but also greater decleration). Most students were not as systematic in their solutions. There is room for some creativity in coming up with reasonable recommendations for the area and length of the cords.
\end{enumerate}

In addition, an Excel file (3-104-T-Excel-BungeeJumping-TeacherVersion.xlsx) with a numerical simulation (Euler's method) is provided. I used this file to aid in grading because it made it easy to enter the parameters from the students' recommendations and check whether all three conditions were satisfied. 

}


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