14.1: Bungee Jumping

Last updated
Save as PDF

Page ID: 84549

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Suppose you want to set the world record for the highest “bungee dunk,” which is a stunt in which a bungee jumper dunks a cookie in a cup of tea at the lowest point of a jump. An example is shown in this video: https://greenteapress.com/matlab/dunk.

Since the record is 70 m, let’s design a jump for 80 m. We’ll start with the following parameters:

Initially, the jumper stands on a platform 80 m above a cup of tea. One end of the bungee cord is connected to the platform, the other end is attached to the jumper, and the middle hangs down.
The mass of the jumper is 75 kg, and they are subject to gravitational acceleration of 9.8 m/s².
In free fall the jumper has a cross-sectional area of 1 m and a terminal velocity of 60 m/s.

To model the force of the bungee cord on the jumper, I’ll make the following assumptions:

Until the cord is fully extended, it applies no force to the jumper. It turns out this might not be a good assumption; we’ll revisit it in the next section.
After the cord is fully extended, it obeys Hooke’s Law; that is, it applies a force to the jumper proportional to the extension of the cord beyond its resting length.

We can write Hooke’s Law as \(F_\mathrm{s} = -k x\) where \(F_\mathrm{s}\) is the force of the spring (bungee cord) on the jumper in newtons, \(x\) is the distance the spring is stretched in meters, and \(k\) is a spring constant that represents the strength of the spring in newtons per meter. The minus sign indicates that the direction of the spring force is opposite to the direction the spring is stretched.

Hooke’s Law is not a law in the sense that it is always true; really, it is a model of how some things behave under some conditions. Almost everything obeys Hooke’s Law when \(x\) is small enough, but for large values everything deviates from this ideal behavior, one way or the other.

In reality, the spring constant of a bungee cord depends on \(x\) over the range we are interested in, but as a starting place I’ll assume \(k\) is constant.

Exercise 14.1

Write a simulation of this scenario, based on these parameters and modeling assumptions. Use your simulation to choose the length of the cord, L, and its spring constant, k, so that the jumper falls all the way to the tea cup, but no farther!

You could start with the length 25 m and the spring constant 40 N/m.