Brisbane City Council
Background
As part of Brisbane’s “New World City” transformation, the Brisbane City Council is proposing
to allow bungee jumping off the Story Bridge. A commercial bungee jump company has
expressed interest, and provided Council with facts and figures concerning the proposal. Your
group has been hired as consultants to verify certain aspects of this information, and to answer
several key questions that have been raised about the proposal.
To answer these questions you will develop a mathematical model of the bungee jumping
process, solve it numerically using MATLAB, and prepare a report with your results to
present to Council. As the proposal is still only in preliminary stages, the scope of this report
is limited to investigating a few key aspects of the proposal (as will be outlined below).
2 Model
The proposal calls for a platform to be installed at the very top of the bridge, from which the
bungee jumps will take place. Suppose this platform is at height
let y represent the distance the jumper has fallen. Hence, y = 0 corresponds to the platform,
and y increases as the jumper falls towards the river. Later, we will also be interested in the
height of the deck of the bridge, which will be distance D from the water level.
y = 0 Figure 1: Illustration of platform height H, deck height D and jumper’s position y, where
the positive direction of y is downwards.
The mathematical model of bungee jumping can be derived by considering the forces acting
on the jumper at all times. There are three forces we must consider:
1. Gravity
2. Drag (also called air resistance): the frictional force due to the air
3. Tension: the pull of the bungee rope when it is taut
As Figure 2 (next page) illustrates, there are four different parts of the jump: falling with a
slack bungee rope, falling with a taut bungee rope, upward bounce with a taut bungee rope,
and upward bounce with a slack bungee rope (after which the cycle repeats). Depending on
which part of the jump we consider, some of the above three forces may change direction, or
be absent completely.
The simplest force to model is gravity: it always acts downwards, and its value is given by
mg, where m is the mass of the jumper, and g is the gravitational acceleration.
The next force we consider is drag. This force always acts in the opposite direction to motion,
so that it is always slowing the jumper down. Its value is given by −c|v|v, where c is the drag
coefficient, and v is the velocity of the jumper. Notice that the strength of the drag force
is proportional to the square of the velocity. The absolute value sign on the first factor of v
ensures it always acts in the opposite direction to motion.
The final force we must consider is tension. When the bungee rope is taut, it exerts a force on
the jumper proportional to how much it has been stretched (this is called Hooke’s law). This
force always acts upwards, pulling the jumper back up. If we let the length of the unstretched
2
mg
c|v|v
direction
of
travel rope
(a) Downward fall, bungee rope slack
mg
c|v|v
k(y-L)
direction
of
travel
r
o
p
e
(b) Downward fall, bungee rope taut
r
o
p
e
c|v|v mg
k(y-L)
direction
of
travel
(c) Upward bounce, bungee rope taut
c|v|v mg
direction
of
travel rope
(d) Upward bounce, bungee rope slack
Figure 2: The four stages of a bungee jump and the forces acting on the jumper.
3
rope be L, then the tension when the rope is taut is given by k(y −L), where k is the “spring
constant”, which measures the elasticity of the bungee rope. When the rope is slack (like at
the beginning of the jump), there is no tension. Altogether, we may write the tension force
as − max(0, k(y − L)). The maximum function ensures that the tension only “switches on”
when y > L (i.e. when the rope is taut) and the minus sign in front ensures that it acts
upwards.
To summarise, the forces acting on the jumper are
gravity mg
drag −c|v|v
tension − max(0, k(y − L))
Now, Newton’s second law of motion says that the sum of these forces must equal the product
of the jumper’s mass and acceleration. Hence, the equation governing bungee jumping is the
following ordinary differential equation (ODE):
m
dv
dt
= mg − c|v|v − max(0, k(y − L)),
where dv
dt
is the jumper’s acceleration. We can simplify this slightly by dividing through by
m, to obtain
dv
dt
= g − C|v|v − max(0, K(y − L)), (1)
where C = c/m, and K = k/m.
3 Numerical method
Equation (1) is too complicated to solve analytically, so you will need to use numerical
methods to find solutions. You have developed numerical methods for solving ODEs in
your lectures and practicals. Solving the bungee jumping equation will require just a slight
modification of these methods. Since equation (1) actually involves two unknowns, v and y,
we need a second equation that relates the two. This relationship is simply that the jumper’s
velocity v is the derivative of the jumper’s position y. Hence for numerical purposes, we can
think of this problem as two ODEs:
dy
dt
= v (2)
dv
dt
= g − C|v|v − max(0, K(y − L)). (3)
We can now apply any of our numerical methods to these two equations. For example, if we let
yi and vi be our numerical solutions for y and v (we won’t bother introducing a new variable
wi to represent the numerical solution, since we have enough different letters already!), Euler’s
method would become:
yi+1 = yi + hvi (4)
vi+1 = vi + h
g − C|vi
|vi − max(0, K(yi − L))
(5)
where h is the subinterval width.
A complete implementation of Euler’s method for the bungee jumping model is given below:
function [t, y, v, h] = euler_bungee(T, n, g, C, K, L)
%euler_bungee Euler’s method for the bungee jumping model
% [t, y, v, h] = euler_bungee(T, n, g, C, K, L) performs Euler’s method on
% the bungee jumping model, taking n steps from t = 0 to t = T.
% The initial conditions are y(0) = 0 and v(0) = 0.
% The inputs g, C, K and L are parameters from the model (see project description).
% The outputs are the time array t, the solution arrays y and v, and the
% subinterval width h.
% Calculate subinterval width h
h = T / n;
% Create time array t
t = 0:h:T;
% Initialise solution arrays y and v
y = zeros(1,n+1);
v = zeros(1,n+1);
% Perform iterations
for j = 1:n
y(j+1) = y(j) + h*v(j);
v(j+1) = v(j) + h*(g – C*abs(v(j))*v(j) – max(0, K*(y(j) – L)));
end
4 Model parameters
You have been provided with the following parameters for the model. Some of these come from
facts about the bridge itself, while others have been supplied by the bungee jump company
assuming an 80kg jumper.
5 Your tasks
Numerical solution
i. Although you have been provided with an Euler method code for this model, the consultancy
contract with the Council requires you to use a second order or higher numerical
method. Hence you must implement the Second Order Taylor Method or the Modified
Euler Method or the Classical Fourth Order Runge-Kutta method for this model. You
only need to implement one of these: you may choose which.
ii. Use your code to obtain the numerical solution for the model and plot the jumper’s
position against time. You have been provided with the set of model parameters in Table
1. The scope of this report focuses on this particular set of parameters only, which are
for an 80kg jumper.
Analysis
1. The bungee jump company suggests that the standard jump will consist of 10 “bounces”
which should take approximately 60 seconds. (Although the jumper will still be in motion
at this point, the jump will be considered to be over, and the the jumper will be gently
raised all the way back onto the platform above.) Do your model results agree with this
timing: 10 bounces in around 60 seconds?
2. The “thrill factor” of bungee jumping is partly determined by the maximum speed experienced
by the jumper. What is this maximum speed and when does it occur in relation
to the overall jump? Answer this question graphically by plotting the jumper’s velocity
against time.
3. Another factor for thrill-seekers is the maximum acceleration experienced by the jumper.
More acceleration equals bigger thrills, but too much acceleration can be dangerous. The
bungee jump company boasts that the jumper will experience acceleration “up to 2g”. Use
numerical differentiation to find the acceleration predicted by your model, and plot the
jumper’s acceleration against time. What is this maximum acceleration and when does it
occur in relation to the overall jump? Is the claim of “up to 2g” acceleration supported
by the model?
4. For the writing of promotional material it is of interest to know how far the jumper actually
travels in the 60 second jump. One way to answer this question is to compute the integral
Z 60
0
|v| dt .
Use numerical integration to compute this integral and hence determine how far the jumper
travels.
5. Part of the proposal is to have a camera installed on the bridge deck, at height D from the
water. As the jumper first passes this point, the camera would take a photo which could
then be offered for purchase afterwards. It is hoped that the model can provide sufficiently
accurate results that the camera could be set to trigger at a predetermined time, for a
given set of model parameters.
The distance the jumper falls from the platform to the deck is H − D. Hence you need to
compute an accurate value for t such that y(t) = H − D. Since you only know y(t) as a
discrete set of points, not as a function, you will need to fit an interpolating polynomial.
6
(a) To begin with, determine the nearest four values yi of your numerical solution that lie
either side of H −D. That is, find values yi
, yi+1, yi+2, yi+3 such that yi
, yi+1 < H −D
and yi+2, yi+3 > H − D.
(b) Fit the interpolating polynomial p(t) through the four points (ti
, yi), (ti+1, yi+1),
(ti+2, yi+2), (ti+3, yi+3).
(c) Use a rootfinding method of your choice to find the value of t such that p(t) = H −D.
(d) Hence, for the model parameters provided, at what time should the camera trigger
in order to capture the image of the jumper?
6. The bungee jump company has suggested a “water touch” option could be considered,
whereby the jumper just touches the water at the bottom of the first bounce. For the
given parameters, how close does the jumper come to touching the water? Investigate how
the bungee rope could be altered (its length, its spring constant, or both) to produce a
true water touch experience for an 80kg jumper, while keeping as close as possible to 10
bounces in 60 seconds. Note: any combination of parameters that produces acceleration
of greater than 2g must be rejected as too dangerous.
6 Report and code
You must submit your findings in a report. This report will be written in MATLAB itself, using
a script file with sections, which is then built into a PDF report using the Publish feature.
A video tutorial on publishing with MATLAB is available here: http://www.mathworks.
com.au/videos/publishing-matlab-code-from-the-editor-69016.html
A partially complete script M-file for this report has been provided: you must complete the
file by adding in the missing text and code.
Your final submission will consist of the report in PDF format as well as all code, including
the script M-file and all other function M-files that you write, which must include at a
minimum, your own functions to perform
• second order (or higher) numerical solution of model ODEs
• numerical differentiation
• numerical integration
• interpolation
• rootfinding
which will be called from the script file in the appropriate places. The script itself should run
to completion without errors, and generate precisely the report that you submit.
7 Marking criteria
This assignment will be marked using the criteria accompanying this document. The group’s
final grade on a 1-7 scale will be a weighted average of the grades for the criteria:
Grade = (5 × Tasks + Clarity + Correctness + Language + Report + Script)/10 .
This grade will then be mapped to a final score out of 30.
7
8 Statement of contribution
Each group member must complete a statement outlining, in that person’s opinion, the relative
contributions made by every member of the group. A template has been provided for this
purpose. It is expected that each group member will contribute equally to the assignment.
Failure to do so may result in lower marks being awarded to students who did not contribute
sufficiently.
Important: if one or more group members are not contributing (e.g. because of illness or
absence) the rest of the group should not increase their workload to get the assignment completed.
Instead, the assignment should be submitted in incomplete form, with the statement
of contributions clearly indicating the workload that was assigned to the non-contributing
student(s).
9 What to submit
One member of the group submits, as a single zip file:
• all M-files written for the assignment;
• the report in PDF format.
Each member of the group submits (not in a zip file):
• a statement outlining the relative contributions made by each group member (use the
template provided).
