20/07/2012
Bungee
Jumping
How can mathematical
modelling determine the
critical length of rope
required for an exciting
yet safe bungee jump?
The Modelling Cycle
The problem
Set up a
Mathematical
Analyse the
Model
Problem
Kevin Lord
Further Maths
Support Programme
Interpret /
Validate the
solution
Set up the Model
The problem
List the Factors Involved
Make Assumptions
To determine the
critical length
of rope required?
Determine Constants
Define Variables
Draw a Diagram
The Model
Assumptions
Constants
Variables
•
•
•
•
The jumper is a particle
The rope is light
Air resistance is negligible
Jump is vertically down from rest
•  N is the coefficient of Elasticity
• h metres is the height of the jump
• Gravity is 9.8 ms-2
• m kg is the mass of the jumper
• l metres is the length of the rope
• x metres is extension of the rope
Hooke’s Law
The Tension in an elastic string is
proportional to the extension of the
l
T
string.
T
x
mg
=
x
l
where x is extension of string
and l is the unextended length of string.
 is the coefficient of elasticity.
It is measured in Newtons.
1
20/07/2012
Experiment to find 
Energy
1. Using a piece of elastic of known length l
The energy a body has due to its motion
Kinetic
2. Vary the weight hanging from the elastic and
measure the extension
l
T
3. Repeat for a range of weights
KE
GPE
x
x
x
x
x
x
mg
=
mgh
The energy stored in a elastic string when
it is stretched
Elastic
4. Plot results and estimate 
using a line of best fit
x/l
½ mv2
The energy a body has due to its
position in a gravitational field
The energy due its height above the Earth
Gravitational
Potential
Note: Take care to stretch the elastic before beginning and do
NOT exceed its elastic limit
=
Potential
EPE
=
½x2
l
mg
The Diagram
Analysis
Another simplifying assumption:- All energy is conserved
u=0
Therefore the Principle of Conservation of Energy
can be applied.
l
h
This states that
Initial Energy
x
v=0
Zero GPE level
½
mu2
KE
=
Final Energy
+
mv2 + mgh
x2
+ mgh
GPE + EPE = ½ KE
GPE + ½
EPE
mgh
=
l2
½ x
l
Analysis
2
20/07/2012
Interpretation
Using the values
length
= 1 N
h =2m
We can simplify for l as a function of m
It is possible to calculate the critical length of rope for any
given mass of jumper
mass
The relationship between length and mass is not a simple one,
but a graph may help to understand this
Validation
length
0.2 < < 2.0
We can predict the critical length and test for various masses
Limitations
•
Energy is clearly not conserved!
There is some air resistance
The string is not perfectly elastic
•
Experimental result for  is likely to be inaccurate
mass
Extension Ideas
Excitement of the Jump depends on the Free Fall
Time – how can this be increased safely?
What forces does the jumper experience at different
points in the jump?
Investigate the energy lost during a jump.
For a given bungee cord is there a maximum weight
of jumper or maximum height of drop?
3