Post on 13-Dec-2015
An Analysis of the Physics Behind Bungee Jumping
Mathematical Modeling
Will Leland, Sanket Prabhu
Tarboro High School, William G. Enloe High School
2008
Outline
• Background/History• Model• Data
– Constants– Equations– Force – Acceleration– Velocity
• Conclusion http://www.vancouverisland.travel/img/wildplay/bungy.jpg
Problem
• How do the spring constant, damping constant, and jumper mass affect the path of a bungee jumper?
http://alexandre.alapetite.net/cv/photos/19990730-alexandre-alapetite-1.jpg
Origin of Bungee Jumping
• Created thousands of years ago, by the inhabitants of Pentecost Island
• A group of 20 young men would take the leap of death
• Used to please the gods in order to have plentiful crops
• The land dive would symbolize the jumper’s transition from a child to a man
New Beginning of Bungee Jumping
• The first modern day bungee jumps were executed on April 1, 1979 by the Oxford University Dangerous Sports Club
• The sport’s popularity quickly spread across the world
• The world record for the highest jump is 216 meters of off the Bloukrans River Bridge
Equipment
• An elastic rope that is usually enclosed in a tough outer cover
• A simple ankle attachment
• A body harness
• Jumping platform
http://www.adrenalindreams.com/Gear%20-%20harness%20GEAR%20SPORTS%20ankle%20logo.gif http://www.adrenalindreams.com/iconbingeepurple.gif
Types of Jumps
• Swallow Dive – classic jump, arms out wide and soar down like a bird
• Water Touchdown – some sites are confident about the length that the cord will stretch, so at the bottom the jumper goes into the water
• Sandbagging – extremely dangerous, you jump with a heavy weight, once you get to the bottom, you let go of the weight, the added elastic energy will make you fly much higher than from where you jumped from
What is Force, Velocity, and Acceleration?
• Force- a push or pull
• Velocity is the derivative of position
• Acceleration is the derivative of velocity
Constants
• K = spring constant - determines elasticity of cord, meaning how far it stretches
• m = mass - determines mass of jumper
• b = damping constant - a constant that is put in to represent the loss of energy
Physics Behind the Jump
• L is the distance from the bridge to the position of the jumper
• l is the length of the cord at rest• While L < l, the only force working on the jumper
is projectile motion• When L > l, the cord starts to exert an upward
force on the jumper• The spring constant factors in as it determines
the magnitude of the upward force.
Equations
• For L<l: • For L > l:
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adtvdtPP
m
Fa
bvmgF
nn
nn
1
21 5.0
adtvv
adtvdtPP
m
Fa
bvlLkmgF
nn
nn
1
21 5.0
)(
Bungee Cord Diagram
http://www.pa.uky.edu/~moshe/phy231/lecture_notes/bungee_forces.html
The Code:
The Model
Assumptions
• Bungee cord is in perfect condition
• Ideal environment so that jumpers only move in one direction
Max vs. Min spring constant (N/m)
-6
-4
-2
0
2
4
6
8
10
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Po
siti
on
Minimum Spring Constant (250N/m)
Maximum Spring Constant(750N/m)
c
Mass= 80 kg
Damping Constant= 25 Kg/s
Spring Constant vs. Period
T = 0.0015k + 2.435
0
0.5
1
1.5
2
2.5
3
3.5
4
300 350 400 450 500 550 600 650 700 750 800
Spring Constant
Perio
d
Max vs. Min Jumper Mass (kg)
-4
-2
0
2
4
6
8
10
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Po
siti
on
Minimum Jumper Mass (60 kg)
Maximum Jumper Mass (100kg)
Spring Constant= 500 N/m
Damping Constant= 25 kg/s
Mass vs. Period
T = 0.0195m + 1.47
0
0.5
1
1.5
2
2.5
3
3.5
4
55 60 65 70 75 80 85 90 95 100 105
Mass
Perio
d
Max vs. Min Damping Constant (kg/s)
-20
-10
0
10
20
30
40
50
60
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Po
stio
n
Minimum Damping Constant(kg/s)
Maximum Damping Constant(kg/s)
Spring Constant= 500 N/m
Mass=80 kg
Normal Constants
-2
0
2
4
6
8
10
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Po
siti
on
Spring Constant= 500 N/m
Damping Constant= 25 kg/s
Mass= 80 kg
Jumper Position vs. Time
-10
-5
0
5
10
15
20
25
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Po
siti
on
Average with 100 kg sandbag
Average with 50 kg sandbag
Spring Constant= 500 N/m
Damping Constant= 25 kg/s
Mass= 80 kg
Force (N) vs. Time
-1500
-1000
-500
0
500
1000
1500
2000
2500
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Posi
tion
Spring Constant= 500 N/m
Damping Constant= 25 kg/s
Mass= 80 kg
Velocity (m/s) vs. Time
-15
-10
-5
0
5
10
15
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Po
siti
on
Spring Constant= 500 N/m
Damping Constant= 25 kg/s
Mass= 80 kg
Acceleration (m/s^2) vs. Time
-15
-10
-5
0
5
10
15
20
25
30
0 5 10 15 20 25 30 35 40 45 50
Time (s)
Po
siti
on
Spring Constant= 500 N/m
Damping Constant= 25 kg/s
Mass= 80 kg
Changes Based on Findings
• Add wind factor, so we would be able to manipulate a z factor as well.
• Work on the rope so that when it came up it would produce slack and fold over
• Model a water touchdown
Summary
• Bungee jumping was created thousands of years ago and still continues today as a popular and exhilarating sport
• Spring constant, damping constant, and mass vary the jumper’s fall by different magnitudes.
Conclusion
• It was found that a high damping constant and mass results in the jumper coming to equilibrium faster
• A larger spring constant limits the jumper’s oscillation amplitude.
• The period looks to have linear relationships with the spring constant and mass
What We Learned
• The basics of VPython, Excel, and PowerPoint
• The physics behind bungee jumping and how to manipulate the parameters
• The long, rich history of bungee jumping
References
• http://library.thinkquest.org/C0123122/historybungee.htm
• http://www.bungeezone.com/history/
• http://www.bungeeamerica.com/nowhr.htm
• http://www.pa.uky.edu/~moshe/phy231/lecture_notes/bungee_forces.html
Acknowledgments
• Special thanks to: Dr. Russ Herman and Mr. David Glasier for their generous aid in class and on this project
• Also thanks to: the 2008 SVSM staff for providing an excellent social and learning environment
• Thanks to our parents for the opportunity