an analysis of the physics behind bungee jumping mathematical modeling will leland, sanket prabhu...

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:

adtvv

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