**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/iconbingeepurple.gif http://www.adrenalindreams.com/Gear%20-%20harness%20GEAR%20SPORTS%20ankle%20logo.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.

**For L<l:** For L > l: Equations

**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

**Mass= 80 kg** Damping Constant= 25 Kg/s

**Spring Constant= 500 N/m** Damping Constant= 25 kg/s

**Spring Constant= 500 N/m** Mass=80 kg

**Spring Constant= 500 N/m** Damping Constant= 25 kg/s Mass= 80 kg

**Spring Constant= 500 N/m** Damping Constant= 25 kg/s Mass= 80 kg

**Spring Constant= 500 N/m** Damping Constant= 25 kg/s Mass= 80 kg

**Spring Constant= 500 N/m** Damping Constant= 25 kg/s Mass= 80 kg

**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