Bungee Jumping Physics

Introduction: Bungee Jumping has become a popular trend in extreme sports. More and more daredevils are participating in this high adrenaline pumping activity around the world. Although many may believe this extreme sport is any other reckless activity, in contrast it is actually a prime example of one of the most exciting and invigorating experiment of physics. More specifically, bungee jumping helps explain and elaborate the complex concept of Hooke’s Law.

History: The origin of Bungee Jumping came about by the ritualistic practices of "land divers" of Pentecost Island in the Pacific Archipelago of Vanautu. However it wasn’t really publicized until members of the Oxford University’s Dangerous Sport Club demonstrated the extreme sport in April 1979. From then on, the sport climbed to new heights and attracted many around the world.

Hooke’s Law: If a force is applied to an elastic spring or prismatic rod, its extension is linearly proportional to its tensile stress and modulus of elasticity. The law holds up to a limit, called the elastic limit.

Hooke’s Law (Simplified): Defines the direct relationship between an applied force and the change in a spring’s length due to that force. After the jumper has passed the stage of freefall, the cord acts as a spring.
Spring Equation: F= -kx
Force= Negative (Spring constant) x (Displacement or Change in the spring’s Length)
*Negative sign does not mean negative value, but direction.
*Associated with the Spring equation is the potential energy to the force. U=-1/2kx^2
*Hooke’s Law can further be generalized in a 3-dimensional aspect as well.

Experiment: In our very own experiment, we can actually determine the k constant (the elastic limit of the cord, each cord has a unique k constant) by dividing the weight in kilograms by 9.8 m/s^2 (gravity) in order to get the force in Newtons, and then dividing the distance change of the cord. This k constant then would be the limit the cord can endure before stretching it to the point where the elasticity would fail.

MATERIALS: Tape
Rubber band (3 kinds)
Meter stick
1000g weight
500g weight
Video camera
Counter top
Hook to hang a rubber band on

SET UP: 

1. Tape the meter stick to the side of a tabletop.
2. Set up the hook in such a way that you can hang a rubber band from it. Keep the hook and meter stick close, however to make sure the weights don't hit the meter sick, keep a certain distance away.
 
PROCEDURE 

3. Place the stiffest rubber band around the hook and observe how far it stretches.
4. Place the 1000g weight around the rubber band, and see how fare it stretches. Take note of the difference.
5. Pick up the 1000g weight so it is parallel to the top of the rubber band, let go of the weight, and Make sure that the rubber band catches both the hook and the weight. Record all of this on the video camera.
6. Place the 500g weight on the rubber band and repeat 4/5.
7. Repeat 3-6 for all three rubber bands, be sure to get it all on camera, as you will analyze the video on a computer.
 
Using mathematics you are able to determine certain things about the rubber bands, such as How far its “stretchability” is, etc.. 
 
OUR RESULTS: 

In our lab study we discovered some obvious truths. The more weight added, the farther the rubber band stretched, the bigger the rubber band, the smaller the stretch. But we discovered something even more. When we tested the biggest rubber band, it stopped the weight much faster than normal. If this was set up as a real bungee jump, the person may have been jumping with a strong piece of string, it would stop them almost immediately, making the jump very uneasy. The smallest rubber band, with the most weight, hit the ground, making that jump fatal. The middle rubber band held tightly, and wouldn’t cause discomfort by shaking or sudden stop, and would be the ultimate bungee jump. Our results came out that to much, or to little, resistance is bad, and for people of different weights and size, different bungee cords must be used.

GRAPH OF OUR RESULTS: A paragraph of information is added below each graph to help you understand what it means:

Note: All of these graph are based off one trial.

This graph that you see above is a sine wave that refers to the movement of a rubber band during our bungee jumps. The physical movement of each bungee jump can be related to a sine wave and an equation that creates that wave. the equation creates the wave and with that equation we can predict what will happen in future bounces of the bungee cord. Amplitude refers to the distace above and below the zero line of the X axis. Frequency is the distance between each wave, phase and offset are where it is positioned physically on the graph.

This graph refers to the position physically of the object during the video. The individual dots were taken from each frame of our video showing the movement over time of our object. Physically you can see speed difference, the larger the distance between two points, the more the speed. You can also see the bounce of our object over time. It started high, then fell past the middle point, then was pulled back up, but not as far as its starting point, then gravity to place again and it fell, the pattern is repeated until the object stabalizes at a certian point over time.

This graph was anaized to show the postition of the object durring its bungee jump, versus its velocity. (position is top, vilocity is bottom). As you can see, the top and bottom points in the position graph shows zero velocity on the bottom graph. durring movement of the position, the velocity gets greatter or smaller durring tha movement. AThe velocity becomes a negative only to show that the object is moving downwords. there is no such thaing as negative vilocity, but the graph shos negative for that reason.

Conclusion: In conclusion our tests results show that the higher the k constant is, the greaterthe force is needed to stretch it. So therefore a good bungee jump requires a good combination of safe resistance of the bungee cord with a good long stretch as well. To relate to the frequency of the bungee jumping itself, the graph (the 1st graph) showed the relative closeness the change of position is to the curve. This demonstrates the frequency of the "oscilllation of the person or weight", which a lower value is desired in bungee jumping because it reduces the chances of injuries.

Data Table of the Rubber Band Experiment.

Related Concepts: Another theory and or concept associated with Hooke's Law, is the vibration motion. The force constant k, effects the frequency of oscillation. This concept relates the actual oscillating effect mentioned in Hooke's Law. Whereas when the projectile is dropped, the force of gravity (the weight of the object) is only acting on it. However as the bungee cord reaches equilibrium, the cord then uses the potential energy gathered by the cord, (spring constant) and sends the individual in the opposite direction. With this up and down motion, it signifies the oscillating movements of a wave and therefore can be related to frequency. So we can use this information to calculate (speed) the frequency to check the measured value of k.

Movies:

Exteme Bungee Jumping

In-Class Experiment

Frontal View of the Experiment

Other Links:

http://www.kettering.edu/~drussell/Demos/SHO/mass.html
http://www.phys.utk.edu/labs/HookesLaw.pdf.
http://www.warren-wilson.edu/~rsmith0001/WebPage2/Simple_Harmonic_Motion.html

Picture Page:

Bungee Picture Page

Saftey Message:

*****This webpage and all its contents do not promote the actions of bungee jumping, but only the physics behind the sport. We are not responsible for any bodily injuries and accidents associated with bungee jumping after reading this page. DO NOT TRY THIS AT HOME!!! And if you do attempt to bungee jump, please consider having a professional instructor present, and up to date safety equipment. ONCE AGAIN WE ARE NOT AT FAULT FOR ANY INJURIES OR YOUR DEMISE!*****

Bibliography:
Smith, Rachel. Simple Harmonic Motion. 5-10-05. 5-24-05
<http://www.warren-wilson.edu/~rsmith0001/WebPage2/Simple_Harmonic_Motion.html>.
(Demonstrates the correlation between Hooke’s Law and the relationship to frequency)

Russell, Dan. The Simple Harmonic Oscillator. Kettering University.
5-24-05 <http://www.kettering.edu/~drussell/Demos/SHO/mass.html>.
(Has a few simulations that provide a good image of the frequency with different lengths of springs)

Menz, Paul. The Physics of Bungee Jumping. Bungee.com. 5-10-05
<http://www.bungee.com/bzapp/press/pt.html>.
(Provides a step by step process of the bungee jumping process)

Hooke's Law. Lockergnome Encyclopedia. 5-22-05
<http://encyclopedia.lockgnome.com/s/b/hooke's_law>.
(Has a few sample calculations)

Nave, Simple Harmonic Motion. Hyper Physics. 5-23-05
<http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html#c1>.
(Provides a insert for values to calculate: Amplitude, Displacement, Angular Frequency and Time)

Solid Mechanics. Wikibooks, 2005. (Has general concepts of Hooke’s Law)
Osler, Margaret. "Hooke, Robert." World Book Encyclopedia. 2000 ed. (Shares brief history of Hooke’s life, and his accomplishments)

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