1. The Bungee Jump: potential energy at work
AiS Challenge
Summer Teacher Institute
2002
Richard Allen  :

2. Bungee Jumping: a short history
The origin of bungee jumping is quite recent, and probably related to the centuries-old, ritualistic practices of the "land divers" of Pentecost Island in the S Pacific.
In rites of passage, young men jump hundreds of feet, protected only by tree vines attached to their ankles
Bungee Jumping was inspired by the vine jumpers of Pentecost Island in Vanuatu (formerly the New Hebrides) in the Pacific Ocean, where it is understood to be both a rite of passage into manhood, and a fertility rite performed to ensure a good yam harvest.

3. A Short History
Modern Bungee jumping began with a four-man team from the Oxford Univ. Dangerous Sports Club jumping off the Clifton Suspension Bridge in Bristol, England, on April 1, 1979 dressed in their customary top hat and tails

4. A Short History
During the late 1980's A.J. Hackett opened up the first commercial jump site in New Zealand and to publicize his site, per-formed an astounding bungee jump from the Eiffel Tower!
Sport flourished in New Zealand and France during 1980s and brought to US by John and Peter Kockelman of CA in late 1980s.

5. The virtual Bungee jumper
A Short History
In 1990s facilities sprang up all over the US with cranes, towers, and hot-air balloons as jumping platforms.
Thousands have now experienced the “ultimate adrenaline rush”.
The virtual Bungee jumper

6. Bungee Jump Geometry
L (cord free length) *
d (cord stretch length)
Schematic depiction of a jumper having fallen a jump height, L + d.

7. Potential Energy
Potential energy is the energy an object has stored as a result of its position, relative to a zero or equilibrium position.
The principle physics components of bungee jumping are the gravitational potential energy of the bungee jumper and the elastic potential energy of the bungee cord.

8. Examples: Potential Energy
Ref:

9. Gravitational Potential Energy
An object has gravitational potential energy if it is positioned at a height above its zero height position: PEgrav = m*g*h.
If the fall length of the bungee jumper is L + d, the bungee jumper has gravitational potential energy,
PEgrav = m*g*(L + d)

10. Treating the Bungee Cord as a Linear Spring
Springs can store elastic potential energy resulting from compression or stretching.
A spring is called a linear spring if the amount of force, F, required to compress or stretch it a distance x is proportional to x: F = k*x where k is the spring stiffness
Such springs are said to obey Hooke’s Law
Hooke’s Law:
If a spring is not stretched or compressed, then there is no elastic potential energy stored in it. The spring is said to be at its equilibrium position.

11. Elastic Potential Energy
An object has elastic potential energy if it’s in a non-equilibrium position on an elastic medium
For a bungee cord with restoring force, F = k*x, the bungee jumper, at the cords limiting stretch d, has elastic potential energy,
PEelas = {[F(0) + F(d)]/2}*d
= {[0 + k*d] /2}*d = k*d2/2

12. Conservation of Energy
From energy considerations, the gravitational potential energy of the jumper in the initial state (height L + D) is equal the elastic potential energy of the cord in the final state (bottom of the jump) where the jumper’s velocity is 0:
m*g*(L + d) = k*d2/2
Gravitational potential energy at the top of the jump has been converted to elastic potential energy at the bottom of the jump.
Conservation of energy: These energy relationships have been used for years to describe events in which loads are suddenly applied to springs. The bungee cord is simply a weak spring yielding large spring deflections and small force magnitudes.

13. Equations for d and k d = mg/k + [m2g2/k2 + 2m*g*L/k]1/2.
When a given cord (k, L) is matched with a given person (m), the cord’s stretch length (d) is determined by:
d = mg/k + [m2g2/k2 + 2m*g*L/k]1/2.
When a given jump height (L + d) is matched with a given person (m), the cord’s stiffness (k) is determined by:
k = 2(m*g)*[(L + d)/d2].
The equation, d2 – (2m*g/k)*d –2m*g*L/k = 0, is just a rearrangement of the basic energy balance equation, m*g*(L + d) = k*d2/2. To solve it, we use the quadratic formula to get
d = {(2m*g/k) +/- [(2m*g/k)2 + 8m*g*L/k]1/2/2
Simplifying and choosing the “+” sign (“-” sign will yield a negative d) gives the above result.

14. Example: a firm bungee ride
Suppose a jumper weighing 70 kg (686 N,154 lbs) jumps using a 9m cord that stretches 18m. Then
k = 2(m * g) * [(L + d)/d2] = 2 * (7 0 * 9.8) *(27/182) = N/m (7.8 lbs/ft)
The maximum force, F = k*x, exerted on the jumper occurs when x = d:
Fmax = N/m * 18 m = N (461.2 lbs),
This produces a force 3 times the jumper weight:
2057.4N/686N ~ 3.0 g’s
This requires a jump height of 27 meters.
F = m*a = m*g = 70*9.8 = 686 N
N/m to lbs/ft2 multiply by
Kilograms to pounds multiply by
Newtons to pounds multiply by
Pounds to kilograms multiply by
Pounds to Newtons multiply by ;

15. Example: a “softer” bungee ride
If the 9m cord stretches 27m (3 times its original length), its stiffness is
k = 2*(70*9.8)*(36/272) = 67.8 N/m (4.6 lbs/ft)
producing a maximum force of
Fmax = (67.8 N/ m)*(27 m) = N (411.5 lbs)
This produces a force 2.7 times the jumpers weight,
N/686 N ~ 2.7 g’s,
and a “softer” ride.
This requires a jump height of 36 meters.

16. Extensions
Incorporate variable stiffness in the bungee cord; in practice, cords generally do not behave like linear springs over their entire range of use.
Add a static line to the bungee cord: customize jump height to the individual.
Develop a mathematical model for jumpers position and speed as functions of time; incorporate drag.

17. Work To Stretch a Piecewise Linear Spring

18. Evaluation
In designing a safe bungee cord facility, what issues must be addressed and why?
Formulate a hypothesis about the weight of the jumper compared to the stretch of the cord as the jumper’s weight increases. Design an experiment to test your hypothesis.
Picture:

19. Reference URLs Constructivism and the Five E's
Physics Teacher article on bungee jumping
Hooke’s Law applet

20. Reference URLs Jumper’s weight vs stretch experiment
Ultimate adrenalin rush movie
Potential energy examples