Transcript Bungee Jumping - Supercomputing Challenge

The Bungee Jump:
potential energy at work
AiS Challenge
Summer Teacher Institute
2002
Richard Allen

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
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
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, performed 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.
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
Bungee Jump Geometry
L (cord free length)
*
d (cord stretch length)
Schematic depiction of a jumper having
fallen a jump height, L + d.
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.
Examples: Potential Energy
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)
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
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
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.
Equations for d and k
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].
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)
= 114.3 N/m (7.8 lbs/ft)
The maximum force, F = k*x, exerted on the jumper
occurs when x = d:
Fmax = 114.3 N/m * 18 m = 2057.4 N (461.2 lbs),
This produces a force 3 times the jumper weight:
2057.4N/686N ~ 3.0 g’s
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) = 1830.6 N (411.5 lbs)
This produces a force 2.7 times the jumpers weight,
1830.6 N/686 N ~ 2.7 g’s,
and a “softer” ride.
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.
Work To Stretch a Piecewise
Linear Spring
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.
Reference URLs
Constructivism and the Five E's
http://www.miamisci.org/ph/lpintro5e.html
Physics Teacher article on bungee jumping
http://www.bungee.com/press&more/press/pt.html
Hooke’s Law applet
www.sciencejoywagon.com/physicszone/lesson/02
forces/hookeslaw.htm
Reference URLs
Jumper’s weight vs stretch experiment
http://www.uvm.edu/vsta/sample11.html
Ultimate adrenalin rush movie
http://www-scf.usc.edu/~operchuc/bungy.htm
 Potential energy examples
www.glenbrook.k12.il.us/gbssci/phys/Class/energ
y/u5l1b.htm