The Bungee Jump: potential energy at work

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The Bungee Jump potential energy at work

• AiS Challenge
• Summer Teacher Institute
• 2002
• Richard Allen
• ?

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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

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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
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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.

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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

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Bungee Jump Geometry
L (cord free length)

d (cord stretch length)
Schematic depiction of a jumper having fallen a
jump height, L d.
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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.

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Examples Potential Energy
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Gravitational Potential Energy

• An object has gravitational potential energy if
  it is positioned at a height above its zero
  height position PEgrav mgh.
• If the fall length of the bungee jumper is L
  d, the bungee jumper has gravitational
  potential energy,
  
• PEgrav mg(L d)

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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 kx where
  k is the spring stiffness
• Such springs are said to obey Hookes Law

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Elastic Potential Energy

• An object has elastic potential energy if its in
  a non-equilibrium position on an elastic medium
• For a bungee cord with restoring force, F
  kx, the bungee jumper, at the cords limiting
  stretch d, has elastic potential energy,
• PEelas F(0) F(d)/2d
• 0 kd /2d kd2/2

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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 jumpers velocity
  is 0
• mg(L d) kd2/2
• Gravitational potential energy at the top of the
  jump has been converted to elastic potential
  energy at the bottom of the jump.

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Equations for d and k

• When a given cord (k, L) is matched with a
  given person (m), the cords stretch length (d)
  is determined by
• d mg/k m2g2/k2 2mgL/k1/2.
• When a given jump height (L d) is matched
  with a given person (m), the cords stiffness (k)
  is determined by
• k 2(mg)(L d)/d2.

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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 kx, 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 gs

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Example a softer bungee ride

• If the 9m cord stretches 27m (3 times its
  original length), its stiffness is
• k 2(709.8)(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 gs,
• and a softer ride.

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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.

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Work To Stretch a Piecewise Linear Spring
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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 jumpers weight
  increases. Design an experiment to
  test your hypothesis.

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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/pressmore/press/pt.html
• Hookes Law applet
• www.sciencejoywagon.com/physicszone/lesson/02force
  s/hookeslaw.htm

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Reference URLs

• Jumpers 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/energy/u
  5l1b.htm