The Bungee Jump: potential energy at work Ais challenge



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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



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.



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





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/02forces/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/energy/u5l1b.htm


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