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Bungee Jumping Physics

  1. Apr 24, 2015 #1
    1. The problem statement, all variables and given/known data

    Hi, just wondering if you could help me with a Physics/Math assignment I'm currently working on. Involves the following questions:

    New Zealand is the home of bungee jumping. One of the major jumps is located on a bridge over the Shotover River near Queenstown. In this case, the bridge is 71 m above the river. Two types of jumps are available — wet and dry. In a dry jump, the person’s fall ends just above the water surface. In a wet jump the person is submerged to a depth of 1 m. Participants jump from the bridge, fastened to an elastic rope that is adjusted to halt their descent at an appropriate level. The rope is specially designed and its spring constant is known from specifications. For the purposes of the problem, we will assume that the rope is stretched to twice its normal length by a person of mass 75 kg hanging at rest from the free end. It is necessary to adjust the length of the rope in terms of the weight of the jumper.
    1. For a person of mass m kg, calculate the depth to which a person would fall if attached to a rope of the type described above, with length l metres. Treat the jumper as a particle so that the height of the person can be neglected. Discuss the assumptions made in this calculation.
    2. If you were the person jumping off the 71 m attraction, find the length of rope needed for a dry jump, where the descent is halted 1 m above the water.
    3. Now find the length of rope needed for a wet jump, where the descent would end 1 m below the surface of the water. Find the speed of entry to the water.
    4. In practice, the bungee rope is attached to the ankles of the jumper. Refine the previous model to allow for the height of the jumper and modify the earlier calculations. Is the difference significant?
    5. At present, the model does not include air resistance. Discuss the changes which would have to be made to the model to include air resistance, which is proportional to the velocity of the jumper. Discuss the difficulties involved with the mathematics of this model.

    2. Relevant equations

    All displayed above.

    3. The attempt at a solution

    I'm mainly stuck upon Question 5... I can calculate the velocity at any point but not sure where to find the maximum velocity to show what the major impacts of air resistance will be??
     
  2. jcsd
  3. Apr 24, 2015 #2
    Are you sure about this? Do you realize that the acceleration of the man falling down is changing throughout the fall?
     
  4. Apr 24, 2015 #3
    Air resistance can be modeled as a frictional force that is proportional to the square of the velocity of the jumper. I just could not let that slide.

    Junior Newton, why don't you draw a free body diagram for the jumper (include air resistance) and then upload it.
     
  5. Apr 24, 2015 #4

    haruspex

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    How air resistance varies with speed is complicated. It approximates a quadratic over some ranges, closer to linear over others. Quadratic probably is more appropriate for a bungee jumper, but if the question says linear then that's what must be used. It will be crucial since the question is asking about the difficulty of solving the ODE.
     
  6. Apr 24, 2015 #5
    I've got a formula. The GPEtotal = GPE +EPE+KE
    and using physics formula, KE =0.5mv^2, can rearange for v at any point
     
  7. Apr 24, 2015 #6
    If I was to find the maximum velocity, using the Air Resistance formula. Would you know how I would find out how it would alter velocity?
     
  8. Apr 24, 2015 #7
    Like I said before, why don't you draw a free body diagram for the jumper using the whiteboard functionality and then post it. Make sure to include air resistance in the diagram. This is crucial to solving these types of problems.
     
  9. Apr 24, 2015 #8
    Good. I too would recommend that you now draw an FBD of the bungee jumper.
    See this wiki article and this hyper-physics link. Also do you know anything about differential equations ?
    Cheers.
     
  10. Apr 24, 2015 #9

    haruspex

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    If you have to take into account air resistance that is not going to work. Air resistance removes energy. The first step is to consider forces on the jumper at an arbitrary point of the descent and find a differential equation for how velocity will change with time (or with distance).
     
  11. Apr 25, 2015 #10
    Differential Equation?
     
  12. Apr 25, 2015 #11

    haruspex

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    Yes, as in dv/dt = acceleration = Fnet/m. Fill in an expression for Fnet. This will include a drag term which will be a function of v.
     
  13. Apr 25, 2015 #12
    Rearanging GPE=GPE+EPE+KE to get v by itself and then deriving to find the maximum velocity. WOuld that work?
     
  14. Apr 25, 2015 #13
    No, you are talking about energy in that equation. Draw an FBD of the bungee jumper, trust me it'll help.
     
  15. Apr 25, 2015 #14
    Not really understanding what you mean?
     
  16. Apr 25, 2015 #15
    Are the only forces acting upon the jumper gravity and air resistance?
     
  17. Apr 25, 2015 #16
    A free body diagram (FBD) of an object is a diagram that shows all the forces acting on the object. It is often used in classical mechanics problems to make things clearer
    and more easier.
    No. This is why I wanted you to realize that the acceleration of the jumper is changing throughout even without air resistance. Remember that he is attached to the bungee rope which is pulling on him in the opposite direction throughout the fall and this force is increasing as the jumper falls down because the tension in the rope is increasing.
     
  18. Apr 25, 2015 #17
    I don't want to go into great detail in my assignment. I would love to show how air resistance changes the velocity at the maximum velocity if possible.

    I'm unsure on how to do this. My idea is to create a formula which i can reanrage to get v as the subject and derive to find the maximum. Would that work?
     
  19. Apr 25, 2015 #18

    haruspex

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    Then I do not see how you are going to answer this:
    Yes, you can get to the terminal velocity without solving a differential equation. But you still need to look at forces to do that, not energy.
     
  20. Apr 25, 2015 #19
    hmmmm......... in that case I think the answer to 5 is simply this:- that air resistance is a force that always opposes the motion of the falling body and it is directly proportional to the velocity of the body.The dependence of air resistance on velocity is itself somewhat uncertain, it is linear for low speeds and small objects, and quadratic for high speeds and bigger objects. And this causes the mathematics governing the motion of the body to become much more complex.

    I can't think up of a simpler answer than that :)
     
  21. Apr 25, 2015 #20

    haruspex

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    That would be a great answer if it were not for the fact that the question as stated says to assume it's linear.
     
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