A Closer Look at Bungee Jumping Physics

[Junior Newton]

Homework Statement

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.

Homework Equations

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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??[/B]

 

[certainly]

I can calculate the velocity at any point

Are you sure about this? Do you realize that the acceleration of the man falling down is changing throughout the fall?

 

[AlephNumbers]

which is proportional to the velocity of the jumper.

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.

 

[haruspex]

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.

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.

 

[Junior Newton]

Are you sure about this? Do you realize that the acceleration of the man falling down is changing throughout the fall?

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

 

[Junior Newton]

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.

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?

 

[AlephNumbers]

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.

 

[certainly]

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

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.

 

[haruspex]

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

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).

 

[Junior Newton]

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).

Differential Equation?

 

[haruspex]

Differential Equation?

Yes, as in dv/dt = acceleration = F_net/m. Fill in an expression for F_net. This will include a drag term which will be a function of v.

 

[Junior Newton]

Yes, as in dv/dt = acceleration = F_net/m. Fill in an expression for F_net. This will include a drag term which will be a function of v.

Rearanging GPE=GPE+EPE+KE to get v by itself and then deriving to find the maximum velocity. WOuld that work?

 

[certainly]

Rearanging GPE=GPE+EPE+KE to get v by itself and then deriving to find the maximum velocity. WOuld that work?

No, you are talking about energy in that equation. Draw an FBD of the bungee jumper, trust me it'll help.

 

[Junior Newton]

No, you are talking about energy in that equation. Draw an FBD of the bungee jumper, trust me it'll help.

Not really understanding what you mean?

 

[Junior Newton]

No, you are talking about energy in that equation. Draw an FBD of the bungee jumper, trust me it'll help.

Are the only forces acting upon the jumper gravity and air resistance?

 

[certainly]

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.

Are the only forces acting upon the jumper gravity and air resistance?

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.

 

[Junior Newton]

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.

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?

 

[haruspex]

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?

Then I do not see how you are going to answer this:

Discuss the difficulties involved with the mathematics of this model.

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.

 

[certainly]

I don't want to go into great detail in my assignment

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 :)

 

[haruspex]

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 :)

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

 

[certainly]

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

Well, sometimes introductory books don't really mean linear when they say "which is proportional to the velocity of the jumper." and they just want to convey the fact that the 2 are related. And want the people studying the book to just explore that relation and find out as much as they can about it. My answer was for that :)

 

[AlephNumbers]

Well, sometimes introductory books don't really mean linear when they say "which is proportional to the velocity of the jumper." and they just want to convey the fact that the 2 are related. And want the people studying the book to just explore that relation and find out as much as they can about it. My answer was for that :)

I think that finishing this question would be a good way to find out more about air resistance.
Textbooks are usually precise in their language. And, as Haruspex already pointed out,

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.

 

[certainly]

I think that finishing this question would be a good way to find out more about air resistance.

Absolutely.
But what can we do if the OP is not interested ?

 

[Ashdawne]

hello, I have an assignment that is similar to this one, would you be able to explain how to make up the eqn for #1
and I've read the comments on the air resistance for #5 and I am still confused

 

[haruspex]

hello, I have an assignment that is similar to this one, would you be able to explain how to make up the eqn for #1
and I've read the comments on the air resistance for #5 and I am still confused

You need to post an attempt first.
What equations and principles have you been taught that might be relevant?

 

[Ashdawne]

You need to post an attempt first.
What equations and principles have you been taught that might be relevant?

I have d= F/k +L for my equation but am unsure about it
where kx= mg
and m is the weight
and x is half of that
then have m*g (9.8)
and rearranged to find the value of k

and for #5 there just seemed like a lot of different equations posted and I am not sure what would be the best to use for air resistance for a bungee jumper

 

[haruspex]

I have d= F/k +L for my equation but am unsure about it
where kx= mg
and m is the weight
and x is half of that
then have m*g (9.8)
and rearranged to find the value of k

and for #5 there just seemed like a lot of different equations posted and I am not sure what would be the best to use for air resistance for a bungee jumper

Please define each of your variables (or I cannot tell whether your equations are correct).
x is half of what?