Bungee jumping and Conservation of energy

[solarcat]

Homework Statement

A person is bungee jumping from the top of a cliff with height H. The un-stretched length of the bungee rope is L. The person comes to a stop just before hitting the ground. The length of the cord is equal to H(amax-g)/(amax+g), where amax is the maximum acceleration upward right before the person hits the ground. Show this equation is physically reasonable by evaluating it when amax = g and amax = infinity.

Homework Equations

Conservation of energy

The Attempt at a Solution

amax = g
L = H(g-g)/(g+g) = 0
This means that the cord starts stretching right as the person jumps.
amax = infinity
L = (infinity -g)/(infinity+g)*H = H
If the length of the cord is equal to H, the cord never starts stretching.
If the cord never starts stretching, then the jumper is always in free-fall, and the acceleration is equal to g. Then shouldn't the results be the other way around?

 

[haruspex]

The length of the cord is equal to H(amax-g)/(amax+g),

It's not clear which length that refers to, but you seem to have interpreted it correctly.

This means that the cord starts stretching right as the person jumps.

Yes, but why should that correspond to a_max=g? I don't think it is at all obvious, but in order to satisfy the question there needs to be some justification.

evaluating it when ... amax = infinity.

Bad question. It should say as a_max tends to infinity, and likewise you need to use limits in the answer. You cannot validly write:

L = (infinity -g)/(infinity+g)*H = H

If the cord never starts stretching, then the jumper is always in free-fall

You are given that the jumper does come to a halt just before hitting the ground, so it cannot be all free fall.

 

[solarcat]

It should say as amax tends to infinity, and likewise you need to use limits in the answer. You cannot validly write: L = (infinity -g)/(infinity+g)*H = H

OK, fair enough. But I have also derived the equation amax = k/m(H-L) - g. This is because when the cord starts stretching, the net force on the jumper ma = Ft-mg. a is maximized when Ft (tension) is greatest, which would be when the cord is stretched the most, which would be at the bottom. Then
m*amax = k(H-L) - mg
amax = (k/m)(H-L) - mg
(infinity + mg) (m/k) = H-L
Smaller values of L result in larger values of amax, and the length can't be negative... But I'm confused, because also, the elastic potential energy at the bottom should equal the initial potential energy, mgH.
So mgH = 1/2 k (H-L)^2
mgH/(H-L) = 1/2 k (H-L)
2mgH(H-L) = k (H-L)
2gH/(H-L) = (k/m)(H-L)
From the last equation, amax + mg = (k/m)(H-L)

amax + mg = 2gH/(H-L)
As amax gets larger and larger, the denominator should get smaller, so H-L = 0 --> H = L

And I'm still not sure why L = H(g-g)/(g+g) = 0

 

[haruspex]

(infinity + mg) (m/k) = H-L

This illustrates the need to treat it as a limits problem. That equation tells you that as amax tends to infinity k tends to infinity, but it does not tell you the relationship between those two trends, so it does not indicate what happens to H-L.

amax + mg = 2gH/(H-L)

That can be turned into the equation you were given. By eliminating k it does allow you to see what happens to H-L.

And I'm still not sure why L = H(g-g)/(g+g) = 0

I guess you mean you do not see a simple reason why a_max=g should correspond to L=0. Maybe you can reason that L=0 means this will be SHM across the entire height H. What would that tell you about the relationship between acceleration at the top and acceleration at the bottom?