# Bungee jumping and Conservation of energy

• solarcat
In summary, the bungee jumper comes to a stop just before hitting the ground and 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.
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?

solarcat said:
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.
solarcat said:
This means that the cord starts stretching right as the person jumps.
Yes, but why should that correspond to amax=g? I don't think it is at all obvious, but in order to satisfy the question there needs to be some justification.
solarcat said:
evaluating it when ... amax = infinity.
Bad question. It should say as amax tends to infinity, and likewise you need to use limits in the answer. You cannot validly write:
solarcat said:
L = (infinity -g)/(infinity+g)*H = H

solarcat said:
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.

haruspex said:
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

Last edited:
solarcat said:
(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.
solarcat said:
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.
solarcat said:
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 amax=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?

## What is bungee jumping?

Bungee jumping is an adventure sport in which a person jumps from a tall structure, such as a bridge or building, while connected to a long elastic cord. The cord stretches and then recoils, allowing the person to bounce up and down until they come to a stop.

## How does bungee jumping relate to the conservation of energy?

Bungee jumping is a perfect example of the law of conservation of energy, which states that energy cannot be created or destroyed, but can only be transferred or transformed. In bungee jumping, the potential energy of the jumper at the top of the structure is converted into kinetic energy as they fall, and then back into potential energy as they bounce back up.

## What safety measures are in place for bungee jumping?

Bungee jumping companies have strict safety protocols in place to ensure the safety of their customers. This includes regular inspections of equipment, weight and height restrictions, and trained staff to assist with the jump. Additionally, the length of the cord is carefully calculated to ensure that the jumper does not hit the ground or any other objects during the fall.

## Is bungee jumping harmful to the environment?

In general, bungee jumping does not have a significant impact on the environment. However, it is important for operators to follow responsible practices, such as using biodegradable materials for cords and avoiding jumping in sensitive ecosystems. Overall, the impact of bungee jumping on the environment is minimal.

## Are there any risks associated with bungee jumping and conservation of energy?

Like any adventure sport, there are inherent risks associated with bungee jumping. These risks can be minimized by following safety protocols and using proper equipment. As for conservation of energy, the main risk is for the jumper to experience a fall that is too long or too short due to miscalculations of the cord length. However, this risk is also minimized through thorough planning and safety measures.

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