# Bungee Jumper Energy equation

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1. Feb 23, 2017

### patrickmoloney

1. The problem statement, all variables and given/known data
A bungee jumper of mass $m$ attaches a light elastic string to his foot, and the other end to railing of a bridge on which he stands. The string has natural length $l_0$ and modulus of elasticity $\lambda$. Write down the energy equation. use conservation of energy to find the maximum distance the jumper descends after stepping off the bridge.

2. Relevant equations
Gravitational potential energy = $mgh$
Elastic Potential energy = $\frac{1}{2}kx^2$
Kinetic energy = $\frac{1}{2}mv^2$

3. The attempt at a solution
I understand the principles of energy conversion during a bungee jump. So during the first interval, stepping of the bridge, the only force acting on the system(bungee jumper) is gravity(free fall). As the bungee jumper falls, the gravitational potential energy is converted into kinetic energy. When he reaches the point where the bungee cord begins to stretch, gravitational potential energy begins to be converted into the elastic potential energy of the cord. Eventually, all of the kinetic energy is also converted into potential energy of the cord, and he comes to rest for a short period of time.

I'm not really sure how to write this in mathematics. I can probably make thing easier by setting up a coordinate system with an origin. If I used the origin at $l_0$ below the bridge since that would 0 some terms in the conservation equation.

I found the relationship between the spring constant and elasticity, which is $$k = \frac{\lambda}{L}$$

The conservation of energy is $$PE_i + KE_i = PE_f + KE_f$$. I'm not really sure how to tie this all together. Do I have to model a differential equation or is it simpler than that?

I apprecitate any help,
Thanks

2. Feb 23, 2017

### PeroK

Putting the origin at $l_0$ below the bridge is a good idea. What does this give you for the initial energy of the system?

Your description is fine and, as you suggest, you just need to translate that into the maths.

3. Feb 23, 2017

### BvU

1. Make a drawing, clearly indicating you choice of coordinate system.
2. Convert $k = \frac{\lambda}{L}$ to that system. Oops, you need k only, so: done.
3. Choose the initial and final points in time for the energy balance -- naturally $t_i = 0$ is the initial point when he/she jumps and $t_f$ is at the lowest point. (the exercise isn't about $t$, but nevertheless.
4. Write the energy balance. What are the kinetic energies (which I guess you call $KE_i$ and $KE_f$ ?

4. Feb 23, 2017

### patrickmoloney

I think I might have made some progress since the post.

Let $h$ be the distance the jumper falls
$PE_i=mgy=mgh$
$PE_f=mgy = mg(0) = 0$

The potential energy of the elastic string is $\frac{1}{2}kx^2$ where $x$ is the amount the string has stretched. At the bottom of the fall the string is stretched by an amount $(h-l_0)$. But the cord doesn't begin to stretch until he falls a distance $l_0$. Therefore the final elastic potential energy is $\frac{1}{2}k(h-l_0)^2 = \frac{\lambda}{2l_0}(h-l_0)^2$

Last edited: Feb 23, 2017
5. Feb 23, 2017

### BvU

What is your coordinate system ? You confuse me with an x and a y -- and perhaps yourself too (if not here then later on in your curriculum)

6. Feb 23, 2017

### patrickmoloney

ok so I've changed it a little bit. The initial position at the top before the jump is a distance $h$ from the bottom when the jumper is at rest momentarily. Should I use a different variable for the length the cord stretches? I currently have it as $x$.

$l_0$ is the unstretched length of the string. Is there an easier way to coordinate this?

7. Feb 23, 2017

### patrickmoloney

the initial kinetic energy is zero and the final kinetic energy zero, since he is at rest both of those times.

is the energy equation $$mgh = \frac{\lambda}{2l_0}(h-l_0)^2$$

Last edited: Feb 23, 2017
8. Feb 23, 2017

### PeroK

How have you defined $h$? Is that a constant or a variable? Don't you think you need a square somewhere?

9. Feb 23, 2017

### patrickmoloney

I defined $h$ to be the distance from the initial position to the final position at the bottom. It's a constant, the only variable of length that changes is the length of the cord, when it extends. I did forget to square the term $(h-l_0)$

10. Feb 23, 2017

### PeroK

Okay. so what next?

11. Feb 23, 2017

### patrickmoloney

I need to find the maximum distance the jumper descends after steeping off the ledge. The maximum distance corresponds to the lowest point of the fall where the velocity is 0 (momentarily).

When he is standing on the top, he is at rest. Therefore his kinetic energy is zero.
When he is at the bottom, he is also at rest so the kinetic energy is also 0 (since the velocity is zero.)

That being said, we need to be careful because there is also potential energy in the string as it stretches.

Do I need to solve the differential equation $$mgh=\frac{\lambda}{2l_0}(h-l_0)^2$$

12. Feb 23, 2017

### PeroK

Solving the equation sounds like a good idea!

13. Feb 23, 2017

### patrickmoloney

\begin{align*} mgh &= \frac{\lambda}{2l_0}(h-l_0)^2 \\ mgh &= \frac{\lambda}{2l_0}(h^2-2hl_0+l_0^2) \\ mgh &= \frac{\lambda}{2l_0}h^2 - \lambda h +\frac{\lambda l_0}{2} \\ \end{align*}

Then I found this quadratic equation that needs to be solved using the quadratic formula
\begin{align*} \frac{\lambda}{2l_0}h^2 - h(mg+\lambda) + \frac{\lambda l_0}{2} = 0 \end{align*}

Is this correct so far? I ended up with

$$h = \frac{(mg+\lambda)l_0\pm l_0\sqrt{mg(mg+2\lambda)}}{\lambda}$$

This looks a little messy though so I'm not sure it's correct.

14. Feb 23, 2017

### haruspex

Looks right.
It is not differential.

15. Feb 23, 2017

### patrickmoloney

Thanks very much for the help.

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