Egg drop experiment with a twist

[trgoostrey]

Hello,
My physics class is doing the classic egg drop experiment, but with a different twist.
We are asked to find the length of the bungee cord needed given a mass and a height to drop it from. The success of the lab is determined by how close we can get to the ground without touching.
We are using a single bungee cord, and not multiple bungees.
This is a common question here, but I haven't been able to determine the equations to use to determine what I need.

I have tried using U1 + K1 = U2 + K2 +Uspring
Where U1 is the initial potential energy (mgh1)
K1 is the initial kinetic energy (K1 = 0)
U2 is the final kinetic energy (mgh2 with h2 being the closest distance off the ground)
K2 is the final kinetic energy (K2 = 0)
And Uspring is the integral of Force with respect to distance F(x) from 0 to the unstretched bungee length - h2

Our F(x) function is 2.823 - 6.696x + 31.64x^2 - 110.9x^3 + 177.3x^4 - 103.3x^5
This is the characteristic that our bungee follows.

Using this, I get the correct value, if the egg wasn't dropped, but rather if the egg was just hanging there.

My question is, what am I doing wrong? and what equations/principles could I use to determine the length of the string that I need?

 

[Zatman]

Are your limits in your integral correct? Think about how you have defined x.

 

[trgoostrey]

Are your limits in your integral correct? Think about how you have defined x.

Should I define it from the unstretched length to the height off the ground? I know that the integral gives me the work done by the bungee, so it would make sense, but I am out of the lab now, and have no way of testing this.

 

[Zatman]

I would imagine you have x defined as the extension of the string, but I don't really know where your equation for F(x) has come from so I can't tell if this is the case or not.

Do you know the modulus of elasticity, $\lambda$, of the string? If yes then you can just use

$E.P.E = \frac{\lambda e^2}{2l}$

where e is the extension. This would be a lot easier and you woudn't need to use an integral.

If you don't know $\lambda$ then assuming your equation for F(x) is correct with x as the extension, you would want to integrate over the entire extension, i.e. x=0 to the distance from where the string first goes taut to the bottom, $h_1 - h_2 - l$, where l is the natural (unstretched) length of the string. Draw a diagram to help you see this. :)

 

[trgoostrey]

Thank you. I will try this and see if it works. Unfortunately I don't know the modulus of elasticity, but I will try integrating it :)