Bungee Jumper Physics Lab Question

[flouran]

Homework Statement

Hi,
So I have a final lab in my college physics class. And the lab procedure is as follows:

Make or use a bungee cord by tying ten or eleven #19 latex rubber bands end-to-end. Attach the upper end high enough so that when 200 grams (0.2 kg) is hung from its lower end, it will almost touch the floor. Begin with a weight of about 0.2 N (0.0204082 kg) and measure the extension of the bungee cord as a function of the applied force up to a maximum extension of 1 to 12 meters. Also hang the Super Hero on the bungee cord and measure the resulting extension. The purpose is to predict, given a particular bungee cord, the minimum height above the floor necessary to ensure (or insure :razz:) that the jumper comes within 5-10 centimeters (0.05 to 0.1 meters) of the floor.

Note: The measured mass of the Bungee jumper is 0.25837 kilograms (258.37 grams).
I have attached my data as an Excel file:
View attachment AP Physics Post Lab.xls.
My question is, what formula does k follow (it is most definitely not linear)?

Homework Equations

Thus, once I find k, it should hopefully be easy to compute this minimum height using energy considerations (please let me know if I am somehow wrong):
U_i + K_i = U_f + K_f,
Since the object is released from rest,
mg(h_{min}-{L_0}) = mg(0.05)+0.5k(h_{min}-0.05-L_0)^2,
where h_{min} is the minimum height (the thing I need to calculate), m is the mass of the jumper = 0.25837 kg. L_0 is the initial length of the bungee which I measured to be 0.395 meters.

The Attempt at a Solution

In the attached Excel file I included a graph of the empirical computation of k (y-axis) versus the attached mass in kilograms (x-axis). As you can see, k asymptotically approaches 4 N/m, but is non-linear. I estimate k to be around 7 N/m if the bungee jumper is attached.

 

[nvn]

flouran: I don't think I would bother with computing k, for the main calculations, although k is informative (for information only).

Your empirical data, although extremely coarse, suggests that a better regression would be P(x) = 17.24*x/(1 + 4.786*x - 1.886*x^2), where P(x) = applied force (N) as a function of x, and x = deflection (m). Thus, U1₁ + K₁ = U1₂ + K₂ + U2₂, where U1 = potential energy, and U2 = strain energy. Therefore,

m*g*h1 + 0 = m*g*h3 + 0 + integral[P(x)*dx],

where h1 = jumper initial height above ground, h3 = jumper final height above ground, and the integral is integrated from 0 to h2 - h3, where h2 = height of unstretched bungee cord lower end above ground (the unknown). Unless you can find an analytic solution to the above integral, which I did not attempt, you could solve the problem numerically, by trial and error, if you wish.