14.2: Bungee Revisited

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In the previous section, we modeled the motion of a bungee jumper taking into account gravity, air resistance, and the spring force of the bungee cord. But we ignored the weight of the cord.

It’s tempting to say that the cord has no effect because it falls along with the jumper, but that intuition is incorrect. As the cord falls, it transfers energy to the jumper.

At https://greenteapress.com/matlab/bungee, you’ll find a paper by Heck, Uylings, and Kedzierska titled “Understanding the physics of bungee jumping”; it explains this phenomenon and derives the acceleration of the jumper, \(a\), as a function of position, \(y\), and velocity, \(v\): \[a = g + \frac{\mu v^2/2}{\mu(L+y) + 2L}\notag\] where \(g\) is acceleration due to gravity, \(L\) is the length of the cord, and \(\mu\) is the ratio of the mass of the cord, \(m\), to the mass of the jumper, \(M\).

If you don’t believe that their model is correct, this video might convince you: https://greenteapress.com/matlab/chain.

Exercise 14.2

Modify your solution to the previous problem to model this effect. How does the behavior of the system change as we vary the mass of the cord? When the mass of the cord equals the mass of the jumper, what is the net effect on the lowest point in the jump?