The Bungee Jumper: A Comparison of Predicted and Measured Values

T he greater-than-g acceleration of a bungee jumper discussed in a previous article in this journal by Kagan and Kott 1 led to many lively discussions among Dutch physics teachers. These inspired me to look for an inexpensive experimental setup, suitable for use in a high school physics class, that can be used to confirm that indeed the acceleration is greater than g. In this paper I describe an exercise to compare the predicted and the measured graphs for the displacement y(t) of the jumper and the force F b (t) exerted by the bungee on the bridge to which it is fastened. In my apparatus, the "bungee" consists of a light chain and the "jumper" is a small piece of brass. Data collection and the calculation of predicted values were carried out using Coach. Theory A real bungee jumper (a person of mass M) is tied to an elastic cord (bungee) of mass m and length L. The other end of the rope is tied to a fixed pointperhaps on a bridge. Starting from rest on the bridge, the jumper accelerates downward to speed v after falling a distance y. Using conservation of mechanical energy, Kagan and Kott show that while the cord is still slack: And differentiating: A solution for y(t) may be obtained in terms of elliptic integrals. 3 This is inappropriate for the introductory-level physics course, however. Numerical integration using, for example, a spreadsheet is certainly preferable. Commenting about the counterintuitive conclusion that a > g, Kagan and Kott write: "Some insight can be gleaned from thinking about the fact that the portion of cord at the bottom of the loop is coming to rest. Since it was moving downward, an upward force is required on this part of the system." This remark put me on the track to find a simple algorithm for determining numerically the displacement y(t) of the jumper and also the force F b (t) exerted by the bungee on the bridge. My analysis is limited to the time during which the bungee remains slack. Algorithm The method consists of N successive loops in which we continually calculate new values for t, y, and F b . We start with the values of the various constants employed in our experimental setup. The length and mass of the chain (bungee) are L = 1.51 m and m = 0.290 kg. The mass of the brass cylinder (jumper) is 0.057 kg. The linear density of the bungee is = m/L. We divide the length L into N (N = 10,000 for the calculations described here) pieces of length dy = L/N, each having mass dy. After the values of the constants have been set, the iterative process may be initiated