Rather than the nonlinear relationship of Eq. (1.7), you might choose to model the upward...

Rather than the nonlinear relationship of Eq. (1.7), you might choose to model the upward force on the bungee jumper as a linear relationship:

F_U = −c′υ

where c′ = a first-order drag coefficient (kg/s).

(a) Using calculus, obtain the closed-form solution for the case where the jumper is initially at rest (υ = 0 at t = 0).

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(b) Repeat the numerical calculation in Example 1.2 with the same initial condition and parameter values. Use a value of 11.5 kg/s for c′.

Equation 1.7:

F_U = −c_dυ²

Example 1.2:

Numerical Solution to the Bungee Jumper Problem

Problem Statement. Perform the same computation as in Example 1.1 but use Eq. (1.12) to compute velocity with Euler’s method. Employ a step size of 2 s for the calculation.

Solution. At the start of the computation (t₀ = 0), the velocity of the jumper is zero. Using this information and the parameter values from Example 1.1, Eq. (1.12) can be used to compute velocity at t₁ = 2 s:

For the next interval (from t = 2 to 4 s), the computation is repeated, with the result

The calculation is continued in a similar fashion to obtain additional values:

The results are plotted in Fig. 1.4 along with the exact solution. We can see that the numerical method captures the essential features of the exact solution. However, because we have employed straight-line segments to approximate a continuously curving function, there is some discrepancy between the two results. One way to minimize such discrepancies is to use a smaller step size. For example, applying Eq. (1.12) at 1-s intervals results in a smaller error, as the straight-line segments track closer to the true solution. Using hand calculations, the effort associated with using smaller and smaller step sizes would make such numerical solutions impractical. However, with the aid of the computer, large numbers of calculations can be performed easily. Thus, you can accurately model the velocity of the jumper without having to solve the differential equation exactly.

Equation 1.12:

Figure 1.4: Comparison of the numerical and analytical solutions for the bungee jumper problem.

Example 1.1: Analytical Solution to the Bungee Jumper Problem

Problem Statement. A bungee jumper with a mass of 68.1 kg leaps from a stationary hot air balloon. Use Eq. (1.9) to compute velocity for the first 12 s of free fall. Also determine the terminal velocity that will be attained for an infinitely long cord (or alternatively, the jumpmaster is having a particularly bad day!). Use a drag coefficient of 0.25 kg/m.

Solution. Inserting the parameters into Eq. (1.9) yields

which can be used to compute

According to the model, the jumper accelerates rapidly (Fig. 1.2). A velocity of 49.4214 m/s (about 110 mi/hr) is attained after 10 s. Note also that after a sufficiently long time, a constant velocity, called the terminal velocity, of 51.6983 m/s (115.6 mi/hr) is reached. This velocity is constant because, eventually, the force of gravity will be in balance with the air resistance. Thus, the net force is zero and acceleration has ceased.

Equation 1.9:

Figure 1.2: The analytical solution for the bungee jumper problem as computed in Example 1.1. Velocity Increases with time and asymptotically approaches a terminal velocity.