Use calculus to solve Eq. (1.21) for the case where the initial velocity is (a) positive a...

Use calculus to solve Eq. (1.21) for the case where the initial velocity is (a) positive and (b) negative. (c) Based on your results for (a) and (b), perform the same computation as in Example 1.1 but with an initial velocity of −40 m/s. Compute values of the velocity from t = 0 to 12 s at intervals of 2 s. Note that for this case, the zero velocity occurs at t = 3.470239 s.

Equation 1.21:

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.