Chapter 28, Problem 50P

The differential equation for the velocity of a bungee jumper is different depending on whether the jumper has fallen to a distance where the cord is fully extended and begins to stretch. Thus, if the distance fallen is less than the cord length, the jumper is only subject to gravitational and drag forces. Once the cord begins to stretch, the spring and dampening forces of the cord must also be included. These two conditions can be expressed by the following equations:

$\begin{array}{l}\frac{dv}{dt}=g-\text{sign}\left(v\right)\frac{c_d}{m}{v}^{2}x\le L\\ \frac{dv}{dt}=g-\text{sign}\left(v\right)\frac{c_d}{m}{v}^2-\frac{k}{m}\left(x-L\right)-\frac{\gamma }{m}vx>L\end{array}$

where $v=\text{ velocity (m/s)}$, $t=$ time (s), $g=$ gravitational constant $\left(=9.81{\text{ m/s}}^2\right)$, $\text{sign}\left(x\right)=$ function that returns –1, 0, and 1 for negative, zero, and positive x, respectively, $c_d=$ second-order drag coefficient $\left(\text{kg/m}\right)$, $m=$ mass $\left(\text{kg}\right)$, $k=$ cord spring constant $\left(\text{N/m}\right)$, $\gamma =$ cord dampening coefficient $\left(\text{N}\cdot \text{s/m}\right)$, and $L=$ cord length $\left(\text{m}\right)$. Determine the position and velocity of the jumper given the following parameters: $L=30\text{ m, }m=68.1\text{ kg}$, $c_d=0.25\text{ kg/m}$, $k=40\text{ N/m, and }\gamma =8\text{ kg/s}$. Perform the computation from $t=0$ to 50 s and assume that the initial conditions are $x\left(0\right)=v\left(0\right)=0$.