Chapter 13, Problem 21P

An object can be projected upward at a specified velocity. If it is subject to linear drag, its altitude as a function of time can be computed as

$z=z_0+\frac{m}{c}\left(v_0+\frac{mg}{c}\right)\left(1-e^{-\left(c/m\right)t}\right)-\frac{mg}{c}t$

where $z=\text{altitude}$(m) above the earth's surface $\left(\text{defined as }z=0\right)$, $z_0=\text{the}$ initial altitude (m), $m=\text{mass}$(kg), $c=\text{a}$ linear drag coefficient (kg/s), $v_0=\text{initial}$ velocity (m/s), and $t=\text{time}$(s). Note that for this formulation, positive velocity is considered to be in the upward direction. Given the following parameter values: $g=9.81{\text{m/s}}^2$, $z_0=100$m, $v_0=55$m/s, $m=80$kg, and $c=15$kg/s, the equation can be used to calculate the jumper's altitude. Determine the time and altitude of the peak elevation (a) graphically, (b) analytically, and (c) with the golden-section search until the approximate error falls below ${\epsilon }_s=1\%$ with initial guesses of $t_l=0\text{and}{t}_u=10$ s.