Chapter 25, Problem 25P

Use the following differential equations to compute the velocity and position of a soccer ball that is kicked straight up in the air with an initial velocity of 40 m/s:

$\begin{array}{l}\frac{dy}{dt}=v\\ \frac{dv}{dt}=-g-\frac{c_d}{m}v\left|v\right|\end{array}$

where $y=$ upward distance (m), $t=$ time (s), $v=$ upward velocity (m/s), $g=$ gravitational constant $\left(=9.81{\text{ m/s}}^2\right)$, $c_d=$ drag coefficient (kg/m), and $m=$mass (kg). Note that the drag coefficient is related to more fundamental parameters by

$c_d=\frac{1}{2}\rho A{C}_d$

where $\rho =$ air density (kg/m3), $A=$ area (m²), and $C_d=$ the dimensionless drag coefficient. Use the following parameter values for your calculation: $d=22\text{ cm,}$ $m=0.4\text{ kg, }\rho =\text{1}{\text{.3 kg/m}}^3\text{ and }{C}_d=0.52$.