Chapter 3.6, Problem 20E

To determine

The relationship between the solutions obtained from the models and the constant solution, $v\left(t\right)={\left(9.8\frac{1}{2}\right)}^{1/r}$. The model of the velocity of a falling body by the initial value problem,

$m\frac{dv}{dt}=mg-bv$, $v(0)=v_0$

under the assumption that the force due to air resistance is $-bv$. However, in certain cases the force due to air resistance behaves more like $-b{v}^r$ where $r$ is some constant that leads to the model,

$m\frac{dv}{dt}=mg-b{v}^r$, $v(0)=v_0$ ... $(1)$

To study the effect of changing the parameter $r$ in equation $(1)$, take $m=1$, $g=9.81,b=2,v_0=0$. Then use the improved Euler’s method subroutine with $h=0.2$ to approximate the solution to $(1)$ on the interval $0\le t\le 5$ for $r=1.0,1.5,2.0$.