Chapter 2.3, Problem 33P

A more realistic model (than that in Problem $32$) of a baseball in flight includes the effect of air resistance. In this case, the equation of motion are,

$\frac{dv}{dt}=-rv,\frac{dw}{dt}=-g-rw$

where, $r$ is the coefficient of resistance.

(a) Determine $v\left(t\right)$ and $w\left(t\right)$ in terms of the initial speed $u$ and initial angle of elevation $A$.

(b) Find $x\left(t\right)$ and $y\left(t\right)$ at any time $t$, if $x\left(0\right)=0$ and $y\left(0\right)=h$.

(c) Plotthetrajectory of the ball for $r=\frac{1}{5},u=125,h=3$ and for several values $A$. How do the trajectories differ from those in Problem $32$ with $r=0$?

(d) Assuming that $r=\frac{1}{5}$ and $h=3$, find the minimum initial velocity $u$ and the optimal angle $A$ for which the ball will clear the wall that is $350\text{ft}$ distant and $10\text{ft}$ high. Compare this result with that in Problem $32(\text{f})$.

Problem $32$.