Chapter 2.3, Problem 32P

Let $v(t)$ and $w(t)$, respectively, be the horizontal andvertical components of the velocity of a batted (or thrown)baseball. In the absence of air resistance, $v$ and $w$ satisfy the equations

$dv/dt=0,dw/dt=-g$.

Show that

$v=u\cos A$, $w=-gt+u\sin A$,

where $u$ is the initial speed of the ball and $A$ is its initial angle of elevation.

Let $x\left(t\right)$ and $y\left(t\right)$, respectively, be the horizontal and vertical coordinates of the ball at time $t$. If $x\left(0\right)=0$ and $y\left(0\right)=h$, find $x\left(t\right)$ and $y\left(t\right)$ at any time $t$.

Let $g=32{\text{ft/s}}^{\text{2}}$, $u=125{\text{ft/s}}^{\text{2}}$, and $h=3\text{ft}$. Plot the trajectory of the ball for several values of the angle $A$; that is, plot $x\left(t\right)$ and $y\left(t\right)$ parametrically.

Suppose the outfield wall is at a distance $L$ and has height $H$. Find a relation between $u$ and $A$ that must be satisfied if the ball is to clear the wall.

Suppose that L = 350 ft and $H=10\text{ft}$. Using the relation in part (d), find (or estimate from a plot) the range of values of $A$ that corresponds to an initial velocity of $u=110\text{ft/s}$.

For $L=350\text{ft}$ and $H=10\text{ft}$, find the minimum initialvelocity $u$ and the corresponding optimal angle $A$ for which the ball will clear the wall.