Chapter 10.7, Problem 59AYU

The Green Monster The left field wall at Fenway Park is 310 feet from home plate; the wall itself (affectionately named the Green Monster) is 37 feet high. A batted ball must clear the wall to be a home run. Suppose a ball leaves the bat 3 feet off the ground, at an angle of ${45}^{\circ }$. Use $g=32$ feet per second2 as the acceleration due to gravity and ignore any air resistance.

a. Find parametric equations that model the position of the ball as a function of time. What is the maximum height of the ball if it leaves the bat with a speed of 90 miles per hour? Give your answer in feet.

b. How far is the ball from home plate at its maximum height? Give your answer in feet.

c. If the ball is hit straight down the left field wall, will it clear the Green Monster? If it does, by how much does it clear the wall?

To determine

To find:

a. The parametric equations that model the position of the ball as a function of time.

Answer to Problem 59AYU

a. $x=\left(v_o\frac{1}{\sqrt{2}}\right)t=\frac{\sqrt{2}{v}_{o}t}{2}$; $y=-16t² + \surd 2\frac{v_o}{2}t + 3$

Explanation of Solution

Given:

The left field wall at Fenway Park is 310 feet from home plate; the wall itself (affectionately named the Green Monster) is 37 feet high. A batted ball must clear the wall to be a home run. Suppose a ball leaves the bat 3 feet off the ground, at an angle of $45°$. Use $g=32$ feet per second² as the acceleration due to gravity and ignore any air resistance.

Formula used:

$x=\left(v_o\text{cos}\theta \right)t$, $y=-\frac{1}{2}gt² + \left(v_o\text{sin}\theta \right)t + h$

Calculation:

$\theta =45°$, $g=32ft/sec²$ and $h=3 feet$.

a. $x=\left(v_o\text{cos}\theta \right)t$, $y=-\frac{1}{2}gt² + \left(v_o\text{sin}\theta \right)t + h$

$x=\left(v_o\text{cos}45\right)t$

$x=\left(v_o\frac{1}{\sqrt{2}}\right)t=\frac{\sqrt{2}{v}_{o}t}{2}$

$h=3 feet$

$y=-\frac{1}{2} \&*\#x00A0;32t² + (v_o\sin 45)t + 3$

$y=-16t² + \surd 2\frac{v_o}{2}t + 3$

To determine

To find:

b. What is the maximum height of the ball if it leaves the bat with a speed of 90 miles per hour? Give your answer in feet.

Answer to Problem 59AYU

b. Maximum height $= 139.1feet$

Explanation of Solution

Given:

The left field wall at Fenway Park is 310 feet from home plate; the wall itself (affectionately named the Green Monster) is 37 feet high. A batted ball must clear the wall to be a home run. Suppose a ball leaves the bat 3 feet off the ground, at an angle of $45°$. Use $g=32$ feet per second² as the acceleration due to gravity and ignore any air resistance.

Formula used:

$x=\left(v_o\text{cos}\theta \right)t$, $y=-\frac{1}{2}gt² + \left(v_o\text{sin}\theta \right)t + h$

Calculation:

$\theta =45°$, $g=32ft/sec²$ and $h=3 feet$.

b. Let $v_o=90$ miles/hr $= 132ft/sec$

$y=-16t² + \surd 2\frac{v_o}{2}t + 3$

$y=-16t^2 + \frac{132}{2}\sqrt{2}t + 3=-16t² + 66\sqrt{2}t + 3$

Height is maximum when $t=-b/2a$

$t=-\frac{66\sqrt{2}}{2 \&*\#x00A0;-16} \approx 2.9168$ sec

$y=-16{\left(2.9168\right)}^2 + 66\surd 2\left(2.9168\right) + 3 \approx 139.1$ feet.

To determine

To find:

c. How far is the ball from home plate at its maximum height? Give your answer in feet.

Answer to Problem 59AYU

c. The ball is $272.25$ feet from home plate at its maximum height.

Explanation of Solution

Given:

The left field wall at Fenway Park is 310 feet from home plate; the wall itself (affectionately named the Green Monster) is 37 feet high. A batted ball must clear the wall to be a home run. Suppose a ball leaves the bat 3 feet off the ground, at an angle of $45°$. Use $g=32$ feet per second² as the acceleration due to gravity and ignore any air resistance.

Formula used:

$x=\left(v_o\text{cos}\theta \right)t$, $y=-\frac{1}{2}gt² + \left(v_o\text{sin}\theta \right)t + h$

Calculation:

$\theta =45°$, $g=32ft/sec²$ and $h=3 feet$.

c. Horizontal distance $x=\left(v_o\text{cos}\theta \right)t$.

From b height is maximum when $t=2.9168$ sec.

$x=\frac{132}{2}\surd 2 \&*\#x00A0;2.9168 \approx 272.25$ feet.

The ball is $272.25$ feet from home plate at its maximum height.

To determine

To find:

d. If the ball is hit straight down the left field wall, will it clear the Green Monster? If it does, by how much does it clear the wall?

Answer to Problem 59AYU

d. The ball will clear the Green Monster by $136.5 - 37=99.5$ feet.

Explanation of Solution

Given:

The left field wall at Fenway Park is 310 feet from home plate; the wall itself (affectionately named the Green Monster) is 37 feet high. A batted ball must clear the wall to be a home run. Suppose a ball leaves the bat 3 feet off the ground, at an angle of $45°$. Use $g=32$ feet per second² as the acceleration due to gravity and ignore any air resistance.

Formula used:

$x=\left(v_o\text{cos}\theta \right)t$, $y=-\frac{1}{2}gt² + \left(v_o\text{sin}\theta \right)t + h$

Calculation:

$\theta =45°$, $g=32ft/sec²$ and $h=3 feet$.

d. The ball will reach the left field fence when $x=310$ feet.

$x=\left(v_o\text{cos}\theta \right)t=66\surd 2t$

$\frac{310}{66\sqrt{2}}=t \approx 3.3213$ sec

Thus, it will take about $3.3213$ sec to reach the left field fence.

$y=-16{\left(3.3213\right)}^2 + 66\sqrt{2}\left(3.3213\right) + 3 \approx 136.5$ feet

Field fence height $= 37$ feet,

The ball will clear the Green Monster by $136.5 - 37=99.5$ feet.