Chapter 7.3, Problem 106AYU

Projectile Motion A golfer hits a golf ball with an initial velocity of 100 miles per hour. The range $R$ of the ball as a function of the angle $\theta$ to the horizontal is given by $R\left(\theta \right)=672\sin \left(2\theta \right)$, where $R$ is measured in feet.

a. At what angle $\theta$ should the ball be hit if the golfer wants the ball to travel 450 feet (150 yards)?

b. At what angle $\theta$ should the ball be hit if the golfer wants the ball to travel 540 feet (180 yards)?

c. At what angle $\theta$ should the ball be hit if the golfer wants the ball to travel at least 480 feet (160 yards)?

d. Can the golfer hit the ball 720 feet (240 yards)?

To determine

To find:

a. At what angle $\theta$ should the ball be hit if the golfer wants the ball to travel 450 feet (150 yards)?

Answer to Problem 106AYU

a. $21°$

Explanation of Solution

Given:

A golfer hits a golf ball with an initial velocity of 100 miles per hour. The range $R$ of the ball as a function of the angle $\theta$ to the horizontal is given by $R\left(\theta \right)=672\sin \left(2\theta \right)$, where $R$ is measured in feet.

Calculation:

a. At what angle $\theta$ should the ball be hit if the golfer wants the ball to travel 450 feet (150 yards)?

$R\left(\theta \right)=672\sin \left(2\theta \right)$

$450=672\sin \left(2\theta \right)$

$\sin 2\theta =\frac{450}{672}$

$2\theta ={\sin }^{-1}\left(\frac{450}{672}\right)$

$\theta =\frac{1}{2}{\sin }^{-1}\left(\frac{450}{672}\right)=21°$

To determine

To find:

b. At what angle $\theta$ should the ball be hit if the golfer wants the ball to travel 540 feet (180 yards)?

Answer to Problem 106AYU

b. $26.7°$

Explanation of Solution

Given:

A golfer hits a golf ball with an initial velocity of 100 miles per hour. The range $R$ of the ball as a function of the angle $\theta$ to the horizontal is given by $R\left(\theta \right)=672\sin \left(2\theta \right)$, where $R$ is measured in feet.

Calculation:

b. At what angle $\theta$ should the ball be hit if the golfer wants the ball to travel 540 feet (180 yards)?

$R\left(\theta \right)=672\sin \left(2\theta \right)$

$540=672\sin \left(2\theta \right)$

$\sin 2\theta =\frac{540}{672}$

$2\theta ={\sin }^{-1}\left(\frac{540}{672}\right)$

$\theta =\frac{1}{2}{\sin }^{-1}\left(\frac{540}{672}\right)=26.7°$

To determine

To find:

c. At what angle $\theta$ should the ball be hit if the golfer wants the ball to travel at least 480 feet (160 yards)?

Answer to Problem 106AYU

c. $22.8°$

Explanation of Solution

Given:

A golfer hits a golf ball with an initial velocity of 100 miles per hour. The range $R$ of the ball as a function of the angle $\theta$ to the horizontal is given by $R\left(\theta \right)=672\sin \left(2\theta \right)$, where $R$ is measured in feet.

Calculation:

c. At what angle $\theta$ should the ball be hit if the golfer wants the ball to travel at least 480 feet (160 yards)?

$R\left(\theta \right)=672\sin \left(2\theta \right)$

$480=672\sin \left(2\theta \right)$

$\sin 2\theta =\frac{480}{672}$

$2\theta ={\sin }^{-1}\left(\frac{480}{672}\right)$

$\theta =\frac{1}{2}{\sin }^{-1}\left(\frac{480}{672}\right)=22.8°$

To determine

To find:

d. Can the golfer hit the ball 720 feet (240 yards)?

Answer to Problem 106AYU

d. No

Explanation of Solution

Given:

A golfer hits a golf ball with an initial velocity of 100 miles per hour. The range $R$ of the ball as a function of the angle $\theta$ to the horizontal is given by $R\left(\theta \right)=672\sin \left(2\theta \right)$, where $R$ is measured in feet.

Calculation:

d. Can the golfer hit the ball 720 feet (240 yards)?

$R\left(\theta \right)=672\sin \left(2\theta \right)$

$720=672\sin \left(2\theta \right)$

$\sin 2\theta =\frac{720}{672}$

$2\theta ={\sin }^{-1}\left(\frac{720}{672}\right)$

$\theta =\frac{1}{2}{\sin }^{-1}\left(\frac{720}{672}\right)$

Angle values not more than one, therefore the golfer not possible to hit the ball 720 feet.