Question

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)?

Short answer

Part a. The ball should be hit at the angle $\theta =21°$ if the golfer wants the ball to travel 450 feet (150 yards).

Part b. The ball should be hit at the angle $\theta =27°$if the golfer wants the ball to travel 540 feet (180 yards).

Part c. The ball should be hit at the angle $\theta =23°$ if the golfer wants the ball to travel at least 480 feet (160 yards).

Part d. Yes, the golfer can hit the ball720feet at$\theta >45°.$

Step-by-step solution

Part (a) Step 1. Given Information

The range R of the ball as a function is $R\left(\theta \right)=672\sin \left(2\theta \right)$, where Ris measured in feet and the initial velocity is $100$miles per hour.

We have to find the angle of the ball if the golfer wants the ball to travel450feet(150yards).

Part (a) Step 2. Finding the angle

By using the function of the ball we get,

$$\begin{gathered} R\left(\theta \right)=672\sin \left(2\theta \right) \\ 450=672\sin \left(2\theta \right) \\ \sin \left(2\theta \right)=\frac{450}{672} \\ \sin \left(2\theta \right)\approx 0.67 \\ 2\theta ={\sin }^{-1}\left(0.67\right) \\ 2\theta =42 \\ \theta =21° \end{gathered}$$

Part (b) Step 1. Finding the angle 

By using the function of the ball we get,

$$\begin{gathered} R\left(\theta \right)=672\sin \left(2\theta \right) \\ 540=672\sin \left(2\theta \right) \\ \sin \left(2\theta \right)=\frac{540}{672} \\ \sin \left(2\theta \right)\approx 0.80 \\ 2\theta ={\sin }^{-1}\left(0.80\right) \\ 2\theta =53.47 \\ \theta \approx 27° \end{gathered}$$

Part (c) Step 1. Finding the angle

By using the function of the ball we get,

$$\begin{gathered} R\left(\theta \right)\ge 480 \\ 672\sin \left(2\theta \right)\ge 480 \\ \sin \left(2\theta \right)\ge \frac{480}{672}\approx 0.71 \\ 2\theta \ge {\sin }^{-1}\left(0.71\right) \\ 2\theta =45.5 \\ \theta \approx 23° \end{gathered}$$

Part (d) Step 1. Finding the golfer can hit the ball 720 feet

To find that the golfer can hit the ball 720 feet we will use the function of a ball.

$$\begin{gathered} R\left(\theta \right)=672\sin \left(2\theta \right) \\ 720=672\sin \left(2\theta \right) \\ \sin \left(2\theta \right)=\frac{720}{672} \\ \sin \left(2\theta \right)\approx 1.07 \\ 2\theta ={\sin }^{-1}\left(1.07\right)\left[{\sin }^{-1}\left(1.07\right)>90°\right] \\ 2\theta >90° \\ \theta >45° \end{gathered}$$

Thus, the golfer can hit the ball at$\theta >45°.$