Chapter 1, Problem 1.73AP

You stand in a flat meadow and observe two cows (Fig. P1.73). Cow A is due north of you and 15.0 m from your position. Clow B is 25.0111 from your position. From your point of view, the angle between cow A and cow H is 20.0°, with cow B appearing to the right of cow A. (a) How far apart are cow A and cow B? (b) Consider the view seen by cow A. According to this cow, what is the angle between you and cow B? (c) Consider the view seen by cow B. According to this cow, what is the angle between you and cow A? Hint: What does the situation look like to a hummingbird hovering above the meadow? (d) Two stars in the sky appear to be 20.0° apart. Star A is 15.0 ly from the Earth, and star B. appearing to the right of star A. is 25.0 ly from the Earth. To an inhabitant of a planet orbiting star A. what is the angle in the sky between star B and our Sun?

Figure P1.73 Your view of two cows in a meadow. Cow A U due north of you. You must rotate your eyes through an angle of 20.0° to look from cow A to cow B.

To determine

The distance between cow $\text{A}$ and the cow $\text{B}$.

Answer to Problem 1.73AP

The distance between cow $\text{A}$ and the cow $\text{B}$ is $12.1\text{m}$.

Explanation of Solution

Given data: The Cow $\text{B}$ is $25.0\text{m}$ apart from the observer. It is at an angle of $20.0°$ from the observer point to the right of cow $\text{A}$. The distance of the cow $\text{A}$ from the observer is $15\text{m}$.

Consider the following figure.

Figure (1)

The expression for the distance between cow $\text{A}$ and the cow $\text{B}$ is,

$AB=\sqrt{O{A}^2+O{B}^2-2\left(OA\right)\left(OB\right)\cos \theta }$

Substitute $15.0\text{m}$ for $OA$, $25.0\text{m}$ for $OB$ and $20°$ for $\theta$ in the above expression.

$\begin{array}{c}AB=\sqrt{{\left(15.0\text{m}\right)}^2+{\left(25.0\text{m}\right)}^2-2\left(15.0\text{m}\right)\left(25.0\text{m}\right)\cos 20°}\\ =12.05\text{m}\\ \approx \text{12}\text{.1}\text{m}\end{array}$

Conclusion:

Therefore the distance between cow $\text{A}$ and the cow $\text{B}$ is $12.1\text{m}$.

To determine

The angle between observer and the cow B.

Answer to Problem 1.73AP

The angle between observer and the cow B is $135°$.

Explanation of Solution

Given data: The Cow $\text{B}$ is $25.0\text{m}$ apart from the observer. It is at an angle of $20.0°$ from the observer point to the right of cow $\text{A}$. The distance of the cow $\text{A}$ from the observer is $15\text{m}$.

From figure (1),

In triangle $OMA$,

$\cos \theta =\frac{OM}{OA}$

Substitute $15\text{m}$ for $OA$ and $20°$ for $\theta$ in above expression.

$\begin{array}{l}\cos 20°=\frac{OM}{15\text{m}}\\ OM=15\cos 20°\text{m}\end{array}$

Thus the value of the $OM$ is $15\cos 20°\text{m}$.

In triangle $AMB$,

$\beta ={\sin }^{-1}\left(\frac{\left(OB-OM\right)}{AB}\right)$

Substitute $15\cos 20°\text{m}$ for $OM$, $25\text{m}$ for $OB$, $15\text{m}$ for $OA$ in above expression.

$\begin{array}{c}\beta ={\sin }^{-1}\left(\frac{\left(25\text{m}-15\cos 20°\text{m}\right)}{12.05\text{m}}\right)\\ =64.8°\end{array}$

The value of the $\beta$ is $64.8°$.

In triangle $AMO$,

$\alpha =180-\left(90+\theta \right)$

Substitute $20°$ for $\theta$ in above expression.

$\begin{array}{c}\alpha =180-\left(90+20°\right)\\ =70°\end{array}$

Thus the value of the $\alpha$ is $70°$.

The expression for the angle between observer and the cow $\text{A}$ is,

${\theta }_{\text{A}}=\alpha +\beta$

Substitute $70°$ for $\alpha$ and $64.8°$ for $\beta$ in above expression.

$\begin{array}{c}{\theta }_{\text{A}}=70°+64.8°\\ =134.8°\\ \approx 135°\end{array}$

Conclusion:

Therefore the angle between observer and the cow B is $135°$.

To determine

The angle between the observer and cow $\text{B}$.

Answer to Problem 1.73AP

The angle between the observer and cow $\text{B}$ is $25.2°$.

Explanation of Solution

Given data: The Cow $\text{B}$ is $25.0\text{m}$ apart from the observer. It is at an angle of $20.0°$ from the observer point to the right of cow $\text{A}$. The distance of the cow $\text{A}$ from the observer is $15\text{m}$.

The expression for the angle between the observer and cow $\text{B}$ is,

${\theta }_{\text{B}}=180-\left(90+\beta \right)$

Substitute $64.8°$ for $\beta$ in above expression.

$\begin{array}{c}{\theta }_{\text{B}}=180°-\left(90°+64.8°\right)\\ =25.2°\end{array}$

Conclusion:

Therefore the angle between the observer and cow $\text{B}$ is $25.2°$.

To determine

The angle between the star $\text{B}$ and the sun.

Answer to Problem 1.73AP

The angle between the star $\text{B}$ and the sun is $135°$.

Explanation of Solution

Given data: The star $\text{B}$ is $25.0\text{ly}$ apart from the observer. The angle of $20.0°$ between the stars from the sky and the distance of the star $\text{A}$ from the earth is $15\text{ly}$.

Consider the following figure.

Figure (2)

The expression for the distance between star $\text{A}$ and the star $\text{B}$ is,

$AB=\sqrt{O{A}^2+O{B}^2-2\left(OA\right)\left(OB\right)\cos \theta }$

Substitute $15.0\text{ly}$ for $OA$, $25.0\text{ly}$ for $OB$ and $20°$ for $\theta$ in the above expression.

$\begin{array}{c}AB=\sqrt{{\left(15.0\text{ly}\right)}^2+{\left(25.0\text{ly}\right)}^2-2\left(15.0\text{ly}\right)\left(25.0\text{ly}\right)\cos 20°}\\ =12.05\text{ly}\end{array}$

Thus the value of $AB$ is $12.05\text{ly}$.

In triangle $OMA$,

$\cos \theta =\frac{OM}{OA}$

Substitute $15\text{ly}$ for $OA$ and $20°$ for $\theta$ in above expression.

$\begin{array}{l}\cos 20°=\frac{OM}{15\text{ly}}\\ OM=15\cos 20°\text{ly}\end{array}$

Thus the value of the $OM$ is $15\cos 20°\text{ly}$.

In triangle $AMB$,

$\beta ={\sin }^{-1}\left(\frac{\left(OB-OM\right)}{AB}\right)$

Substitute $15\cos 20°\text{ly}$ for $OM$, $25\text{ly}$ for $OB$, $15\text{ly}$ for $OA$ in above expression.

$\begin{array}{c}\beta ={\sin }^{-1}\left(\frac{\left(25\text{ly}-15\cos 20°\text{ly}\right)}{12.05\text{ly}}\right)\\ =64.8°\end{array}$

The value of the $\beta$ is $64.8°$.

In triangle $AMO$,

$\alpha =180-\left(90+\theta \right)$

Substitute $20°$ for $\theta$ in above expression.

$\begin{array}{c}\alpha =180-\left(90+20°\right)\\ =70°\end{array}$

Thus the value of the $\alpha$ is $70°$.

The expression for the angle between earth and the star $\text{A}$ is,

${\theta }_{\text{A}}=\alpha +\beta$

Substitute $70°$ for $\alpha$ and $64.8°$ for $\beta$ in above expression.

$\begin{array}{c}{\theta }_{\text{A}}=70°+64.8°\\ =134.8°\\ \approx 135°\end{array}$

Conclusion:

Therefore, the angle between the star $\text{B}$ and the sun is $135°$.