Q76P - University Physics with Modern Physics

Question

Ricardo and Jane are standing under a tree in the middle of a pasture. An argument ensues, and they walk away in different directions. Ricardo walks$26.0 m$in a direction$60.0°$west of north. Jane walks$16.0 m$in a direction $30.0°$  south of west. They then stop and turn to face each other.

(a) What is the distance between them? 

(b) In what direction should Ricardo walk to go directly toward Jane?

Step-by-Step Solution

Answer

a) The distance between Ricardo and Jane is $22.71 m$  .

b) The direction in which Ricardo should walk is$37.58°$in the southeast direction.

Step 1: Significance of the law of cosine to evaluate the distance and the direction

The law of cosine mainly relates the triangle’s lengths of its sides to the cosine angle of the triangle.  It is used for identifying the angle of the triangle.

 

Step 2: Identification of the given data

The given data can be listed below as:

 

• Ricardo walks to a distance of $26.0 m$ to$60.0°$west of north.
• Jane walks to a distance of $16.0 m$ to $30.0°$ south of west.

Step 3: Determination of the distance and the direction in which Ricardo should walk

The free body diagram that shows the distance between Jane and Richard and the distance covered by them and the angle between them has been provided below,

a) 

 

From the law of cosines, the distance between Ricardo and Jane is expressed as:

 $C= \sqrt{a^2+b^2-2ab\cos y}$

Here, c is the distance between Ricardo and Jane, a and b are the distance covered by Jane and Richard, which are $26.0 m and 16.0 m$respectively, and $\gamma$ is the angle between the persons, which is $60°$ .

 

Substituting the values in the above expression, we get,

$$\begin{gathered} c=\sqrt{{\left(26.0m\right)}^2+{\left(16.0m\right)}^2-2.26m.16m.\cos 60°} \\ =\sqrt{676m^2+256m^2-416m^2} \\ \approx 22.71 m \end{gathered}$$

Thus, the distance between Ricardo and Jane is 22.71m .

 

b)

 

From the law of sine, the direction in which Ricardo should walk is expressed as:

$\frac{\sin r}{b}=\frac{\sin 60°}{c}$

Here  $r$is the angle of the direction in which Ricardo should walk in the southeast direction.

 

Substituting the values in the above equation, we get,

$$\begin{gathered} \frac{\sin r}{16.0m}=\frac{0.866}{22.71m} \\ \sin r =0.610 \\ r={\sin }^{-1}\left(0.61\right) \\ r\approx 37.58° \end{gathered}$$

Thus, the direction in which Ricardo should walk is$37.58°$ in the southeast direction.