Question

While exploring a cave, a spelunker starts at the entrance and moves the following distances in a horizontal plane. She goes $75.0\text{m}$north,$250\text{m}$ east, _{$125\text{m}$}at an angle $\theta =30.0°$north of east, and south. Find her resultant displacement from the cave entrance. Figure P3.21 suggests the situation but is not drawn to scale.

Short answer

$R=358.5\text{m},\theta =2°\text{(South of East)}$

Step-by-step solution

Define the displacement

Displacement is defined as the body moving from its position.If a body moves in two separate directions (x and y), the resulting displacement is:

$S=\sqrt{x^2+y^2}$

Find the displacement in both axes

In x component

$$\begin{gathered} \sum x=250+125c\mathrm{os}\left(30°\right) \\ =358.3\text{m} \end{gathered}$$

In y component

$$\begin{gathered} \sum y=75+125\sin \left(30°\right)-150 \\ =-12.5\text{m} \end{gathered}$$

The resultant magnitude is

$$\begin{gathered} R=\sqrt{\sum x^2+\sum y^2} \\ =\sqrt{{(358.3)}^2+{(-12.5)}^2} \\ =358.5\text{m} \end{gathered}$$

Find the direction

The resultant direction

$$\begin{gathered} \tan \theta =\frac{\sum y}{\sum x} \\ \tan \theta =\frac{-12.5}{358.3} \\ {\tan }^{-1}=\left(\frac{-12.5}{358.3}\right) \\ \theta =-1.99° \\ \approx -2° \end{gathered}$$

Hence, the resultant displacement from the cave entranceis $R=358.5\text{m},\theta =2°\text{(South of East)}$.