Question

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1. a hiker begins a trip by first walking 25.0 km southeast from her car. she stops and sets up her tent for the night. on the second day, she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest rangers tower. (a) determine the components of the hikers displacement for each day.

Explanation:

Step1: Define coordinate system

Let east = positive $x$-axis, north = positive $y$-axis.

Step2: Day 1 displacement components

Southeast = 45° south of east.
$x_1 = 25.0\cos(45.0^\circ) = 25.0 \cdot \frac{\sqrt{2}}{2} \approx 17.7$ km
$y_1 = -25.0\sin(45.0^\circ) = -25.0 \cdot \frac{\sqrt{2}}{2} \approx -17.7$ km

Step3: Day 2 displacement components

60.0° north of east:
$x_2 = 40.0\cos(60.0^\circ) = 40.0 \cdot 0.5 = 20.0$ km
$y_2 = 40.0\sin(60.0^\circ) = 40.0 \cdot \frac{\sqrt{3}}{2} \approx 34.6$ km

Answer:

Day 1:
$x$-component: $\boldsymbol{17.7\ \text{km}}$ (east), $y$-component: $\boldsymbol{-17.7\ \text{km}}$ (17.7 km south)
Day 2:
$x$-component: $\boldsymbol{20.0\ \text{km}}$ (east), $y$-component: $\boldsymbol{34.6\ \text{km}}$ (north)