Question

In an attempt to escape his island, Gilligan builds a raft and sets to sea. The wind shifts a great deal during the day, and he is blown along the following straight lines:
2.5 km 45° north of west; then
4.70 km 60° south of east; then
5.1 km straight east; then
7.2 km 55° south of west; and finally
2.8 km 10° north of east.

Answer 1

Answer on Question #71048, Physics / Mechanics | Relativity

Question. In an attempt to escape his island, Gilligan builds a raft and sets to sea. The wind shifts a great deal during the day, and he is blown along the following straight lines:

2.5 km 45° north of west; then

4.70 km 60° south of east; then

5.1 km straight east; then

7.2 km 55° south of west; and finally

2.8 km 10° north of east.

What is his final position relative to the island?

Solution.

His journey consists of five vectors. Let's convert the angles first to be relative to $+x - axis$. We get

A: 2.5 km 135°

B: 4.70 km 300°

C: 5.1 km 0°

D: 7.2 km 235°

E: 2.8 km 10°

So

$$\vec{A} = 2.5 \cdot \cos 135{}^\circ \vec{i} + 2.5 \cdot \sin 135{}^\circ \vec{j} = -1.77\vec{i} + 1.77\vec{j}$$ $$\vec{B} = 4.7 \cdot \cos 300{}^\circ \vec{i} + 4.7 \cdot \sin 300{}^\circ \vec{j} = 2.35\vec{i} - 4.07\vec{j}$$ $$\vec{C} = 5.1 \cdot \cos 0{}^\circ \vec{i} + 5.1 \cdot \sin 0{}^\circ \vec{j} = 5.1\vec{i}$$ $$\vec{D} = 7.2 \cdot \cos 235{}^\circ \vec{i} + 7.2 \cdot \sin 235{}^\circ \vec{j} = -4.13\vec{i} - 5.90\vec{j}$$ $$\vec{E} = 2.8 \cdot \cos 10{}^\circ \vec{i} + 2.8 \cdot \sin 10{}^\circ \vec{j} = 2.76\vec{i} + 0.48\vec{j}.$$

Finally

$$\vec{S} = \vec{A} + \vec{B} + \vec{C} + \vec{D} + \vec{E} = 4.31\vec{i} - 7.72\vec{j}$$ $$|\vec{S}| = \sqrt{4.31^2 + 7.72^2} = 8.84 \text{ km}.$$ $$\tg \alpha = \frac{y}{x} = \frac{-7.72}{4.31} = -1.791 \quad \rightarrow \quad \alpha = -60.8{}^\circ \text{ (or } 60.8{}^\circ \text{ south of east)}$$

Answer. $|\vec{S}| = 8.84 \text{ km}; 60.8{}^\circ \text{ south of east}$

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