Question

You are on a treasure hunt and your map says, "walk due west for 52 paces, then walk 30.0° north of west for 42 paces, and finally walk due north for 25 paces." What is the magnitude of the component of your displacement (a) due north and (b) due west?

Answer 1

The magnitude of the component of displacement due north is 46 paces, and the magnitude of the component due west is approximately 88.6 paces, using vector addition and trigonometry.

Step-by-step explanation:

To find the magnitude of the component of your displacement due north and due west, you can use vector addition techniques that break down each paced direction into its northward and westward components.

Firstly, walking due west for 52 paces means you have a displacement of 52 paces purely in the westward direction.Next, walking 30.0° north of west for 42 paces means you have to break this displacement into north and west components. You can use trigonometry: the northward component is 42 paces * sin(30.0°) and the westward component is 42 paces * cos(30.0°).Finally, walking due north for 25 paces adds directly to the northward component.

The total northward displacement, (a), is the sum of the northward components: 25 paces + (42 * sin(30.0°)) paces.
The total westward displacement, (b), is the sum of the westward components: 52 paces + (42 * cos(30.0°)) paces.

After calculating, we get:

(a) Total northward displacement = 25 + (42 * 0.5) = 46 paces(b) Total westward displacement = 52 + (42 * cos(30.0°)) = 52 + (42 * 0.866) ≈ 88.6 paces