Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition)

Published by Pearson
ISBN 10: 0133942651
ISBN 13: 978-0-13394-265-1

Chapter 3 - Vectors and Coordinates Systems - Exercises and Problems: 40

Answer

We should walk 385 paces at an angle of $65.4^{\circ}$ north of west.

Work Step by Step

Let $T$ be the position of the treasure from the origin: $T = 100\hat{i}+500\hat{j}$ Let $R$ be the displacement vector along the yellow brick road: $R = (300~sin(\theta))\hat{i}+(300~cos(\theta))\hat{j}$ $R = (300~sin(60^{\circ}))\hat{i}+(300~cos(60^{\circ}))\hat{j}$ $R = 260\hat{i}+150\hat{j}$ Let $d$ be the vector we need to walk to get to the treasure: $R+d = T$ $d = T-R$ $d = (100\hat{i}+500\hat{j})-(260\hat{i}+150\hat{j})$ $d = -160\hat{i}+350\hat{j}$ We can find the distance we need to go: $d = \sqrt{(-160)^2+(350)^2}$ $d = 385~paces$ We then find the angle $\theta$ north of west: $tan(\theta) = \frac{160}{350}$ $\theta = arctan(\frac{160}{350})$ $\theta = 24.6^{\circ}$ We should walk 385 paces at an angle of $24.6^{\circ}$ north of west.
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