Question

A man is lost in the woods. He wanders 3.0km N, then 7.0km E, then 7km S, then 4km W. What is the magnitude and direction of his resultant displacement?

Answer 1

A man is lost in the woods. He wanders $3.0\mathrm{km}$ N, then $7.0\mathrm{km}$ E, then $7\mathrm{km}$ S, then $4\mathrm{km}$ W. What is the magnitude and direction of his resultant displacement?

Solution:

Consider separately vertical movement (north and south) and horizontal movement (east and west):

Vertical: man wandered 3 km north and 7 km south, therefore he wandered:

$$7 k m - 3 k m = 4 k m \text {s o u t h}$$

Horizontal: man wandered 7 km east and 4 km west, therefore he wandered 7 - 4 = 3 km east.

$$7 k m - 4 k m = 3 k m e a s t$$

So man wandered 4 km south and 3 km east.

By the Pythagorean theorem we find the magnitude of resultant displacement:

$$S = \sqrt {3 k m ^ {2} + 4 k m ^ {2}} = 5 k m$$

Direction of resultant displacement: of rectangular triangle we can find the arc tangent of the angle $a$ :

$$a = \tan^ {- 1} \left(\frac {4}{3}\right)$$

Answer: magnitude of resultant displacement: $S = 5$ km

Direction of resultant displacement: $\tan^{-1}\left(\frac{4}{3}\right)$ degrees from east towards south.