An explorer is caught in a whiteout (in which the snowfall is so thick that the ground cannot be distinguished from the sky) while returning to base camp. He was supposed to travel due north for 4.5 km, but when the snow clears, he discovers that he actually traveled 7.8 km at 45° north of due east. In what direction must he travel (counterclockwise from due east)?

Question

An explorer is caught in a whiteout (in which the snowfall is so thick that the ground cannot be distinguished from the sky) while returning to base camp. He was supposed to travel due north for 4.5 km, but when the snow clears, he discovers that he actually traveled 7.8 km at 45° north of due east. In what direction must he travel (counterclockwise from due east)?

Answer 1

First you know he started from a point (0,0) and wanted to end up 4.5km north of that point. So essentially he needed to be at (0,4.5). Now he traveled NE at 45° from (0,0) with a magnitude of 7.8km. Draw a triangle to help illustrate this now... find his total x displacement and his total y displacement. Use your Cos and Sin laws to solve this.

After that, you can compare that with where his base camp should be, which has a displacement of x=0 and y=4.5. After that all you do is use a tan law to find what degree he needs to travel to get back to camp.

Answer 2

To determine the direction the explorer must travel to return to the base camp, we can use the concept of vector addition.

First, we need to understand the components of the explorer's actual movement. We know that he traveled 7.8 km at 45° north of due east.

Now, let's break down this movement into its northward and eastward components. To find the northward component, we can use trigonometry:

northward component = distance * sine(angle)

northward component = 7.8 km * sine(45°)

northward component = 7.8 km * 0.707 (approximately)

northward component = 5.51 km (approximately)

To find the eastward component, we can use the same logic but replace sine with cosine:

eastward component = distance * cosine(angle)

eastward component = 7.8 km * cosine(45°)

eastward component = 7.8 km * 0.707 (approximately)

eastward component = 5.51 km (approximately)

Now, we have the northward and eastward components of the actual movement. To return to the base camp, the explorer needs to cancel out these components by traveling in the opposite direction for the same distances.

Since the intended direction was due north, to cancel out the northward component (5.51 km), the explorer needs to travel due south for the same distance (5.51 km).

Similarly, since the explorer traveled at 45° north of due east, to cancel out the eastward component (5.51 km), the explorer needs to travel 45° counter-clockwise from due east, which is west-northwest (WNW).

In conclusion, the explorer needs to travel due south for 5.51 km and then continue in a direction of 45° counter-clockwise from due east, which is the west-northwest direction, to return to the base camp.