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Science / Geography

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Losing Your Bearings

1. Begin journey at 40 degrees N by 77 degrees W.

2. Travel 4200 miles with a compass bearing of 130 degrees.

3. Travel 2800 miles with a compass bearing of 180 degrees.

4. Travel 5250 miles with a compass bearing of 90 degrees.

5. Travel 1750 miles with a compass bearing of 0 degrees.

6. Where is your location using latitude and longitude coordinates?

1 degree latitude = 70 miles


I know how to do this, however, your 1 degree latitude is just wrong. It could be that at some latitude, say the starting point, however, as soon as you go N or S it changes. Steps 3, and 5 is a considerable change in latitude, so the 1 degree = fixed value is wrong.
As one goes N or S, the conversion of miles to latitude difference changes. So even in step1, and step 4, it has to be considered. In step 4, you are not at the starting point anymore.

So how does one work this? The easy way is to plot it on a globe. The wrong way to work it is to plot it on a mercator projection (flat map).
The complicated way to solve it is to use spherical trignometry, which I doubt you have been introduced to.
http://star-www.st-and.ac.uk/~fv/webnotes/chapter2.htm
As you can see, the problem is that as the problem states the travel as straight paths, the real path traveled in each step was a curve on the surface of the Earth, not a straight line. Any attempt to measure it as a straight line is wrong. When the distances are large, as in this problem, the error is very large.
Plot this on a globe.


A reminder: Due to the curvature, step one 4200miles at a compass bearing 130 will make a spiral upwards as it is drawn on a sphere. You will have to be careful drawing the line segments to make it a constant angle with Longitude lines.
Wouldn't life be simple if you could use a flat map and ignore the fact that it is totally a wrong representation of Earth's surface?

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