latitude / longitude maths question?

Question

hey, i am having a lot of trouble with a question on latitudes/longitudes and distances travelled.

i know the latitude and longitude of 2 points on earth.
apparently, early navigators could only determine their latitude, so they tended to sail to the desired latitude then head east or west.

suppose i do this travelling from plymouth (latitude 50° 47' N, longitude 4° 05' W) to miami beach (latitude 25° 47' N, longitude 80° 17' W)

so i head south from 50° to 25° then head west. i worked this out as being 6050 nautical miles, could somebody tell me if this is correct??

but anyway, the main problem i have is, what if i headed west first from the more northerly point, then turned and headed south, would this be the same distance??????

thinking about it, i'd say its a shorter route, but i dont know how to back this up with calculations. can anybody give me a hand with this??
cheers

Answer 1

Firstly, your number is wrong. The southward traverse is 25 degrees exactly, at 60 miles per degree, so 1500 miles. Secondly, the distance east or west on a rhumb line [constant compass direction] will be the change in longitude multiplied by the cosine of the latitude. Since the cosine is smaller farther north, the distance going westbound first will be shorter -- except that you will run into the coast of North America before you get as far west as you want to go. Finally, nowadays you can sail the great circle distance, which is given by a particularly simple formula: the distance S (as an angle; mulltiply degrees by 60 for nautical miles) is given by:
cos S = sin L1 sin L2 + cos L1 cos L2 cos D, where:
L1 and L2 are the latitudes of the endpoints;
D is the difference in longitude.
A scientific calculator can solve this easily.

Answer 2

Yes, you are correct supposing to travel long. first. Think about doing this near the pole, in might be just a few steps to the desired longitude. to check your calc. of 6050 nm: it would be 25 deg. times 60 miles, plus 55 deg. times 60 miles times cos 25. = 4490 n.miles

Answer 3

definite, you're maximum ideal supposing to commute lengthy. first. imagine about doing this close to the pole, in will be very few steps to the wanted longitude. to envision your calc. of 6050 nm: it would want to be 25 deg. situations 60 miles, plus fifty 5 deg. situations 60 miles situations cos 25. = 4490 n.miles