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Old 06-21-2014
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How many miles?

Hi Folks,

I'm wondering if any of you expert navigators out there could solve a simple geometry problem for us...

A person sets sail from one side of the Earth to the exact opposite along the equator, aiming for a very small port, but they are off course by just one degree the entire voyage. By how many miles would they miss their precise destination?

Thank you!
Bob
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Old 06-22-2014
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Re: How many miles?

If I remember my spherical trig correctly, one could use the spherical law of sines.
The angle between the intended course and actual end point will be 90 degrees or pi/2 radians
Spherical law of sines says A/a =B/b=C/c where A, B, C are arc lengths of a triangle and a, b, c are angles in radians.
So, a is 1 degree or .0174 radians (from 2pi radians in a circle) so with a tiny bit of algebra you get:

A=2Ba/pi (B is half the circumference of earth or12000 miles)
A= 2*12000*.0174/3.14=133 miles.
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Old 06-22-2014
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Re: How many miles?

How close would you be simply using the formula for arc length in planar geometry?
s=r*theta where s is arc length, r is distance(12,000 miles) and Theta is in radians (.0174) gives 12000*.0174 = 208 miles, a diff of about 67 miles.
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Old 06-22-2014
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Re: How many miles?

Do NOT trust my answer, this is off the top of my head.
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Old 06-22-2014
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Re: How many miles?

Frogwatch,

THANKS for taking the time! Great answer.

(I wish I could spin off brilliance like that from the top of my head!)

Bob
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Old 06-22-2014
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Re: How many miles?

No brilliance, simply familiarity. I use this stuff for a living so I remember it.
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Old 06-22-2014
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Re: How many miles?

And for us simpletons, you could borrow a TLAR Navigation formula that I use in aviation every day. One degree over one mile yields 100 foot difference. So, miles traveled times 100 divided by 6000 (6000 to convert to miles'ish).

12000*100/6000=200 miles off course

For those unfamiliar with TLAR Navigation... That Looks About Right
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Old 06-22-2014
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Re: How many miles?

Thanks, Panpan!
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