Great Circles, Nautical Miles - May 27-31, 1996
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This is a long one, but I learned something and wanted to share it with you all.When I was in Seattle and there was that really cool earthquake, we were trying to figure out how far it was from the epicenter to my friend Steve's house where I was staying. The best way we could figure it out was by using latitude and longitude, but I didn't know how latitude or longitude was converted into distance easily. Steve, who has a better sailing background than I do, said that one minute of latitude was a nautical mile. So we figured things out from there.
I was curious about how the nautical mile came about, so when I got home I did a little reading in my 1929 Encyclopaedia Britannica. I learned a bunch of neat stuff. For example, a meter is supposed to be one ten-millionth of the distance between the north pole and the south pole. Turns out they were off by a bit, but it's really close considering the technology they had available to them!
The same thing works with the nautical mile. It was originally defined to be the length of one minute of arc of a great circle of the earth. Now, a great circle on a sphere is a circle whose center is the center of the sphere. Basically, it's the biggest circle you could draw on a sphere - maybe that's why they're called "great" circles? And we know that circles can be split into degrees, and also into minutes (and even into seconds, but we won't worry about that right now). So they defined a nautical mile as one minute of arc of this great circle.
As with the meter, measuring the earth back then wasn't as easy as it is now, and it turns out that the earth isn't really a sphere - it's sort of squashed. So a great circle drawn around the equator is going to be bigger than a great circle drawn around the poles. How do you figure out what the nautical mile's really going to be?
A figure of 1.852 kilometers was agreed upon for the nautical mile. Let's figure out how close that is to the right figures for the equatorial great circle and the polar great circle.
The great circle around the equator has a radius of 6378 kilometers. The great circle around the poles has a radius of 6357 kilometers. Figure out what the nautical mile would be for each of these circles (remember, it's the length of one minute of arc of the great circle), and tell me if you think 1.852 kilometers is a good compromise.
Annie says: Solutions
I think this is a cool problem - it's not that tough, really, but it's interesting the way all of the distances and measurements actually come from somewhere, and aren't just random lengths picked for no good reason. Here is a something from a Web page (which you can find at http://www.essex1.com/people/speer/metric.html) I found after reading my encyclopaedia (a Web page and a 1929 encyclopaedia on the same night!).
"Used in nearly every country in the world the International System was devised by French scientists in the late 18th century to replace the chaotic collection of units then in use. The goal of this effort was to produce a system that did not rely on a miscellany of separate standards, and used the decimal system rather than fractions.Below are highlighted solutions. All the names of those who submitted correct solutions and most of the solutions are also available.To obtain a standard of length a quadrant of the earth (one-fourth of a circumference) was surveyed (actually only in part) along the meridian that passes through Paris. The distance from the equator to the north pole along this meridian was divided into 10-million parts and the result was called the meter. Platinum-iridium alloy bars were cast and marked with this length as physical standards of comparison. Modern scientists have redefined the meter in terms of a number of waves of the orange-red light given off by the element krypton (using a light source similar to a neon light).
The nautical mile used in modern navigation, in relation to which boat speeds and wind velocities are measured (one KNOT is one nautical-mile-per-hour), is defined as one minute of latitude. A degree of latitude therefore is 60 nautical miles. The quadrant of the earth measured by the French, being 90 degrees, measures 90x60 or 5400 nautical miles. Therefore: 5400 nautical miles exactly equal 10-million meters, or 10,000 kilometers.
Aviation maps (WAC), scaled to one-millionth actual size, can be measured with an ordinary "ruler". One millimeter on the map equals one kilometer on the ground. But, curiously, one sixteenth of an inch is also almost exactly equal to one nautical mile on the ground!"
Brent Tworetzky Grade: 9 School: J.P. Taravella High School (Broward County, Fla.) This question is just inserting the radius into a formula. That formula is 1 * (2)(pi)(radius) {one minute * the diameter of the great circle} ------------------- {divided by} (60)(360) {total minutes * total degrees} Thus, by substituting 6378 and 6357, we get approximately 1.855 and 1.849, respectively. 1.852 is a good approximation, as it is the arithmetic mean of our two solutions.
Jackie Benn and Shannon Firth Mt. St. Joseph's Academy, Flourtown, PA Grade 9 We know that the equator's radius is 6378 km and that the radius of the circle that goes through the poles is 6357 km. So the first step here is to find the circumference of each circle, we can do this by plugging in the radii to the formula for circumference which is 2piR. When we do this, we get for the circumferences: Equator - 40,074.156 km, and for the poles - 39,942.209 km. Okay, now the second step to the problem: Remembering what you had said, that one minute of latitude was a nautical mile, and knowing that 1 degree of a circle is 1/360 and that a minute is 1/60 of an hour, we multiplied 1/360 by 1/60 and we got 1/21600. Now to find 1 nautical mile of the equator circle and the circle of the poles, we multiplied the two different circumferences by 1/21600. So this is what it looked like: nautical mile for the equator - 40,074.156 km/21600. This equaled 1.855285 km for a nautical mile traveling across or latitude-wise or perpendicular to the equator. 1 nautical mile for the circle of the poles would look like: 39,942.209km/21600 and this would equal 1.8491763 km. We think 1.852 would be a pretty good guess for a nautical mile because when we averaged the two numbers we got for the two different nautical miles, we got 1.8522307. Very close! Also, knowing that people don't always travel completely in the directions of latitude or longitude is another reason why we averaged the two numbers.
Adam Morley Grade: 8 School: Newport High School There are 360 degrees around the earth and there are 60 minutes in every degree. So there are 21600 minutes around the earth. If the radius of the great circle around the earth at the poles is 6357km, then the diameter is 12714km. Since the way to find the circumference of a circle is d*pi, the circumference around is 39942.2089977km. Then you divide that by 21600 (because that is the distance around in minutes) and you get 1.84917634249km per minute. Then we take the radius 6378km for the equator and double it for the diameter to get 12756 and multiply by pi to get 40074.1558892km around at the equator. Then you divide by 21600 to get the distance around in minutes and get 1.8552849948km per minute. You average those two together and get 1.8522306686km per minute average. That rounds to 1.852, which is what a nm is. Yes, I do think it is close enough.
Brian Gordon Dartmouth '92 One minute is 1/60 * 1/360 of the circumference of each of these circles. So for each, I multiplied 2 * pi* radius, and then divided by 21,600. Here's what I got: equator: 6378 * 2 * pi / 21600 = 1.855 km poles: 6357 * 2 * pi / 21600 = 1.849 km Sounds like 1.852 km is a pretty good compromise to me. --bri
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