First or all, let's pretend the Earth is a perfect sphere (it's not, but it puts the math within my realm of understanding.) Now according to my National Geographic map of the world, the earth's equatorial circumference is 24,902 miles. A person standing on the equator travels all the way around that 24,902 mile-long circuit in a 24 hour period. So, to find the speed of a person at the equator, we simply divide 24,902 miles by 24 hours, which equals 1037.5833333333... miles per hour. So, the equatorial person is traveling at about 1038 mph.
x, y, and z are sides of a triangle formed by points X, Y, and Z. Let's call the circumference of the earth "C." The radius (distance from the center to any point on the surface) of the earth will be known as "r." In a perfect sphere, all radii are equal, so z = r.
The circumference of any circle = 2pi r, so C = 2pi r.
We know C (24,902 miles) so we can find r with the formula r = C/2pi.
Since z = r, z = C/2pi
Now, in order to find the speed traveled at a particular lattitude, we need to find the circumference of the circle formed when a plane parallel to the equator intersects the sphere of the Earth at the lattitude in question. To do that, we must first find the radius of this circle. Let's say that we want the speed that I'm traveling here in Boston. Boston is at point Y on our diagram, so the radius we need is line x.
Triangle XYZ is a right triangle, which means it has one right (90 degree) angle (angle YZX.) The interior angles of all triangles add up to 180 degrees, so if we have one 90 degree angle in triangle XYZ, we know that the sum of the other two angles must equal 90 degrees.
If the angles and the hypotenuse of a right triangle are known, the lengths of the two other sides, or "legs," can be caluclated using sine or cosine. The length of a leg equals the length of the hypotenuse times the sine of the opposire angle or the length of the hypotenuse times the cosine of the adjacent angle (the angle next to the leg other than the right angle itself.)
x = z(cosXYZ)
Since z = r = C/2pi
Then x = (C/2pi)(cosXYZ)
Now, remember, what we're really trying to find is not the radius of the Boston circle, but the circumference. So:
CircBos = 2pi x
CircBos = (2pi)[(C/2pi)(cosXYZ)]
The two 2pi's cancel each other out, and we're left with:
CircBos = C(cosXYZ)
Now that we have the circumference at Boston's lattitude, we can find the speed I'm traveling by dividing the circumference by 24 hours:
Speed = CircBos/24
Speed = [C(cosXYZ)]/24
Speed = (24,902 miles [ cos(latitude) ] ) / 24 hours
Now all we have to do is plug in a latitude. Boston is at about 42 degrees north.
Speed = (24,902 miles [cos(42)]) / 24 hours
Speed = [24,902 miles (0.7431..)] / 24 hours
Speed = 18505.7924... miles / 24 hours
Speed = 771.0746... miles per hour
So, I'm traveling at about 771 mph. That's about 266 mph slower than someone at the equator!
Using the above formula, all we need is a latitude and we can calculate the speed for a any place around the globe. www.indo.com/distance is a useful site for finding your city's latitude. Here are the speeds for a number of different locations, organized north to south:
You can see how the speed increases as one nears the equator in this graph:
Finally, if you don't feel like doing the math (and why wouldn't you? It's fun!) here's a handy speed calculator my friend Scott put together. Just enter your latitude (minutes and seconds are optional, use whole numbers for degrees and minutes) and find out how fast you're going.
Remember, these numbers are not in precise accordance with reality, since the earth is not a perfect sphere and I rounded off latitudes to a whole degree when compiling the chart above. Also, people at the poles are not really motionless as the chart suggests. The earth's tilt of the earth's axis varies slightly from 22.2° and 24.5°. The angle of tilt cycles back and forth between these extremes once roughly every 41,000 years. There is also a "wobble" in the earth's axis of rotation where the axis, while maintaining the same degree of tilt (or range of degree, rather) revolves around an invisible center, completing one full wobble every 25,700 years. Of course, this means the poles are only moving very slowly relative to the equator, but this does introduce another source of error for all speeds of rotation around the globe. Of course, the above is all relative as we are all moving with the earth around the sun, with the solar system around the galaxy, and with the galaxy through the universe.
www.indo.com/distance supplies latitude and longitude for any city around the globe. Will also calculate the distance between any two cities.
Geometry Reference Archives
Online Conversion Convert between miles and kilometers.
Mapquest World Atlas
Earth Rotation Answers to questions about the movement of the earth, including the wobble in the earth's axis.
Earth and Moon Viewer Full color satellite image composites of the earth showing day and night or from above any point on the earth's surface. Also has similar images of the moon.