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Einsteinian Velocities

I always learned that 1+1 = 2. Let's say you have a friend that's walking one mile per hour away from a designated starting point. Now let's say you're starting at the same point, but walking in an opposite direction from your friend, also at one mile per hour. After one hour, the two of you will be two miles apart from each other. So far, so good.

Let's presume this time that you and your friend are again starting at the same point and heading in opposite directions. But instead of traveling at one mile per hour, let's have you and your friend travel at 186,000 miles per second -- the speed of light. In formulas, the letter "C" is traditionally used to represent the speed of light, so after one second, we'd expect you and your friend to be this far apart after one second: C+C = 2C = 372,000 miles.

But alas, according to the text book in front of me (Exploring the Cosmos by Louis Berman), C+C = C. Even though both parties are traveling away from each other at the speed of light (186,000 miles per second), the two parties are only 186,000 miles (not 372,000 miles) apart after one second. How can this be? My dear reader, I'm just as baffled as you are -- if not more so. According to the text (and are texts ever wrong?), we need another formula when talking about velocities approaching the speed of light. Here it is, whereby A = the velocity of the first party and the B = the velocity of the second party and C = the velocity of light:

Relative Velocity of A+B = A+B divided by 1 + AxB/CxC

So for two objects traveling at C, the speed of light, we plug in the numbers:

Relative Velocity of C+C = C+C divided by 1 + CxC/CxC
Relative Velocity of C+C = 2C divided by 1 + 1/1
Relative Velocity of C+C = 2C divided by 1 + 1
Relative Velocity of C+C = 2C divided by 2 = C, the answer

I give an example used by the text, that of two spaceships approaching each other at speeds of 100,000 miles per second and 150,000 miles per second. Surprisingly, they don't approach each other at 250,000 miles per second -- as one might think. Let's use the formula, with A = 100,000 miles/second, B = 150,000 miles/second, and C=186,300 miles/second, to find out.

Relative Velocity of A+B = A+B divided by 1 + AxB/CxC
Relative Velocity of A+B = 100,000 + 150,000 divided by 1 + 100,000 x 150,000/186,300 x 186,300
Relative Velocity of A+B = 250,000 divided by 1 + 15,000,000,000/34,708,000,000
Relative Velocity of A+B = 250,000 divided by 1 + 0.432177
Relative Velocity of A+B = 250,000 divided by 1.432177
Relative Velocity of A+B = 174,559.43 miles per second

I don't know how to derive this formula, but would love to learn one of these days. It was the Michelson-Morley experiment that alerted astronomers to this mysterious dimension of space and time.

by Bruce McClure

Einstein's Velocity Formula Proven Wrong?

July 2004 Feature: Blue Moon