The Addition of Velocities
How Special Relativity and the Constant Speed Affect Measurement
Jan 29, 2008
There are very few people (not counting physicists) in the world who would dispute the basic premise behind the law of the addition of velocities. It’s a simple enough principle that one generally learns it very early in school.
The Simple Law of the Addition of Velocities
In short, the law is this:
When two objects are traveling independent of one another, their speed relative to each other can be determined by adding their velocities, either positive or negative.
In layman’s terms: If two jet planes are traveling toward each other in a high-stakes game of chicken at a speed of 500 km/h each, one can find their speed relative to each other by adding their speeds together. Thus, in this instance, the planes are heading toward a collision at a combined speed of 1000 km/h. Pretty basic stuff. That is, until one adds Einstein’s theory of Special Relativity into the mix.
Einstein based his thoughts regarding this principle on his newfound theories oftime and space dilation, which create a logical impossibility when one desires to travel at or above the speed of light. This being considered, Einstein realized that this was also the case in terms of the addition of velocities.
The Relativistic Law of the Addition of Velocities
For example, take the previous analogy involving planes, and turn them into spaceships, each traveling at two-thirds the speed of light (about 200,000 km/s). According to the law of the addition of velocities, if you happened to be standing in the cockpit of one of these swiftly-moving crafts, you would measure the other ship approaching you at about 400,000 km/s – far greater than the speed of light. Now, according to Einstein’s theory, this was a logical impossibility, as nothing can even be observed to travel faster than light.
In fact, Einstein realized, this phenomenon required an entirely new mathematical equation in order to properly define. Where the old equation was simply A + B = S (where “A” is the first object’s speed, “B” is the second object’s speed, and “S” is their observed combined speed), the new equation, which took into account Relativity, looked more like this:
S = (A + B)/(1 + AB/c ²)
When one factors the previous example into this new equation (where “A” and “B” still equal the speeds of the two objects, “c” is the constant speed of light and “S” is the observed combined speed), one finds that, low and behold, no longer does the combined speed exceed the speed of light. According to Special relativity, someone standing in the cockpit of a spaceship traveling toward another spaceship, each at a speed of 200,000 km/s, would view the other ship approaching at a mere 276,923 km/s; well under that cosmic speed limit created by light.
While this may on the surface sound like mere conjectures (as, admittedly, does much of theoretical physics) it is important to realize that this theory was not simply a product of Einstein’s prodigious brain. It did not exist in a vacuum. Einstein gave credit for the experimental verification of this new theory of the addition of velocities to the great French experimenter Armand Hippolyte Fizeau, who in the 19th century created an experiment which measured the speed of light relative to water moving through a tube (it’s surely far more complicated than it sounds). The results of Fizeau’s experiment’s were mysterious at the time, but Einstein’s new method of adding velocities put these mysteries to rest, while at the same time showing skeptics that there was real, physical evidence for the theories he was espousing.
The Addition of Velocities and You
While the relativistic equation for the addition of velocities does hold true for all speeds and at all times, it should be noted that at any speed considerably less than the speed of light, it really makes very little difference. The equation is constructed in such a way that when any common speeds are inserted, one finds very, very little discrepancy between the equation and real-world measurements. For example, if two cars are driving toward each other at 60 km/h each, the equation spits out the fact that in reality, they will be observed to only be traveling at a combined speed of 119.999995 km/h, which is immeasurably close to what would normally be assumed. So have no fear, this new law really has very little affect on anyone in this world.
It is just interesting, to say the least.
References:
Einstein, A. (1905). On the Electrodynamics of Moving Bodies. Annalen der Physik .
Einstein, A. (1961). Relativity: The Special and the General Theory - A clear Explanation that Anyone can Understand. New York, NY: Random House.
Gardner, M. (1962). Relativity Simply Explained. Mineola, NY: Dover Publications, Inc.
Davies, P. (1995). About Time - Einstein's Unfinished Revolution. New York: Simon & Schuster.
