|MLA Citation:||Bloomfield, Louis A. "Question 1398"|
How Everything Works 20 May 2018. 20 May 2018 <http://howeverythingworks.org/print1.php?QNum=1398>.
At this point, you might assert that velocities do add and that objects should be able to reach any speed. But that's not the case. The modern, relativistic understanding of the universe says that even at these small speeds, velocities don't quite add. To the stationary observer, the second runner travels at only 9.9999999999999994 mph and the third runner at only 14.9999999999999988 mph. As you can see, when two or more velocities are combined, the final velocity isn't quite as large as the simple sum. What that means is that the velocity you observe in another object is inextricably related to your own motion. This interrelatedness is part of the theory of relativity—that observers who are moving relative to one another will see space and time somewhat differently.
For objects traveling close to the speed of light, the failure of velocity addition becomes quite severe. For example, if one spaceship travels past the earth at half the speed of light and the people in that spaceship watch a second spaceship pass them at half the speed of light in the same direction, then a person on earth will see the second spaceship traveling only four-fifths of the speed of light. As you can see, relativity is making it difficult to reach the speed of light. In fact, it's impossible to reach the speed of light! No matter how you combine velocities, no observer will ever see a massive object reach or exceed the speed of light. The only objects that can reach the speed of light are objects without mass and they can only travel at the speed of light.
So while the counting numbers obey simple addition and go on forever, velocities do not obey simple addition and have a firm limit—the speed of light. The additive counting numbers are an example of a mathematical group that extends infinitely in both directions, but there are many examples of groups that do not extend to infinity. The group that describes relativistic, real-world velocities is one such group. You can visualize another simple limited group—the one associated with walking around the surface of the earth. No matter how much you try, you can't walk more than a certain distance northward. While it seems as though steps northward add, so that 5 steps north plus 5 steps north equals 10 steps north, things aren't quite that simple. Eventually you reach the north pole and start walking south!