davidlones365 said:
...wait, I still fail to see the need of imposing time dilation and contraction if the formula used to calculate velocities at near light speeds is different. That's the source of my dilemma. I'm going in circles. The only reason explained for things such as time dilation to occur is to in effect "warp" the perception of the "speed" of the light particles to match the "constant" speed of c.
Your best bet will be to retrace Einstein's logic from the start. Time dilation, length contraction, the velocity addition formula, and even E=mc^{2} all follow from a more fundamental set of equations, the Lorentz transforms which relate measurements of time and space taken by one observer to those taken by another observer moving with respect to the first observer.
Einstein's argument is set out in his 1905 paper "On the electrodynamics of moving bodies" (
http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf and many other places on the web), but here's the Cliffs Notes version:
1) We expect that the laws of physics don't change just because you happen to moving. This is basically common sense: even if you think you are standing still, your speed through space changes from midnight to noon as the speed of the rotation of the Earth's surface is added to or subtracted from its speed around the sun... but you don't expect experiments to give different results because of this. There is also a ton of experimental evidence to support this piece of common sense.
2) The laws of electricity and magnetism say that the speed of light in vacuum is equal to c; it's a value that we can calculate from these laws. So if the laws of physics don't change with motion, and the speed of light can be calculated from these laws, we might start to suspect that the speed of light shouldn't change with motion. Furthermore, there is another ton of experimental evidence, starting with the Michelson-Morley experiment, that tells us that the world really does work this way, the speed of light in vacuum really is equal to c for all observers regardless of their relative motion.
3) So Einstein had good reasons for assuming that the speed of light would be constant for all observers, and seeing where this assumption took him. First, he derived the Lorentz transformations from this assumption; you can find the derivation in Appendix A of his book "Relativity: The Special and General Theory" (google it). These formulas relate measurements of time and space taken by one observer to those taken by another observer moving with respect to the first observer.
4) From the formulas, you can derive all the bizarre-sounding bits of special relativity: Time dilation, length contraction, the velocity addition formula, and even E=mc^{2} and all that good stuff. Of course this doesn't mean that it's
right, just that it's logical. However...
5) We have piled up an amazingly huge amount of experimental data showing that the world really does work this way, that the bizarre-sounding predictions are correct, atom bombs go boom, moving clocks run slow, velocity addition works as advertised. So now we believe we have a good theory, that Einstein's chain of logic really is valid.
One final note: Because speed is defined as distance traveled divided by time, you shouldn't be surprised to find that any formula that calculates speeds can be recast as a formula involving distance and time, and vice versa. Thus, you shouldn't be thinking of time dilation and length contraction as stuff that was made up just to make the velocity addition formula come out right; instead all three are consequences of the way that time and space behave for observers moving relative to each other.
Final final note: I'll let someone else explain why they're "Lorentz transforms" instead of "Einstein transforms". This post is already too long.