The relativistic factor (being (1-(v^2/c^2))^.5) is simply derived from applying Pythagoras' right angled triangle rule to different frames of reference of a single photon bouncing between two mirrors. The maths is quite simple. In the frame of reference of the mirror-photon system, the photon is bouncing directly up and down. When the system is moving at a velocity v, perpendicular to the photon movement, the photon appears to be moving in a zig zag pattern. The photon will hit each mirror at the same time regardless of the frame of reference, but cannot travel faster than the speed of light. Somehow it is travelling more distance in the same time, while not speeding up. It can do this because the distance it is travelling has been contracted.
Your OP was pretty much bang on. The relationship v = d/t does still hold, and so to keep v below a maximum constant, the length must be contracted/time dilated. They are the same phenomenon. You're asking the right questions in your OP.
In an example format: Say you travel 10 ly at half the speed of light. From the standpoint of someone back home, you travel the 10 light years in 20 years. From your perspective, you only travel for 17.3 years (relativistic factor = (sqrt0.75)), due to time dilation. But you were still travelling at .5c, so after travelling for 17.3 years you've covered 8.65 light years (17.3*0.5), which by no accident is= sqrt(0.75) * the distance as it appears from home. The length has contracted by the factor of sqrt(0.75) to keep the relationship v=d/t. This is why it is called the relativistic factor, not just the time dilation factor.
As for the speed of light to be impossible for any object with mass to reach, it is because the equation for kinetic energy, when taking into account relativistic effects, doesn't look as simple as Ek = 1/2mv^2. It looks like: Ek = mc^2/((1-(v/c)^2)^.5) - mc^2
As the velocity approaches c, the energy needed to make it go just a little bit faster increases hugely (dividing by number close to 0). You can see this from the equation quite easily. (As v->c, v/c -> 1, so 1-v/c ->0). Thus you can never accelerate any object (with mass) to light speed, as it would require infinite energy. What someone said before about it would if given enough time is wrong. To accelerate a mote of dust to close to light speed would require the entire universe, and then some, to turn into energy. To accelerate it to exactly light speed would require an amount that is by definition impossible. Photons can travel at the speed of light because they don't have any mass. Incidentally, a photon's entire travels throughout the expanse of the universe takes place in no time at all and covers no distance, from its perspective.
Edited for maths goof.
This post was edited by TIMMY213 on Apr 8 2011 04:30am