Whoa, I wish I got here sooner.
First of all, the way mass is generally defined in physics, mass does not depend on speed. One can define mass in other ways such that it does depend on speed, but these definitions are not generally as useful. Mass is most usefully defined as that portion of an object’s or system’s energy which cannot be transformed away: That is to say, no matter what your reference frame, the energy of the system will always be at least equal to its mass.
What does change with speed, however, is momentum and energy, and furthermore, they don’t change in the manner described by Newton. The total momentum of a massive particle is not p = mv, as Newton had it, but rather p = [symbol]g[/symbol]mv. Likewise, the total energy is not E = mc[sup]2[/sup], but E = [symbol]g[/symbol]mc[sup]2[/sup]. For low speeds (compared to the speed of light), [symbol]g[/symbol] is close to 1, so in that limit, it’s a reasonable approximation to use the Newtonian formulas. But as the speed approaches the speed of light, [symbol]g[/symbol] approaches infinity, meaning that it would take an infinite amount of energy and momentum to bring a massive particle to the speed of light.
With our stack of rockets, a person in the reference frame where the whole thing started off would see the first rocket travelling at 7 miles/second. But he would see the second rocket travelling at a hair less than 14 miles/second. 14 miles/second is still much less than c, so the correction would be very small at this point, and the observer would probably be justified in ignoring it. But this small correction will build up, such that for the later rockets, they’ll be going a significant amount less than n*7 miles per second. If I’m not mistaken, our observer will measure the last rocket to be going at about .707c, which is quite a respectable clip, but still less than c. Our observer will also measure the length of the last rocket (and everything on board) to be .707 times the length it had at rest, and will measure a clock on board that rocket to be ticking off 1.41 seconds for every 1 second on his own wristwatch.
Meanwhile, if we put an observer on one of the rockets, he’ll measure himself to be his usual height, and he’ll measure his watch to be running at its normal rate, and in all other ways he’ll feel and appear to himself to be perfectly normal. He’ll see the rocket behind him falling behind at a rate of 7 miles per second, and the rocket ahead of him will be moving away from him also at 7 miles per second. Again, he’ll see more distant rockets as moving away from him at a speed lower than he would have expected, and he’ll measure a difference in their metersticks and clocks. An observer on the last rocket, if he looked back at Earth, would see heights on Earth to be squished, only .707 times their normal height, and he’ll see an Earth clock ticking off 1.41 seconds for each second on his own watch.