I agree. More details: http://en.wikipedia.org/wiki/Velocity-addition_formula

If you add two velocities that are less than c, you can never get c, you always get something smaller.

0.8c + 0.3 c ~= 0.887 c (It's smaller than c)

At low speeds the effect still exists, but it's negligible. There is an important difference between "it doesn't exist" and "it's so small that if we forget it we will be fine".

It's so small that you must use floating point numbers with too many digits to calculate it. It's more useful to calculate the difference between the "relativistic" sum and the "classical" sum. Another way is to aproximate the formula with a Taylor expansion

dif = (X+Y) / (1+(X * Y/c^2)) - (X+Y) ~= - (X + Y) (X * Y/c^2)

0.8mph + 0.3 mph = 11mph - 5.9 * 10^-39 mph

The correction is veeeeery small.

> The correction is veeeeery small.

Indeed it is. The only place I can think where such a small correction matters is in the GPS system. The accuracy of the GPS system depends on tracking down and eliminating every possible source of error, and to maximize position accuracy, two relativistic effects are accounted for:

1. The time dilation caused by the satellites' velocity, from Special Relativity.

2. The rate of time passage at orbital altitude compared to that at the surface, which reflects the gravitational well effect of General Relativity.

The two effects move in opposite directions, but they don't cancel out. The applied correction (to the orbiting clocks) is very small indeed, but enough to avoid a gradual but serious decline in positional accuracy over time.

Does anyone know of a general & special relativity simulator? That is, a program where I can setup various "thought experiments", by placing clocks and masses at various point in space, where they can have various acceleration profiles, and can communicate with each other, and run the scenarios? I never really understood the twins "paradox" and the explanations for the source of the asymmetry seemed hand-wavy and "because".