High Speed Equations

[mendhak]

Generally, we take relative velocity as

a-b

However, at higher velocities (specifically in the case of special relativity cases) it's

w = (a-b)/(1-ab/c^2)

or

w = (a+b)/(1+ab/c^2)

That's as far as I know. I had to take it for granted. Could someone take the time out and explain it to me? Thanks.
-Frog.

[DiGiTalErRoR]

It goes with the notion nothing can travel faster than the speed of light, which is c.

Since c^2 is quite a large number, ab/c^2 is a very small number in most cases. So small for most purposes it rounds out to 0. So 0+1, you get /1 which is the number(a+/-b) itself.

[mendhak]

Yes, I understood that. But how did 'they' arrive at that equation?

[krtxmrtz]

It's a bit too lengthy to explain here. Could be summarized as follows.

The theory of relativity makes these two apparently contradictory facts compatible:

1. The speed of light (in vacuo) is always the same regardless of the reference frame.
2. The "principle of relativity", according to which physical phenomena occur according to the same laws in 2 diferent "Galilean" reference frames (galilean means these frames have only a rectilinear motion -no rotation- with constant velocity)

This is only possible by means of the so-called Lorenz transformation, which replaces the classical Galilean transformation of coordinates between 2 such frames. Using this transformation in the law of addition of velocities you come out with:

V = (v1 + v2) / (1 + v1*v2/c**2)

Of course, if the involved velocities v1 and v2 are much less than the velocity of light c -which is usually the case- the last term is almost zero and you are left with the classical expression:

V = v1 + v2

I'd recommend the book entitled "Relativity" by Einstein himself. It is very clearly explained and probably you can find it for a reasonable amount in paperback.

[si_the_geek]

I would recommend NewScientist.com, as they have an article posted in the last few weeks that states that the speed of light has been broken, thus implying that this equation may not be entirely true.

[mendhak]

Originally posted by si_the_geek
I would recommend NewScientist.com, as they have an article posted in the last few weeks that states that the speed of light has been broken, thus implying that this equation may not be entirely true.

Are you referring to the news about the two guys who used basic lab equipment to make pulses travel at 4 times the speed of light in a coaxial cable... I believe it was University at Murfreesboro...
is that the one?

[mendhak]

Originally posted by krtxmrtz
[B]
2. The "principle of relativity", according to which physical phenomena occur according to the same laws in 2 diferent "Galilean" reference frames (galilean means these frames have only a rectilinear motion -no rotation- with constant velocity)

This is only possible by means of the so-called Lorenz transformation, which replaces the classical Galilean transformation of coordinates between 2 such frames. Using this transformation in the law of addition of velocities you come out with:

V = (v1 + v2) / (1 + v1*v2/c**2)

Hmm. I wasn't aware of "Galilean" reference frames, and I may have brushed over the Lorenz transformations. I'll try a search on that, else get back here.
Thanks.

I'd recommend the book entitled "Relativity" by Einstein himself. It is very clearly explained and probably you can find it for a reasonable amount in paperback.

Uhm... ROFL.

[si_the_geek]

Originally posted by mendhak
Are you referring to the news about the two guys who used basic lab equipment to make pulses travel at 4 times the speed of light in a coaxial cable... I believe it was University at Murfreesboro...
is that the one?

That sounds familiar, it could well be!