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Einstein

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Einstein

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jcsd

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v = (u + v')/(1 + uv'/c^{2})

This is how to sum two velocites, u and v', in SR where c is the speed of light in the vacuum.

If we input the velocities into the equation as a fraction of c, then all the c's oin the bottom of the equation cancel and we're left with:

v = (0.5c + 0.5c)/(1 + 0.25) = 0.8c

Therefore you will observe the other person traveling at 0.8c or 4/5 of the speed of light relative to you.

This is how to sum two velocites, u and v', in SR where c is the speed of light in the vacuum.

If we input the velocities into the equation as a fraction of c, then all the c's oin the bottom of the equation cancel and we're left with:

v = (0.5c + 0.5c)/(1 + 0.25) = 0.8c

Therefore you will observe the other person traveling at 0.8c or 4/5 of the speed of light relative to you.

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jcsd

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I'll just add for speeds a lot lower that c the two velocities do add algerbraically, that is :

v â‰ˆ u + v'

for values of u and v' significantly smaller than c.

For example, taking two objects that both appear to be traveling ~650 mph (10^{-6} c) in oppoiste directions to a 'stationary' observer the difference in observed velocities will only be about 1 part in a trillion.

v â‰ˆ u + v'

for values of u and v' significantly smaller than c.

For example, taking two objects that both appear to be traveling ~650 mph (10

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SEG9585

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If 2 photons were moving towards each other, both moving at the speed of light:

v = (c + c)/(1 + c^2/c^2) = 2c/2 = c

The photons would observe each other moving at the speed of light? If that's the case, how can it tell whether the other photon is moving at all of not?

v = (c + 0)/(1 + 0/c^2) = c/1 = c

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jcsd

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According to the transformation, both photons would observe each other traveling at the speed of light, but reference frames at this speed are forbidden anyway.

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selfAdjoint

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Originally posted by jcsd

I'll just add for speeds a lot lower that c the two velocities do add algerbraically, that is :

v â‰ˆ u + v'

for values of u and v' significantly smaller than c.

For example, taking two objects that both appear to be traveling ~650 mph (10^{-6}c) in oppoiste directions to a 'stationary' observer the difference in observed velocities will only be about 1 part in a trillion.

Actually the same rule holds for low speeds - it holds for all speeds, but if you try it for speeds around a billionth of c, which are the speeds we encounter (feet per second) then the Lorentz formula becomes normal addition to a high degree of accuracy. But there is no cutoff where relativity addition stops and the familiar addition begins.

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quartodeciman

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The relative velocities ought to be calculated by a difference formula:

u' = (u - v)/(1 - uv/c

.

So, for the velocity of the +c photon relative to the -c photon,

u' = ((+c) - (-c))/(1-(+c)(-c)/c

For the velocity of the -c photon relative to the +c photon,

u' = ((-c) - (+c))/(1-(-c)(+c)/c

. The relative velocities would be equal in magnitude, but oppositely directed.

There is a problem with all this. In order to derive composition theorems like this, guaranteeing that the postulates of special relativity are enforced, it is necessary to assume implicitly that all relative velocities between observers have magnitudes less than c. So, it is not a foregone conclusion that the two photons define reference frames for which the composition formula is valid. But it does appear to give "appropriate" answers. One photon gets a relative velocity of +c and the other gets a relative velocity of -c.

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