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Relativity

"Relativistic" Combination of Velocities : Detailed Proof

Suppose we know the velocity of B relative to A (u) and also the velocity of C relative to A (v); what is the velocity of C relative to B, (v’)?

The diagram is drawn "from A’s point of view".

A sends flashes of light to B and B sends flashes of light to C. B sends a flash to C at the same instant as he/she receives flashes from A. It is clear that C will receive both sets of flashes simultaneously. We can therefore write

kAC = kAB×kBC

 
and rearranging gives

in general we have seen that

so, in this case, to find the velocity of C relative to B (v’) we have

which leads to the following complicated-looking (but really very simple) equation...

which becomes

After a bit more messing about, we get

which leads to

 


 

© David Hoult 2008