PARENTS:
 LORENTZ TRANSFORMATIONS 
 RELATIVITY WORKSHOP 
 WORKSHOPS 
 ARCHIVE    
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Speed cumulation. Georges.
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Let Xa,Xb,Xc referentials moving with relative
speeds Vab,Vbc,Vac.
We can write:
{Xa}={cta,xa1,xa2,xa3}
{Xb}={ctb,xb1,xb2,xb3}
{Xc}={ctc,xc1,xc2,xc3}
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Let further:
[Cab],[Cbc],[Cac] matrices of pseudo-rotation
raspectively Xa/Xb,Xb/Xc,Xa/Xc.
We have;
{Xb}=[Cab]{Xa}
{Xc}=[Cbc]{Xb}
thus:
{Xc}=[Cbc][Cab]{Xa}=[Cac]{Xa}
thus:
[Cac]=[Cbc][Cab]
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Now:

[Cbc]=
|shALPbc..chALPbc..0..0|
|chALPbc..shALPbc..0..0|
|0........0........1..0|
|0........0........0..1|

[Cab]=
|shALPab..chALPab..0..0|
|chALPab..shALPab..0..0|
|0........0........1..0|
|0........0........0..1| 

and
[Cac]=[Cbc][Cab]=
|sh(ALPbc+ALPab)..ch(ALPbc+ALPab)..0..0|
|ch(ALPbc+ALPab)..sh(ALPbc+ALPab)..0..0|
|0................0................1..0|
|0................0................0..1|

But:
[Cac]=
|shALPac..chALPac..0..0|
|chALPac..shALPac..0..0|
|0........0........1..0|
|0........0........0..1|

so that:
ALPac=ALPbc+ALPab
(pseudo-rotation angles add, not speeds).   
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Setting:
thALPab=Vab/C
thALPbc=Vbc/C
thALPac=Vac/C
we have:
th(ALPbc+ALPab)=
thALPac=(thALPbc+thALPab)/(1+thALPbc*thALPab),  
or
Vac/C = (Vbc/c + Vab/c)/(1+VbcVab/c^2) 
or
Vac=(Vbc + Vab)/(1+VbcVab/c^2)
Which is the SR speed cumulation formula.

Let's note that 

1.For slow speeds the denominator can be
approximated as 1 and the SR formula reduces
to the Galileo-Newtonian: Vac=(Vbc + Vab).

2.For one of the cumulated speeds, say Vbc,
approaching c Vac approaches c:
Vac-->(c + Vab)/(1+c*Vab/c^2) = c
The same holds of course for both cumulating
speeds approaching c.
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Georges.
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PARENTS:
  LORENTZ TRANSFORMATIONS 
 RELATIVITY WORKSHOP 
 WORKSHOPS 
 ARCHIVE