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SPEED CUMULATION


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}

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]

Now:

[Cbc]=
|shΦbc....chΦbc....0..0|
|chΦbc....shΦbc....0..0|
|0........0........1..0|
|0........0........0..1|

[Cab]=
|shΦab....chΦab....0..0|
|chΦab....shΦab....0..0|
|0........0........1..0|
|0........0........0..1| 

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

But:
[Cac]=
|shΦac....chΦac....0..0|
|chΦac....shΦac....0..0|
|0........0........1..0|
|0........0........0..1|

so that:
Φac=Φbc+Φab
(pseudo-rotation angles add, not speeds).   

Setting:
thΦab=Vab/C
thΦbc=Vbc/C
thΦac=Vac/C
we have:
th(Φbc+Φab)=
thΦac=(thΦbc+thΦab)/(1+thΦbc*thΦab),  
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.