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Subject: Re: [SR] Usefulness of real velocities in accelerated relativistic frames 
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From: Richard Hachel <r.hachel@tiscali.fr>
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Le 13/03/2024 à 21:59, "Paul B. Andersen" a écrit :
> Den 12.03.2024 20:42, skrev Richard Hachel:

>>> It is obviously normal to keep the proper acceleration constant,
>>> and then Vrm  ≠ (1/2)Vri


We know that in accelerated frames of reference the average speed is 
proportional to the instantaneous speed.
 
Let Vrm=(1/2)Vri
 
We are talking about real relativistic speeds (Richard Verret's hobby).
 
But we know that relativistic physicists do not use this notion, and it is 
important to remind them of the relationship between real speed and 
observable speed.
 
Vr=Vo/sqrt(1-Vo²/c²)
 
This immediately leads us to 
Vom/sqrt(1-Vom²/c²)=(1/2)Voi/sqrt(1-Voi²/c²)
 
and therefore to:
 
Vom²(1-Voi²/c²)=(1/4)Voi²(1-Vom²/c²)
 
Let Vom²=(1/4)Voi²+(3/4)Voi².Vom²/c²
 
Hence 
Vom=(1/2)Voi/sqrt(1-(3/4)Vom²/c²) 
and, conversely,
Voi=2.Vom/sqrt(1+3Vom²/c²)


R.H.