Galilean 
Galilean Simplification
Frames in Relative Movement
by Ricardo L. Carezani
Let us examine two parallel frames of coordinates x, y, z, t and x', y', z', t'. Frame F is moving with relative velocity v with respect to frame F '. Using the Galilean transformation, the abscissa x of point P in F is given in the above figure by
x' = x + vt
y' = y
z' = z
t' = t (1)
The velocity of P observed by frame F' is
v' = v (2)
Introducing a third frame of reference F '' with respect to which frame F is moving with relative velocity v1, the abscissa of P in F '' is
x'' = x' + vt
x'' = x + v t + v1 t = x + (v + v1) t (3)
We are using three coordinate reference frames, F, F ', and F '', and we shall try to demonstrate that two of them are useless, inoperative, and superfluous by doing the following analysis.
First: It is impossible to measure the abscissa of P in F from F' independently of the relative velocity v, supposing that P is moving with velocity v with respect to F, because P will be at rest with respect to F', and, in this particular case, F' is useless because it cannot observe a phenomenon at P.
Second: We can also suppose that P is moving within F with relative velocity vx, or with a relative velocity u with respect to F', because no one in F ' could distinguish the velocity vx from v, even though F' receives signals from P and they are transmitted to F', the observer in F' will measure the relative velocity u, because if the frame F first transmits v and then v+vx, the observer in F ' will only think that the relative velocity v increases until u. It would be easier for the observer in F ' to receive directly the signals from P. So, the reference frame F is unnecessary.
It is formally possible to demonstrate that either the frame F '' or F is useless and inoperative, as follows.
The velocity of the point P moving in F with velocity vx, measured from F' is
v' = vx + v (4)
x being the coordinate of P in F when t=0, its coordinate at a later time t will be
x1 = x + vx t (5)
Its coordinate in F' is
x' = x1 + v t
x' = x + vx t + v t = x + (vx + v) t (6)
The equations (6) and (3) are totally equivalent. Is it not the same to say that P moves with respect to F with relative velocity vx, giving equation (6), as to say that P moves with respect to F' with relative velocity v, giving equation (3)?
But equation (6) results only in applying "two systems of reference", F and F'; F'' then is not operative, and is useless. Equation (3) is obtained by applying "three systems of reference" but the point P is at rest in F, v' = v, and consequently frame F is useless, and inoperative.
If point P is moving together with F, and observer in F will never observe a phenomenon at P; that is to say, it is impossible to make the autoobservation of point P.
Even so, we may insist upon the following: Can velocity vx and v be distinguished from each other while we are observing the physical phenomenon. No. In each phenomenon there exists the phenomenon itself and the observer. The relativity of space and time is given by light speed, which is constant or, at least, has a finite value. Can we divide space in such a manner as to have an arbitrary quantity of reference frames?. A physical phenomenon naturally presupposes two objects: the observer and the object under observation. If the process is relative  and really it is  it should be to both, always due to their mutual interaction.
Consequently we shall define as a "physical; system" one that consists of a moving point P (the observed) and a "reference frame of coordinates" (the observer). A brief, formal and complete exposition is the following (1).
Let us consider three parallel frames F, F' F'', their origins coincident at instant t=0, and point P of abscissa x on F. If F moves relative to F' with velocity v and F' in respect to F'' with velocity v1, the coordinate of P on F'' is
x'' = x' + v1 t
x'' = x + (v1 + v) t (7)
During this exposition we have presumed a relation of P with F even though P is at rest with respect to F. This, in our point of view, is not logical due to the fact that self observation of a point is absolutely impossible. In other words, with point P and a geometrical system F linked to it, we have no physics whatsoever. We need another system, one not linked to P, (and which we may individualize by its origin) in which we place an observer describing the physical alternatives which P is undergoing. In this case, frame F is superfluous.
Equation (7) maintains its value if we suppose that at time t=0, the point passes the origin of F''. To make it easier, we shall run up the primed values in the following manner: F' shall be F, and F'' will be F', v is the velocity of P with respect to F, v1 the velocity of F with respect to F' therefore (7) shall read"
x' = x + v1 t
x' = (v1 + v) t (8)
which is equal to (7) when we only consider two systems. The preceding analysis of the systems in relative movement leads to the following definition: a physical system consists of two points in relative movement, P and F or the origin of F. Up until now this has been known as a system F on one side, and point P at rest on the other. This new concept or conclusion will be used to analyze what happens in special relativity.
Derivation 2:
The Frames Derivation


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Storm In Physics (2005) 
2nd book published by Dr. Carezani.

