special relativity-Galilean transformation

special relativity-Galilean transformation

 

Frames of reference, events and transformations

Before proceeding further with the analysis of relative motion the concepts of reference frames, events and transformations need to be defined more closely.
Physical observers are considered to be surrounded by a reference frame which is a set of coordinate axes in terms of which position or movement may be specified or with reference to which physical laws may be mathematically stated.
An event is something that happens independently of the reference frame that might be used to describe it. Turning on a light or the collision of two objects would constitute an event.
Suppose there is a small event, such as a light being turned on, that is at coordinates in one reference frame. What coordinates would another observer, in another reference frame moving relative to the first at velocity along the axis assign to the event? This problem is illustrated below:

 

Frame (O) is always at rest, and frame (O') is always moving parallel to x axis with velocity (u). 

What we are seeking is the relationship between the second observer's coordinates for the event and the first observer's coordinates for the event . The coordinates refer to the positions and timings of the event that are measured by each observer and, for simplicity, the observers are arranged so that they are coincident at t=0. According to Galilean Relativity:

This set of equations is known as a Galilean coordinate transformation or Galilean transformation.

These equations show how the position of an event in one reference frame is related to the position of an event in another reference frame. But what happens if the event is something that is moving? How do velocities transform from one frame to another?

The calculation of velocities depends on Newton's formula: . The use of Newtonian physics to calculate velocities and other physical variables has led to Galilean Relativity being called Newtonian Relativity in the case where conclusions are drawn beyond simple changes in coordinates. The velocity transformations for the velocities in the three directions in space are, according to Galilean relativity:

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1) Inertial system:

they are systems which are in a uniform motion relative one to another.

 2)The proper system:

An inertial system  which contains the observer at rest.
                                                                                                                                              

 3)An event:

it is an occurrence at certain position and certain time. (x,y,z,t)

 4) the invariance property:

 it's invariant under transformation between s & s'.

f(x',y',z',t')=0

after transformation

f(x,y,z,t)=0

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 Galilean transformation:

 

 at t=t'=0 then  O ~= O'

After time t: frame S' moves a distance = ut.

x = ut'+x',      y = y',      z = z',       t = t'.

x'=x-ut,         y' = y,      z' = z,       t' = t.

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x = ut'+x' 

dx/dt = dx'/dt' + u

Vx = Vx' + u

where Vx is the velocity of particle with respect to (wrt) frame S.  

Vx' is the velocity of particle (wrt) frame S'.

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y = y'

dy/dt = dy'/dt'

Vy = Vy'

Vz = Vz'

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Vx = Vx' + u, u is constant

dVx/dt = dVx'/dt' + 0 

ax = ax' 

ay = ay'

az = az'

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proofing that Newton low of motion are invariant by using Galilean transformation: 

1)Vx=Vo +ax*t

Vx'+ u =Vo + u +ax'*t'

 Vx'=Vo'+ax'*t'

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²⁾

V² =V_o + 2a_x *x

(V_x’+u)²= (V_o’ +u)²+2a_x *(x’+ut’)

V_x’²+2uV_x’+u²=V_o²+2uV_o’+u²+2a_x *x+2u t’ a_x’

V_x’²=V_o’²+2a_x ’*x’