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Frames of Reference
A frame of reference is a set of axes for measuring distances and a clock for measuring time.  
Each observer will be considered to be situated at the origin of his/her own set of axes.  
An inertial frame of reference means a frame in uniform motion (ie not accelerating)*.  
   
Consider two observers, A and B, in uniform relative motion, as shown below.  
The motion is parallel to their x axes.  
At time t = 0 (diagram on the left, below), A and B are very close together (ok, it looks as if they are occupying the same space but don't be pedantic!)  
   
 
   
At time zero, A and B will obviously give the same x co-ordinate for the point p, but at any other time, t, (diagram on the right) they will give different x co-ordinates to indicate the position of p.  
   
If the velocity of B relative to A is u, then in time t the distance between A and B changes by ut.  
   
A's coordinates for p are x, y and z and B's are x', y' and z' (but I assume you'd already guessed that from the colours on the diagram...)  
   
As the motion is parallel to the x axis we can see that, at any time t  
y' = y  
z' = z  
but  
 x' = x - ut  
where ut is the distance between A and B.  
These statements are often called the "Galilean transformations" (for one dimensional motion) because they tell us how to "transform" measurements made by one observer into measurements made by another observer who is in motion relative to the first (Galileo Galilei)  
   
Now, suppose that p moves with velocity v relative to A.  
We can transform this velocity into B's frame of reference simply by dividing the x transformation equation by t.  
 x'/t = x/t - ut  
which gives  
v' = v - u
   
* in the context of Galilean Relativity or Einstein's Special Theory of Relativity  
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