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
coordinate for the point p, but at any other time, t, (diagram on
the right) they will give different x coordinates 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 
