A frame of reference is simply a set of axes and a clock.

Each observer will be considered to be at the origin of his/her set of axes.

An inertial frame of reference (usually) means the frame of reference of an observer who is in uniform motion.

Galilean Relativity

Consider two observers, A and B, in uniform relative motion, as shown below.
........

If the velocity of B relative to A is u, then in time t the distance between A and B changes by ut.

At time zero, A and B will give the same co-ordinates for the point p, but at any other time, t, they will give different co-ordinates to indicate the position of p.

Let A’s measurements be represented by x, y, and z.

Let B’s measurements be represented by x’, y’, and z’.

In the case shown in the diagram, it is clear that

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.

Now, suppose that p moves with velocity v relative to A.

Dividing the x transformation equation by t gives
...

...

N.B.
It is conventional to use the letter S to represent a frame of reference. So, we might say the diagrams above represent observer A at rest in the frame S and observer B at rest in frame S’.

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