From my head to yours
Two frames can have a relative velocity and since any velocity is allowed there is an infinite number of possible frames. Rotating frames have a necessary relative acceleration (think centripetal acceleration), so if an object is at rest in one frame it cannot remain at rest in the other. A simple example of this can be shown by holding a ball in your hand and spinning around. In this situation you define a rotating frame relative to the earth, and if you let go of the ball you will see that it flies outwards, even though there is no horizontal force on the ball. Therefore you are not an inertial reference frame. However, if you were on a train moving at a constant speed with respect to the earth and simply placed the ball on a table, the ball would not move.
How does one transform coordinates between two inertial reference frames? Let's consider two families of observers, A and B, moving with respect to each another with a relative speed \(v\). The two inertial reference frames corresponding to each family have coordinates \((t_A,x_A)\) and \((t_B,x_B)\). Which of the following statements must be true? (\(X,T\)are arbitrary functions below.)