Let (t,x) be coordinates of some point P in some inertial frame O. Suppose that at (0,0) there is a particle that has velocity u and acceleration a in the frame O. If the origo of a frame O' is attached to this particle, then what are coordinates (t',x') of the point P in the frame O'? Or does that acceleration have any effect on the coordinates? It would seem reasonable, if correct answer is that one must use coordinates that transform in the usual way similarly as without acceleration, but always choose the value of u that is correct instantaneously.