But we could also represent this movement in a different way. We could choose two mutually perpendicular axes OX and OT as in Fig. 12, OX representing the 'spatial' actual direction of the movement and OT, which we have heretofore used to represent a second 'spatial' co-ordinate, now representing the 'time' co-ordinate.
are taken indefinitely small, in the limit our 'moving' point would be represented by a static line inclined to the X axis. We could then speak either of our 'moving' point, or else not use the term 'moving' but speak of infinitely many static points, each given by two numbers, one representing a distance, the other 'time'. Our 'moving' point would become a static world-line. The reader should notice that in this case we have structurally changed our language from dynamic to static, and raised the dimension. Our mathematical 'moving' 'point', which had no dimension in our one-dimensional 'space', is in our two-dimensional space-time represented by a static one-dimensional line.
In this example we had uniform translation. We did not introduce acceleration. The distances were proportional to the 'times', hence our line was 'straight' and inclined to the X axis at a constant angle.
Using such space-time representation we see that a point when it is not 'moving', but is stationary, is represented by a line parallel to the 'time' axis T, as shown at A on Fig. 13. Our point A is getting older, so to speak, but does not 'move'. In the next case, the point B does not 'move' until it is some seconds old, when at B' it begins to 'move' with constant velocity. Point C 'moves' in the beginning at one constant velocity until C where it acquires a certain different velocity and the direction changes.
In Fig. 13, D represents a point experiencing a series of sudden changes of velocities. The graph is a succession of short straight lines forming a broken line or open polygon. As the changes of velocity occur more and more frequently the sides of our polygon become smaller and smaller; and in the limit, as the changes of velocity become continuous, our broken line becomes a smooth curve E.
Motion with continuously changing velocity is called accelerated or retarded motion. The rate of change of velocity is called acceleration and is