# Relativity: The Special and General Theory

The Theorem of the Addition of Velocities Employed in

Classical Mechanics

Let us suppose our old friend the railway carriage to be travelling along the rails with a

constant velocity v, and that a man traverses the length of the carriage in the direction of

travel with a velocity w. How quickly or, in other words, with what velocity W does the

man advance relative to the embankment during the process ? The only possible answer

seems to result from the following consideration: If the man were to stand still for a

second, he would advance relative to the embankment through a distance v equal

numerically to the velocity of the carriage. As a consequence of his walking, however, he

traverses an additional distance w relative to the carriage, and hence also relative to the

embankment, in this second, the distance w being numerically equal to the velocity with

which he is walking. Thus in total be covers the distance W=v+w relative to the

embankment in the second considered. We shall see later that this result, which expresses

the theorem of the addition of velocities employed in classical mechanics, cannot be

maintained ; in other words, the law that we have just written down does not hold in

reality. For the time being, however, we shall assume its correctness.