Page:Relativity (1931).djvu/39

VI

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 he 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 addi-