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Einstein

Relativity: The Special and General Theory
Albert Einstein: Relativity
Part I: The Special Theory of Relativity
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.
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