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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.
 
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