The Theorem of the Addition of Velocities Employed in TheoremOfTheAdditionOfVelocitiesEmployedIn 2006-03-11 23:19:42 2006-03-11 23:19:42 Topic Relativity: The Special and General Theory % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \subsection{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.