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robphy
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The “Addition of Velocities” formula (more correctly, the “Composition of Velocities” formula) in Special Relativity
[tex]\frac{v_{AC}}{c}=\frac{ \frac{v_{AB}}{c}+\frac{v_{BC}}{c} }{1 + \frac{v_{AB}}{c} \frac{v_{BC}}{c}}[/tex]
is a non-intuitive result that arises from a “hyperbolic-tangent of a sum”-identity in Minkowski spacetime geometry, with its use of hyperbolic trigonometry.
However, I claim it is difficult to obtain this by looking at the Galilean version of this formula and then motivating the special-relativistic version.
Instead, one should start with the Euclidean analogue (in what could be mistakenly called the “addition of slopes” formula… “composition of slopes” is better),
then do the special-relativistic analogue, then do the Galilean analogue (to obtain the familiar but unfortunately-“our common sense” formula).
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Composition of slopes is also called the stacked wedge analogy.
 
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