The triangle trick

In the multiplicative form of velocity composition, velocities appear with signs that depend on who is measuring what. A very common mistake is getting confused about which velocity goes in the numerator and which in the denominator of each Doppler factor. Fortunately there is a graphical trick that makes everything automatic and removes any ambiguity about signs.

The triangle of the three reference frames: one vertex per frame, one directed segment per velocity (from the first subscript to the second). The sought velocity $u_{C(A)}$ is the side directed $A \to C$.

The rule. Draw a triangle with one vertex per reference frame and one directed segment per velocity, always from the first subscript to the second. To compose the velocities, start from the sought vertex (the one with the unknown subscript) and reach the other in two different ways: directly, along the unknown segment, or passing through the third vertex.

Every time you traverse a segment in the direction it is oriented, write $\sqrt{(1 + \beta)/(1 - \beta)}$, where $\beta$ is the speed of that segment in units of $c$. Every time you traverse it in the opposite direction, write the reciprocal root $\sqrt{(1 - \beta)/(1 + \beta)}$. Multiply the two roots of the long path, set them equal to the root of the direct segment, and you have the right equation, with all signs in place, without needing to memorise any convention. The triangle is the formula.

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Topics: Special relativity Concepts: Relativistic composition of velocities Skills: Changing reference frame

Related exercises: Problem — Composition via Doppler · Worked exercise — Condor versus rocket · Problem — Composition of velocities (electron)