General Formula for Sum of Velocities

Does anyone have a nice general formula for the sum of velocities?

We know that the sum of two velocities $x,y$ is $\dfrac{x+y}{1+\dfrac{xy}{c^2}}$ where $c$ is the speed of lights. I went ahead and calculated the sum of three velocities $x,y,z$ , expecting to get something like $\dfrac{x+y+z}{1+\dfrac{xyz}{c^3}}$ but instead I got $\dfrac{x+y+z+\dfrac{xyz}{c^2}}{1+\dfrac{xy+yz+zx}{c^2}}$ . I went further to calculate the sum of $4$ velocities $w,x,y,z$ , to get the ugly expression $\dfrac{w+x+y+z+\dfrac{wxy+xyz+yzw+zwx}{c^2}}{1+\dfrac{wx+wy+wz+xy+xz+yz}{c^2}+\dfrac{wxyz}{c^4}}$

So can anyone else generalize? I do not see an obvious pattern yet, except for the fact that those different terms look suspiciously like something in Vieta's Formulas...

#Generalize #Physics #Velocity #SpecialRelativity

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$