Problem — Velocity composition (electron)

Problem

In the laboratory frame an electron travels at $u' = 0{,}9\,c$ along $+x$, while a second particle travels at $v = 0{,}7\,c$ along $+x$. What is the electron’s velocity as measured in the second particle’s frame?

Solution

Relativistic velocity composition formula. The electron’s velocity in the second particle’s frame is

$u = \frac{u' - v}{1 - \dfrac{u'\,v}{c^2}}$

Substitution. With $u' = 0{,}9\,c$ and $v = 0{,}7\,c$:

$u = \frac{0{,}9\,c - 0{,}7\,c}{1 - (0{,}9)(0{,}7)} = \frac{0{,}2\,c}{1 - 0{,}63} = \frac{0{,}2\,c}{0{,}37}$

$\ev{u \approx 0{,}54\,c}$

The naive Galilean difference would give $0{,}2\,c$; the relativistic correction in the denominator increases the result to $0{,}54\,c$, but it still stays below $c$: no composition of sub-luminal velocities can exceed $c$.

Links

Topics: Special relativity Concepts: Relativistic velocity composition Skills: Changing reference frame