Worked exercise — Condor versus rocket

Problem

A condor flies at $v_c = +0{,}7\,c$ relative to a planet. A rocket travels at $v_r = -0{,}8\,c$ relative to the same planet (in the opposite direction). At what speed does the condor travel in the rocket’s reference frame?

Solution

Setup. Let $u$ be the condor’s velocity in the rocket’s frame. The rocket sees the planet receding at $v = +0{,}8\,c$ (relative to the rocket, the planet has velocity opposite to that of the rocket relative to the planet). In the planet’s frame the condor has velocity $u' = v_c = 0{,}7\,c$. We apply the composition law: $u = \frac{u' + v}{1 + u'v/c^2} = \frac{0{,}7\,c + 0{,}8\,c}{1 + 0{,}7\cdot 0{,}8} = \frac{1{,}5\,c}{1{,}56} \approx 0{,}96\,c$

$\ev{u \approx 0{,}96\,c}$

Comparison with Galileo. Simple Galilean addition would give $u_\text{Galileo} = 0{,}7\,c + 0{,}8\,c = 1{,}5\,c > c \quad\text{(impossible!)}$ a speed greater than that of light. Einstein saves the day: the denominator $1 + u'v/c^2 = 1{,}56 > 1$ automatically brakes the result, which stays at $0{,}96\,c$ — very fast but still below $c$. The speed of light behaves like an insurmountable asymptote: two velocities close to $c$ compose into something still close to $c$, never beyond.

Links

Topics: Special relativity Concepts: Relativistic velocity composition Skills: Changing reference frame · Limiting-case and sci-fi-physics analysis