Special theory of relativity: Velocity addition

Question:
The speed of light is measured in a liquid that has refractive index of $n$. If the liquid is moving at $v$, what is the speed of light in the liquid detected in the laboratory frame?

Answer:
The addition of two velocities from the lab frame in terms of relativity is
\[
v_{\mathrm{added}} = \frac{v_1+v_2}{1+\frac{v_1v_2}{c^2}}
\]
The speed of light in liquid is given by
\[
v' = \frac{c}{n}
\]
In this case, $v_1=\frac{c}{n}$ and $v_2=v$.
\begin{eqnarray}
v_{\mathrm{added}} &=& \frac{\frac{c}{n}+v}{1+\frac{\frac{c}{n}v}{c^2}} \\
  &=& \frac{c/n+v}{1+v/nc}  \\
  &=& \frac{c(vn+c)}{nc+v}
\end{eqnarray}

Powered by Hirophysics.com