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 \(v_2=v\).

\[  v_{\mathrm{added}} = \frac{\frac{c}{n}+v}{1+\frac{\frac{c}{n}v}{c^2}} \]

\[  = \frac{c/n+v}{1+v/nc}  \]

\[  = \frac{c(vn+c)}{nc+v} \]