A nanodrop of water is traveling at a speed 0.5\(c\) relative to the lab frame when a beam of light
traveling in the same direction as the drop enters it. What is the speed of light in the water
relative to the lab frame? The index of refraction of water is n = 4/3
\( (A) \dfrac 12 c\)
\( (B) \dfrac 23 c\)
\( (C) \dfrac 56 c\)
\( (D) \dfrac {10}{11} c\)
\( (E) c\)
In the frame of reference where the nanodrop is at rest, the beam will travel at the speed of light
in vacuum until it reaches the water. Once inside, the speed of the beam will slow down to
\( u = \dfrac cn = \dfrac{3c}{4}\)
From the perspective of the lab, we need to add the velocity of the water plus the velocity of the beam
inside it. The sum needs to be done relativistically, as follows
\( u_{lab} = \dfrac{\dfrac 12 c + \dfrac 34 c}{1+\dfrac{\dfrac 12 c \times \dfrac 34 c}{c^2}} = \dfrac{10}{11}c \)
Answer: D