## Solution to 2001 Problem 80

 We use the velocity addition formula:\begin{align*}v_{AC} = \frac{v_{AB} + v_{BC}}{\sqrt{1 + v_{AB} v_{BC}/c^2 }}\end{align*}This formula can be derived directly from the Lorentz transformations, and therefore it does not only apply to particles with mass. It applies to paths through space parametrized by time that travel at less than or equal to the speed of light. Here $v_{AC}$ is the velocity of the path as seen in reference frame $C$, $v_{AB}$ is the velocity of the path with respect to reference frame $B$, and $v_{BC}$ is the velocity of reference frame $B$ relative to reference frame $C$. In the case of this problem, reference frame $B$ is the tube and reference frame $C$ is the lab. So, $v_{BC} = 0.5 c$, $v_{AB} = c/n = 3c/4$, so\begin{align*}v_{AC} = \frac{v_{AB} + v_{BC}}{\sqrt{1 + v_{AB} v_{BC}/c^2 }} = \frac{5 c/4}{1 + 3/8} = \boxed{\frac{10 c}{11}...Therefore, answer (D) is correct.