Relativity Question: What is the Relative Velocity and Time Elapsed?

[Domnu]

Problem
Two spaceships, each measuring 100 m in its own rest frame, pass by each other traveling
in opposite directions. Instruments on board spaceship A determine that the front of spaceship B requires 5x10^-6 sec to traverse the full length of A.

(a) What is the relative velocity v of the two spaceships?
(b) How much time elapses on a clock on spaceship B as it traverses the full length of A?

Answers
a) Well, the observer in A, in his frame, sees that B takes 5e-6 sec to go 100 m, so this means that the relative velocity, v, of the two spaceships is $$\boxed{100/(5 \cdot 10^{-6}) = 2\cdot 10^7 \text{m/s}}$$.

Homework Statement

b) We know that observer B will still observe the same relative velocity as A, by symmetry. Now, from B's reference frame, A travels at $$2\cdot 10^6 \text{m/s}$$ through $$100 \text{m}$$, so B also measures time $$5 \cdot 10^{-6} \text{sec}$$.

Is my work above correct? Part b) seems wrong, because both measure the same time... doesn't this usually not happen?

 

[Domnu]

Ah, I think I see my mistake in part b)... there is a length contraction that B sees when A moves past, so the time isn't the same. Is this correct?