Relative velocity of two spaceships passing each other

[inkskin]

Here's a weird question where I don't quite know if length contraction or time dilation play a role.

Say I have 2 spaceships moving towards each other. rest length of spaceship A is La and spaceship B is Lb. The pilot in A determines that time taken for the spaceship B to traverse the length of A is T,. What is the relative velocity B wrt A?
Also, according to nose of B, what's the time taken to traverse the length of A?

The way i see it, in the frame of reference of A, the pilot A is at rest, hence relative vel, is La/Tx. Does time dilation play any role at all here?
Also, the second part is just Tx/gamma, right? where gamma can be find once we find the relative vel from part 1

Does this make sense?

 

[Nugatory]

By "the time taken for spaceship B to traverse the length of A", do you mean the time between when B's nose lines up with A's nose and when B's nose lines up with A's tail? Or do you mean the time between when B's nose lines up with A's nose and when B's tail lines up with A's tail? If you mean the first, then you have the relative velocity calculation right and you can get gamma and everything else from there.