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?

 

[Moetasim]

Your work for part a) is correct. However, your reasoning for part b) is incorrect. In special relativity, time is relative and can be dilated or contracted depending on the relative velocity of the observers. In this scenario, observer A sees that it takes 5x10^-6 sec for the front of spaceship B to traverse the full length of A. However, from the perspective of observer B, who is traveling at a different velocity, the time it takes for the front of A to traverse the full length of B will be different. This is due to time dilation, where the time interval between two events is longer for the observer with the slower relative velocity. Therefore, the correct answer for part b) is that the clock on spaceship B will measure a longer time interval than the clock on spaceship A. In order to calculate the exact time elapsed on B's clock, we would need to know the relative velocity of A from B's perspective.