Special relativity spaceships time question

[thenewbosco]

Here is the problem and my solution,

Two spaceships each measuring 100m 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 takes 5x10^-6 s to traverse the full length of A.
a) what is the relative velocity of the two spaceships
b) how much time elapses on a clock on spaceship B as it traverses the full length of A?

So for this i thought that since A is doing the measurement, it takes 5 x 10^-6 s for B to pass 100 m. Since according to A, it is still 100m. then the velocity is just 100/5 x 10^-6. is this correct?

then i took this speed and put it into the time dilation equation with the 5x10^-6s to get the answer for part b.

is this correct and if not how should i go about this problem, thanks.

 

[P3X-018]

The first one seems correct.
In b if you by time dilation equation mean $T = \gamma T_0$ where T is the time obsered in b, then this will be incorrect. I had a similar problem (thread: Relativity). You must remember that the events don't occur at the same location, that is $\Delta x \neq 0$ and $\Delta x' \neq 0$. You have $\Delta x =100$ and $\Delta t =5\times 10^{-6}$ in A's reference frame, and since B is moving with a relative velocity v, you must use the Lorentztransformation to transform this time $\Delta t$ to B's referenceframe, which will be

$$\Delta t' = \gamma(v)\left(\Delta t-\Delta x\,\frac{v}{c^2}\right)$$

You can also solve this problem by considering the event from B. In B referenceframe the length of A's spaceship will be $L = L_0/\gamma(v)$, and since A is moving with a relative velocity v with respect to B, the time it will take to for B to traverse THAT length of A, will be

$$\Delta t' = \frac{L}{v} =\frac{L_0}{v\gamma(v)}$$