I don't see what this has to do with my scenario, which doesn't even require a 2nd ship. The crux of my scenario is that the laser beams route is twice as long (due to mirrors and zig zag bouncing) as the path of the .51C speed spaceship.
Woops, I misread the quote that I was responding to. Sorry about that. I think you're right. But I think it's interesting to take another look at why you're right.
I think a good way to think about this is to still look at it as though your ship is stationary. In this example, it is the mirrors that are moving past you at .51c. Let's say the mirrors are 1 light second (300,000km) apart.
Here's a diagram of what it looks like in the mirror's reference frame, if I were to use the example as you give it:
light:
.......^..........^
...../...\......../
..../.....\....../
.../.......\..../
............\../
..............v
You:
----------------->
My problem is that there are two dimensions of movement, and I can't easily figure out how to solve the problem that way. So, I think works out the same if we look at it with only one dimension, but where it happens like this:
(blue dashes are where the light travelled only once, red coloured dashes are places where the light has to cover that distance twice. Black dashes are just for formatting and should be ignored. Again this is from the mirrors' reference frame)
light:
----------\
----|
----------\
---------|
----------\
You:
--------------------->
The dashes are mirrors that are only very very slighlty tilted (so little that I don't have to take it into account), the straight lines are mirrors that direct the light perpendicular to your line of movement (from the mirror's reference frame of course). The light goes forward 300,000 km, then is reflected back 150,000km, where it hits a second mirror, turning it back for 300,000km again, etc. This will still double it's path, and thus allow you to outrun it.
But all of that is from the mirror's reference frame. What does it look like from yours?
Now, I think I've got this right, from your perspective what happens is that the path of the light beam is distorted. You are stationary, and a set-up of light emiter and mirrors is moving toward you at .51c. When the light emiter passes you, it emits a burst of light.
The light moves away from you at c.
After going something less than 300,000km, it hits a mirror which is moving toward it at .51c. The reason that it didn't have to travel 300,000km is that the mirror covered part of that distance in that time.
Now it moves back toward you at c. It travels back more than 150,000km where it hits another mirror. The reason that it travelled more than 150,000km is that the mirror moved away from it during that time.
Here's the important part - if you do the math, it will turn out that
it traveled back more than the "something less than 300,000km" that it had already travelled. In fact, it will have passed by your ship.
Now it travels toward you again, at c. This time,
before it reaches you, the next mirror which has been wizzing up toward it intersects it, again after it's gone something less than 300,000km. It starts traveling away from you again at c.
As you can see, even though in your reference frame, you aren't moving, the light will be continuing to get further and further away from you, even though it is aways travelling at c.
I think that with your original zig-zag set-up it works out the same because the angle that the light is reflected will be changed. But I'm not quite sure. Can anyone enlighten me about this?
Anyway, I mostly just did all that as a fun exercise, but I do think it helps a lot to ask yourself "what reference frame am I talking about?"
