Two spaceships pass each other/fire rocket

[Leb]

Homework Statement

This is not exactly a homework question. Special relativity question.

See attach. 1.
Two spaceships are passing each other, having their fronts marked F and F'. When F' sees the other's rear passing the front of F', F' fires a rocket. Clearly it misses, since the F SS is contracted. But in the frame of F, F' is contracted and thus, the rocket fire will hit F.

What will happen ?

(I am not sure on the speed of the rocket... I assume it must be near c)

Homework Equations

Lorentz transforms, various four vectors...

The Attempt at a Solution

The only way this could happen in my opinion is if the rockets were at an angle to each other (see attach. 2). That is, if the contracted spaceship covered distance AC in the same time the rocket traveled distance AB.

I have never encountered rotations in SR, only in GR (maybe it's only because of the course specifics in my uni).

But spaceships being at an angle seems to be wrong, because of the way ships see each other. That is being parallel, not at some angle...

 

[Quinzio]

Aha, you have entered the magic world of special relativity, where thing are not what they seem.
There's no angle between the ship. The reason is simultaneity.
SR basically says two thing:
1 You can't go faster than light (and the more you want to reach it, the more energy you need).
2 I don't care in which inertial frame you are

No.1 can be digested in some ways. No.2 hurts more.
In no.2 Einstein says that in your inertial frame you have to stretch and deform your space time to make things happen as in mine. That is, if F' doesn't hits F with the rocket, the same must happen in all frames.

All the rest in SR is a consequence of this two basic facts.
One weird consequence is that what is simultaneous for me is not simultaneous in your inertial frame.
Hard to digest, but true.
In F inertial frame, the rocket was fired "before". Before what ? Before it was fired in F' time.
In F' frame, the crossing of F back and the rocket fire was almost simultaneous, but it took a finite time.
An electrical command traveled at speed light from front to back and fired the rocket. Nano seconds.
Not for F. For F it was even faster and so the rocket was fired before F reached the back of F'.

 

[Leb]

Thank you for your reply, but I cannot get my head around this problem. I am confused about length contraction and time dilation. Does F' know, that time for F runs slow ? Or does F know that he will actually cover some shorter distance, than the 'actual' (I suppose actual distance means the length of F' as measured in its(F''s) stationary frame.

Because when I try to approach this problem by using time dilation argument here is what figure out:
Assume in the frame where F' is stationary, it takes the electrical signal 10 time increments to reach the back of F' and launch the rocket. Now, since F cannot travel faster than light, there is no way (as viewed from stationary F') that he can move past the whole length of F'. So the only option for the ship F to avoid being hit, is to 'have a slow running clock'. Assume that only 5 time increments have passed for F. That F's front will not reach F''s rear (he could be as close as you like, but will never ever reach it in time).

As viewed from F's frame (sorry about the '''s should have just taken A and B...), the signal will seem to reach the rear of F' faster than the speed of light, but again, assuming we know about time dilation, the 10 time increments in F''s frame will become 20 time increments, enough time for F to escape being hit.

So maybe this is correct (assuming we know about time dilation) ?
But what about explaining this in terms of length contraction ? How do I 'convert' time dilation into length contraction ? I mean, I always remember the standard text-book example about muon, and it always states, that the in muon's rest frame, the distance between atmosphere and the Earth is contracted (because the Earth is moving towards the muon), so he will need less time to cover that distance than expected by an observer on the ground (or should I say, for the Earth to cover the distance ?) I could not find an analogy here, since I do not really understand what contracts here. F' ? Then, forgetting about time dilation, the signal would reach the rear even faster and would hit F...

EDIT

OK, so I have done some reading and found that ''effect due to dilation in one frame may instead be attributed to length contraction as measured in another frame''. So there is 'symmetry' of the two... Now I just need to figure this out for the two spaceship example above...