Two rockets, A and B each of proper length Lo, approach each other along the x axis with speed v. their noses cross at Ta = Tb = 0. A man with a gun at the tail of rocket A is aiming perpendicular to the relative motion, and shoots at the very moment he see's the nose of A cross the tail of B. What happens ? Does he, or does he not, hit B ? (knowing that in A's reference he missed because B is contracted, and in B he hits because A is contracted)
The problem also states that you can neglect the time of travel of the bullet, and the time of communication between the nose and the tail of A.
The Attempt at a Solution
Ok so I got this question in my relativity test yesterday as a variation of the barn-pole paradox, and I guess I got thinking a bit too far but to me it seems as though you cannot neglect the time of communication between the nose and the tail of A. (Although you can neglect the time of travel of the bullet since this could all be happening in 1D). My reasoning being that it actually changes the outcome of if the bullet hits or not.
As I read the problem, there is clearly a causal relationship between the events 1: ''Tail B crosses nose of A'' and 2: ''A shoots'' and these events are related by a time-like space, ie there exists no frame of reference where event 2 happens after event 1. Therefore, B cannot observe A shoot before his tail crosses A's nose.
This is the point where I started thinking that the time of communication is crucially important since it leads to different outcomes, and A will hit or not hit B will depend on the speed of B, since A will miss B if the light from A's tail arrives before B's nose, and he will hit if B's nose arrives before the light from A's tail. The critical speed being at 1/sqroot(2) (at this speed, B's nose arrives exactly ay the same time as the light).
Finally, the other problem I have thinking about this is that for A, if he misses, he will shoot in front of B whereas for B, if A misses he will have shot behind B. Should I have a problem with this ? Can I still give a coherent answer to this question while neglecting the time of communication ?