Special relativity and simultaneity: Spacecraft and Target

[LCSphysicist]

Homework Statement

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Relevant Equations

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I made an exercise whose question is like that:

A spacecraft moves with speed v at the axis x. There are two target at the axis x, in which the distance between them is L at the ground frame. Suppose that the spacecraft shoot the target simultaneously when it passes by the target. A|so, neglect the time between the spacecraft shoot and the hit of the target. Is there a frame in which both targets are hitted simultaneously? If yes, what is the velocity of the frame?

Now, i have found an answer that i am 99% sure is wrong, but just want to make sure and, also, would like to know the speed of this frame for both hits simultaneously:

I have decided to name two events:
Event A: Spacecraft passes the first target/hit the first target.
Event B: Spacecraft passes the second target/hit the second target.

Using lorentz transformation, $$\Delta t' = \gamma ' (\Delta t - \beta' \Delta x) = \gamma ' (\Delta t - \beta' v \Delta t) = 0$$
$$\implies$$
$$1 - v'v = 0, v<1 \implies v'>1$$

So there are no frame in which the target are hitted simultaneously.

I am aware of the lost of simultaneously, so i am sure i am wrong because probably in the frame S' the missle is not fired and wouldn't hit the target at the same time the spacecraft passes the target.

But the problem i am having is to understand how could i find the velocity of S' without i know the missle speed? Someone can help me?

 

[anuttarasammyak]

Homework Statement:: .
Relevant Equations:: .

So there are no frame in which the target are hitted simultaneously.

Surely. If there were such a frame, there must be two same rockets in that frame.

 

[italicus]

Homework Statement:: .
Relevant Equations:: .

Suppose that the spacecraft shoot the target simultaneously when it passes by the target. A|so, neglect the time between the spacecraft shoot and the hit of the target. Is there a frame in which both targets are hitted simultaneously?

This supposition is wrong. The proper distance between the targets is L; the contracted distance, wrt the spacecraft , is$L_c=L/\gamma$, thus it takes the proper time $\tau={L_c/v}$ for the spacecraft to go from the first to the second target.