So there is this very simple situation in one of my exercices:
In the earth's frame of reference a tree is at the origin and a pole is at $x=20$km. Lightning strikes at both the tree and the pole at $t=10$ microseconds. The lightning strikes are observed by a rocket traveling in the positive x-direction at $0.5c$.
1) At what time does the lighting strikes take place in the rocket's reference frame?.
I understand the concepts of time dilation, length contraction and etc, but the questions bring me confusion sometimes because they are not very well formulated in my sense. In this exercise I have difficulty understanding what do they really actually mean by 'the time in the rocket's reference frame.'
First it could mean that in the earth's reference frame what is the dilated time that an observer in A (earth) would measure for B (spaceship). An analogy could be that an observer in A measures that it takes his twin 16 years (dilated time) to age by the proper time of 8 years. So the proper question would be what is the dilated time $(t')$ that observer A measures, if it follows correctly the analogy. So we stay in Earth's reference frame and we are only measuring $t'$ as measured by an observer A (and not the time that take place in the rockets frame of reference which is different is my sense as explained below.)
Now a second meaning could be what is the proper time that someone traveling in the spaceship in his OWN frame of reference measures. Following the analogy, the time that it takes someone to go back to earth in the spaceship is 8 years because he measures his own proper time (which is different from the dilated time measured by an observer A on earth).
So when we use the equation $t'= \gamma(t-vx/c^2)$ or the one for position what do we really mean by $t'$? What I think is it is $t'$(dilated time) as measured in frame A because that is what we do in time dilation for example: When the twin measures proper time 8 and gamma factor 2 so $t'=16$ but here we are still measuring dilated time of B IN Earth's frame of reference A and not proper time in the spaceship reference frame B.
So here is my confusion. Does in the question they just 1) what they really mean is at what time does the lightining take place in spaceship B as measured by frame A.
So how do I get over this confusion?