Question About Invariant Interval in Special Relavitiy

[mmmboh]

This isn't a homework question, rather I believe I could use the invariant interval to check if my answers are right sometimes, but we are skipping this part in my class.

Anyway I did this problem where there were two trains, one of length L moving to the right with velocity v, and another of length 2L moving to the left with velocity v, and I had to find how long it took the trains to pass each other (so the time between the front of the trains meeting, and the back of the trains meeting), I had to find the time in each of the trains' frames, and the ground frame. Anyways I have my answers, but I would like to check to see if they are correct using the invariant interval. in c²t'²-x'²=c²t²-x², I believe the times are just how long it took the trains to pass each other in the respective frames, but I am confused on what I should use for the x's?
In the ground frame, the time I got for the trains to pass each other is 3L/2vγ, I guess I have to choose an origin, so when the front of the trains meet would obviously make the most sense. So would x for the ground frame just be (3L/2vγ)*v=3L/2γ? And what about x' of the train on the left let's say, in the left trains frame am I suppose to use -L for x' since that's the place the back of the trains meet in it's frame? and then in the right train's frame it would be -2L?

 

[jbsteele]

I'm sorry if this is confusing, I'm not very good at explaining things. But any help would be greatly appreciated!Yes, your reasoning is correct. In the ground frame, the x coordinate of the point at which the two trains pass each other is 3L/2γ. This is because it takes a time of 3L/2vγ for them to move a distance of 3L in the ground frame. In the left train's frame, the x' coordinate of the point at which the two trains pass each other is -L. This is because the left train has moved a distance of -L in its own frame during the same time that the two trains have moved a distance of 3L in the ground frame. Similarly, in the right train's frame, the x' coordinate of the point at which the two trains pass each other is -2L. This is because the right train has moved a distance of -2L in its own frame during the same time that the two trains have moved a distance of 3L in the ground frame. Now, you can plug all these values into the invariant interval equation: c2t'2-x'2=c2t2-x2. Let t=3L/2vγ, t'=3L/2vγ, x=3L/2γ and x'=-L (for the left train), and x'=-2L (for the right train). You should find that the equation holds true and that your answers are correct.