Two rockets are sent off at t=0, one from x=0 and the other at x=4. The rocket leaving from x=0 is moving at .8c and the rocket leaving x=4 is moving at .2c. When the paths of the two rockets meet, they send a light signal to x=0. Read off the coordinates in the S frame and in the S' frame and check to see that the space and time differences between events 3 and 4 satisfy the invariant rule. Event 3 is the light signal being sent out and event 4 is the light signal arriving at x=0.
The S' frame is moving at .6c.
x' = [tex]\gamma[/tex](x-vt)
t' = [tex]\gamma[/tex](t - vx/c^2)
invariant rule: (t4 - t3)^2 - (x4 - x3)^2 = (t'4 - t'3)^2 - (x'4 - x'3)^2
The Attempt at a Solution
So after drawing all world lines, I came up with the coordinates (3.5, 2.75) for event 3 and (0, 6.3) for event 4 in the S frame by looking at the graph. I am confident in these coordinates.
In the S' frame, I came up with (2.8, 1.1) for event 3 and (-6, 9.8) for event 4 in the S' frame. This is where I think there may be a mistake. These are just based off reading the graph, so they are approximate.
Now when I check to see if it satisfies the invariant rule,
(6.3-2.75)^2 - (0-3.5)^2 = (9.8-1.1)^2 - (-6 - 2.8)^2
.3525 = -1.75
Clearly this is not correct. I understand there will be some error since I am just eyeballing the coordinates from the graph, but this seems way off. Does anybody see where I went wrong?