Ok let's take your simplest diagram first.
B1 reaches planet x in 4.35s of its time and 10s earth time where 5.65s of earth's time is the alignment of B1's perspective of the end of the race and the earth's perspective of the end of the race due to relativity of simultaneity.
Oh I see what you're saying in the 2nd diagram. The earth's velocity gives B1 a push of .9c in the vertical direction while the thrusters give B1 a horizontal direction of .9c. This will seemingly make B1 follow a longer diagonal path to planet x. The problem with this reasoning is the thrusters give B1 a relative velocity of .9c to both the earth and x but what is earth's and planet x's vertical velocity relative to? Absolutely nothing, literally because you can't have any velocity relative to nothing (the vacuum of space). If you shot a vacuum bottle into space, light from inside the bottle would not go any faster than light outside the bottle, it would be like there was no bottle. So far as B1, earth and x are concerned, there is no longer diagonal path because they are all part of the same inertial frame and that inertial frame's velocity is irrelevant to any outside inertial frame. You, as an outside observer, are imposing your perspective on this scenario. Your question should be restated as to what are the times, velocities and distances relative to you, not to earth. You are outside this inertial frame looking down upon it on a computer screen.
I should note at this point that relativity says that B1's velocity shortens the path it must travel from earth to x and it's velocity also makes its clock run slower as compared to earth's clock. I, of course, do not accept this explanation. The path doesn't shorten, the clocks' rates are the same and what causes earth's time to be longer is the relativity of simultaneity. The ship is not separated from the start and end points so its clock does not suffer the same time penalty as the earth's clock so it can traverse the full, uncontracted distance in its own unpenalized time.
Let's make B1 a horizontal beam of light from earth to planet x. Light, being a wave, can't be pushed upward by the movement of its source which is the earth at .9c. Once launched from its source, light's speed, being an electromagnetic wave, is determined by the permittivity and permeability of the electromagnetic medium it moves through which is space in this case. Within the inertial earth/x frame, the light travels straight from earth to planet x but from your outside perspective, you'd expect to see the light miss planet x because x would have moved out of the way. This can't happen because perspective, which is illusion, can't alter reality. If the light hits planet x in one inertial frame, it must hit planet x in all observing frames. You must see the light beam take a longer diagonal path to x and hence it must take more of your time to do that in order to keep c constant from all perspectives. So your perspective creates this illusion of a longer diagonal distance requiring more of your time to traverse whereas for light B1, the distance is the much shorter proper distance and takes much less of the ship's proper time to keep c constant. Perspective can't alter reality so I conclude perspective is illusion but Einstein concluded perspective must be reality.
I don't use this explanation, I use relativity of simultaneity and the universality of proper time rate for all inertial frames. Relativity of simultaneity is part of relativity and it is used to explain how length contraction truly works, yet time dilation is the flip side of length contraction and no one uses relativity of simultaneity to explain how time dilation truly works. This is absolute madness to me. Instead Einstein came up with this absurd idea that perspective is reality.
Now let's say the ship B1 goes so near c that we can consider it to be c. Unlike light, its velocity is not dependent on a medium so it gets a push in the vertical direction at .9c from earth and it goes at c horizontally due to its thrusters. So it seemingly travels a longer diagonal path to x than it did when earth/planet x weren't "moving" (ask yourself moving relative to what; space?). So the ship, which is matter, traces out the same diagonal path, to you on the outside, as the light beam did. From the earth's perspective it's a different story. It's the story and Minkowski diagram I gave you yesterday. Unfortunately almost no one knows how to read a Minkowski diagram. Actually, I'd have to think really hard on how to draw the Minkowski diagram of the earth/x/B1 frame from your outside perspective. You're an observer but there are no real velocities relative to you as you're observing a computer screen. It's more like you're observing closing speeds. The story would be quite different if I made you the reference frame and drew earth/x speeding away from you and B1 speeding from earth to x. I'd have to think about that.
Edited by ralfcis, 13 March 2020 - 11:38 AM.