Two starships are initially at rest, starship B is 1km behind starship A?

Question

>==B==>. . . . . 1km. . . . . >==A==>

At time t=0 both ships inginte their motors.
Both ships are identical and their motors provide constant proper acceleration a=10m/s².

What is relative velocity of the starships (as seen by their captains), when they reach half speed of light v = c/2?

Note:
paradoxically the answer is not zero.

Please assume that starship B is not affected by exhaiust of starship A.

The proper distance between the ships is indeed increasing:
R = L/√(1-(v/c)²)

In proper time τ the ship A will gain extra proper velocity ∆v = aτ.

For observer at rest the new velocity will be
u = (c/2 + ∆v)/(1+ 1/2 ∆v/c) =
c(c + 2aτ)/(2c + aτ) ~
c/2 + 3/4 aτ

The distance will increase accordingly by
∆R = L∆(1/√(1-(v/c)²)) =
L/c² v/√(1-(v/c)²))³ * (3/4 at) =
L/√3c at


The realtive speed of the ships is
La/(√3c) = 19μm/s

Answer 1

I'm not well versed in general relativity, but I'll hazard a guess.

To Joe, an outside observer (one who's at rest with respect to ships' INITIAL (pre-launch) frame), both ships seem to have equal velocity when measured at any given instant T.

But the problem is, the "instant T" is really two DIFFERENT instants from the point of view of the two ships.

For example, say Joe sets up two stationary posts A and B 1km apart (way down the road). From his point of view, Ship A reaches post A at the same instant that Ship B reaches post B; and they're both going at speed c/2 at that moment. Each ship agrees that it's going c/2 relative to Joe at the moment it reaches its respective post.

But from the ships' points of view, the two posts are only 866 meters apart (due to length contraction). So from their points of view, ship "A" (the leading ship) reaches post "A" a little BEFORE ship "B" reaches post "B".

In other words, from the ships' points of view, Ship B had to travel 134 meters farther to achieve speed c/2 than Ship A did. This means Ship B perceives itself as traveling just a bit slower than Ship A.

[EDIT: On second thought, I don't like the above explanation. It seems that the proper time on Ship A when it reaches post A, must be the same as the proper time on Ship B when it reaches post B. Still, from the perspective of either ship, Ship A reaches its post "first" (because of the length contraction between the posts). This must mean that both ships perceive that Ship A's clock is running faster than Ship B's.]

Another way to look at it is this: From Joe's perspective the two ships are constantly separated by 1 km. But (also from Joe's perspective), each ship has shrunk by 13% due to length contraction.

When the ships use their shrunken rulers to measure the distance between each other, the distance looks like more than 1 km. So, they know they started out at 1 km apart; but now they perceive themselves as being 1.15 km apart. Their only explanation is that Ship A is "pulling away" from Ship B; in other words, they have a nonzero relative velocity.

As far as actual numbers, I don't know enough to do the calculation. But I will say that the ships perceive themselves as separating (i.e. "A" is going "faster").

Answer 2

A trick question.

Both rockets produce identical thrust but ship B is traveling through the exhaust gasses of A. That will slow it (B) down. How much slower cannot be determined without more data but A will attain half light speed sooner than B.

Your question is implies that both ships have identical acceleration but that's not the case. The question also implies the both ships reach the target speed at the same time -- also not the case.

Answer 3

Relative to each other? Since they are accelerating at a constant and they both began at rest, they, relatively, do not have velocities.

Answer 4

First, starships cannot be at rest in space;-}
you mean they are staying apart relative to each other.
Since 1km is insignificant to light-speed,
They will remain 'at rest' relative to one another.

Answer 5

relative to one another there is no velocity because they are going at the same speed.

Answer 6

Relative to each other? 0 m/sec

Answer 7

zero