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Just to link it up for Moontanman, the bits I bolded above are worked examples (reverse order) of what I described in the 3rd post:

It was, because otherwise one rocket's relative speed to the other one is equal and opposite to the converse, no matter what relativity principle you use (Galilean or Einsteinian). It's exactly as Swansont said with 0.5, 0.5, giving 0.8 (in units of c) It's perhaps an illuminating exercise to do it with 0.99999 and 0.99999. It gives (0.99999+0.99999)/(1+0.99999*0.99999) = 0.9999999999 (in units of c) which is practically just c. But, and here's what interesting, with small velocities as compared to c. 0.00001, 0.00001, it gives (0.00001+0.00001)/(1+0.00001*0.00001)=0.00002000000000 which is so close to the simple addition of velocities that nobody could tell the difference. That's why our intuition tells us velocities are additive.

Fantastic indeed. Thank you. The Suite muffled by the voices was great for setting the mood. I've long felt that all music gravitates towards Bach... or emanates from it. Or something like that. I feel that music before Bach is a preamble to Bach. And music after Bach is a corollary to Bach. Even atonal music seems like an attempt to break the shackles of Bach while still doing music. Like 'how little Bach can one get without making just noise?' I'm very partial about Bach, you see. I'm very Bach-centred. So thank you.

RIP Dickey Betts

It's always about who is measuring what relative to what.

Thank you, exactly what I was wondering but I didn't realize the station was necessary.

For Moontanman: in the above, that 0.5c is as measured by the space station, and each rocket to the station; and the 0.8c is as measured by each rocket to the other rocket. For the space station, the closing speed of the rockets (which is not the speed of either rocket) is c.

If two rockets each approached the space station at 0.5c, from opposite directions, they would be approaching each other at 0.8c (.5 + .5)/(1 + 0.5*0.5) (Galilean addition would give you c)

If you don't summon any other observer, then it's 0.99999c relative to each other, as Swansont and others said or implied, and/or/thus I'm missing the point. / / 🤷‍♂️

Just come across this Flashmob video from Lausanne, in the course of revising the bass line in my favourite chorus from Bach's St. John Passion: The cellist plays part of one of Bach's Cello Suites and ends on a (baroque pitch) G, from which the basses can get the C they need to start the fugue. The conductor pretends to be a waiter delivering beer to the next table, until the moment arrives. They sing it very well, especially given the acoustics of a busy restaurant. Pretty cool, I thought. This chorus is in 3/4, with real JSB swing, syncopation and drive. Fantastic music.

Impartial observer... not what I had in mind, I was just thinking of the two rockets approaching each other and how they would see each other. The relative speeds of the cars was what I was getting at, I really didn't realize I wasn't being clear on that. I understand there is no track to measure anything by in relativity.

No, you've got it, that's the point. You obviously don't add the velocities, as you would in classical dynamics.

I think you really need to get clear on the difference between measuring closing speed (which is the speed of a gap, a nothing), and the speed of a thing. And it depends a lot on who is observing. Say the fastest car in the World can do 500 km/h. Stick two of them on a track facing each other and run them, at top speed. The gap between (from the point of view of the track) them is decreasing at 1000 km/h. Hang on! That's faster than the car can go! But that 1000 km/h isn't the speed of either car according to the track. Sure, if the track considers it from the point of view of one of the cars, then the other is getting closer at 1000 km/h, but that reference point is moving according to the track. It's an illusion, if you like. Relativistic addition comes in (in this scenario) when you consider the point of view from one of the cars. Each car can consider itself as still, and the other car moving towards it. But note that the track is also moving towards it! The track is moving towards each car at 500 k/h, and the other car is moving towards it at 500 km/h relative to the track. And that's where you cannot (at relativistic speeds where it starts to matter) just add the 500 and 500. Each car will consider the other car approaching at 999.99999 km/h. The track considers the closing speed as 1000 km/h, and the cars consider the other is approaching at 999.99999 km/h. These are different numbers.

Moon, I think you're trying to think of an "impartial" observer who's sitting on some dock of the aeronautical bay, so to speak, and watches both approaching each other at 0.99999c. Then they would see each other approaching at higher than that, but never c or higher. You must run the Einstein velocity transformation formulas that to see how much. The relative speed from their POV would indeed be closer to c than that 0.9999c (or however many pieces of c she/he sees them from the dock. If you actually run the calculations, you'd find, I don't know, something like 0.999999999999999999999999999999c relative to each other (you actually must run the calculation if you want to know how many 9 digits closer to c). I think you're implicitly thinking of this "impartial observer" but failing to say so, and causing some amount of understandable confusion. Is that so? Does that help?

The observer is on each rocket, the rocket judges its own speed by its departure point, from my point of view it looks like the combined speeds would be greater than c, I understand this is not possible except in my frame of reference but what would the two space craft measure each others speed as?

You are confusing yourself by being imprecise. You cannot make your mind up on whose point of view is involved. Who is the observer, one of the rockets? Or someone else? (And: As noted before, someone else (not in the rockets) can consider the rockets as approaching each other at more than c, closing speed can be as much as 2c.)

My thoughts on this is that the rockets would measure a combined speed but still less than c . Out side or before relativity the two space craft would have been thought to approach each other as faster than c but relativity precludes that so I thought they would see each other approaching at a speed faster than .9999 c... something like .9999999 c or something closer to c but not faster than c.

No they can measure speed relative to themselves, as in your first post. In your first post, rocket A (to give it a name) measures rocket B to be coming at 0.99999c. And vice versa.

Ok, so they couldn't measure their relative speeds without the station to compare it to? I am trying guys

The existence of the space station doesn't affect anything. But if that's what speeds are being measured against, it changes your scenario. You started off saying each rocket measures the other as going some speed. Well then that's just what they measure. But if you change the scenario so that each rocket measures some speed from themselves to the station, and the other rocket as at some speed relative to the station, then yes, each rocket needs to use the proper formula to determine the speed of the other rocket relative to themselves.

Evidently I am too slow to get this one. I understand the two space craft would be approaching the space station at .9999 c but what would they measure each others speed at?

There are some attempts in that direction, mostly targeted at toxic VOCs. From what I recall, it seems that likely microbes are responsible for oxidizing some of the VOCs and breaking them down . But I don't think that algae were shown to do that. Conversely, I vaguely remember that some algae actually release VOCs (though I cannot recall whether those were in any form harmful). The provided link paints a very poor picture of the capabilities of the company, considering they are conflating CO2 capture with capture of harmful substances (via photosynthesis, no less). Failing that much at basic biochemistry does not inspire confidence. The blurb also seem to suggest that this is just an exhibit, likely putting some algae (or even just a green liquid) into a stand. A real bioreactor for cyanobacteria or algae needs quite a bit more to work. And randomly growing cyanobacteria can also produce toxic microcystins. So there is also that. From what I remember the carbon yield (for fuel or plastic production) was also rather low. I am also skeptical that oxygen production from those volumes would be significant, but I may be wrong.

OK I give up.

Would the space station influence the speed the two space craft measure on each other?

You can only define a speed relative to some other object. So far the only objects you have mentioned are the two spacecraft. Is 99.999% of c measured relative to the other spacecraft or to something else? If the former, you have already stated the answer. If the latter you need to say what that something is. You could for example say each space craft approaches the same space station, from diametrically opposite directions, at 99.999% of c relative to the space station.