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Starship technology thoughts

We don't get any time indication for that trip, other than 1 min 40 s remaining after the Jenolan's systems start to fail and after LaForge has managed to convey this to the E-D.
But at this time they're not yet in transporter range, which puts a lower limit of their distance at 40,000km. Assuming the entered transporter range a second later (and everyone forgot to mention this to Picard). This puts the lower limit of Enterprise' velocity at this point to be 100km/s, at which speed the flight from the star to the hatch would have taken almost ten days. Even more problematically, it means Enterprise would have been flying for still another one minute and forty seconds AFTER beaming Geordi and Scotty off the Jenolan. If you assume transporter range was reduced for some reason, then Enterprise' speed becomes lower, thus the trip out of the sphere would have taken even LONGER; the missing time between the moment the Jenolen wedges the door open and "The plasma intercooler's gone!" would be something like two weeks.

The real issue is that from the time Enterprise enters transporter range to the time it leaves the sphere is, IIRC, about fifteen seconds. Enterprise would have been traveling about 1.3C at the time, which would put its total flight time slightly more than five minutes, which seems to be more or less consistent with how the scene was filmed: at that speed the scene gap is reduced to something like two or three minutes between "Ensign, Set a course!" and "The plasma intercooler's gone!" Any much longer than that and the progression becomes kinda weird.

Probably much more important is the Jenolan signaling the door from a distance of 500,000km and yet still being able to fly in there and jam itself into the opening before the doors closed. If the ship is quick enough to do that--cheating the known laws of physics all the way--then scifi-movie physics probably applies and impulse engines can very well accelerate to ANY speed, even FTL speeds, as long as you've got enough runway.
 
But at this time they're not yet in transporter range, which puts a lower limit of their distance at 40,000km.
We don't really know the range of TNG transporters. In "Tin Man" and perhaps "Schitzoid Man", several orders of magnitude might have to be added to that; in various adverse conditions, even a hundred kilometers might be too much. And being confined in the same space with an angry star would probably count as "adverse", even if one ignores the problematique of having to penetrate the Jenolan's shields.

If you assume transporter range was reduced for some reason, then Enterprise' speed becomes lower
Why? The 1 min 40 s is not an indication of transporter range in any way. It's merely a timespan for covering a distance that's markedly greater than transporter range, and shouldn't pose any limitations on our analysis at all. It's perfectly possible the ship took, say, seven minutes to cover the entire 100,000,000 kilometers (0.8c), of which 24,000,000 kilometers were remaining when the Jenolan began to fail, and the threshold of 40,000 or less was approaching (Data would announce it suitably in advance) when Picard ordered transporters activated and torpedoes launched.

It's also rather likely that the helm reduced speed on her own volition (or even by Picard's earlier, cut-for-time command) at the last few seconds to achieve these difficult tactical goals. That's not just a timekeeping issue, but also a means of explaining the non-relativistic visuals.

On a more general note, this "Relics" sequence is an action/chase sequence - one where multiple cuts for time could and should be expected. We didn't see even a hundred seconds' worth of action, certainly...

Probably much more important is the Jenolan signaling the door from a distance of 500,000km and yet still being able to fly in there and jam itself into the opening before the doors closed.
True. Then again, we've got some pretty humongous doors there, despite the usual VFX conceits. From "starting to close" to "a gap a few transport ship lengths wide" might realistically take quite a few seconds. And who knows, perhaps the doors even hesitated when they saw a starship accelerating towards them after all? ;)

But we're still talking about multi-thousand-gee accelerations. The availability of which as such isn't a problem, because no TNG plot has really hinged on lack of acceleration. Even in ST:NEM, Picard did manage to ram his enemy before any evasive moves could be made, so perhaps he simply opted not to do 125,000 gee because he knew the outcome would be guaranteed even at half a gee? (He didn't want to obliterate the Scimitar at the cost of his own life - he wanted a survivable crash, with a possible grand finale of evacuation followed by scuttling explosion if the ramming alone didn't stop Shinzon.)

Timo Saloniemi
 
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