Thank you both for your responses.
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Originally Posted by Glom
You can't really bring everything to a full stop. There is no such thing. Relativity does away with it. If you mean full stop relative to Earth, this would be very difficult and stupid considering they'd begin falling directly to the centre of the planet. It's also unnecessary since as far as TD&E was concerned, speeds were slow anyway, because we are only concerned about speeds relative to objects involved in the procedure.
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That's pretty much what I thought. It's just that I've never heard anyone explain that before. When the missions were originally run, I was just a kid, and I have no recollection of reporters saying anything of the sort. So I filled in the rest on my own -- why wouldn't they stop? After all, you wouldn't change a car tire while you drive down an expressway at 60mph, would you? In my tiny little pea-brain, I figured that space was little different than a road. I then carried that assumption with me for some 35 years. It wasn't until I read a book about relativity that it occurred to me that stopping (relative to Earth) would be a really dumb idea.
But I just needed someone to tell me that no, they didn't actually stop.