I don’t think it was a time machine, but rather, a machine that caused things to stop moving, absolutely. There were some dreadful technical gaffes in the story (besides the whole absolute motion thing)
The story is called The Billiard Ball.
Also wrong, Mangetout… the machine was billed as an ‘anti-gravity device’ 
With the explanation being something along the lines of - gravity is a function of the curvature of space-time. To completely negate the effects of gravity, you need to ‘stiffen’ a specific patch of space-time so that it has no curvature. This apparently has the effect of making travel at speeds less than C impossible, in the story.
The ball enters the stiffened anti-gravity zone, instantly accelerates to lightspeed, (taking a shortcut through the law of conservation of mass-energy along the way )and then slows down slightly from friction as it leaves the zone, but still rocketing along at at least .999c
So nothing to do with time machines or the motion of the Earth, really. Fun story nevertheless.
(Waits to see if someone else will attempt to top him as the uber-Asimov nerd)
It seems to require the concept of an absolute location in space, which, it is my understanding, does not exist. That’s why I think it’s impossible.
I’m sure you’re right. Thanks.
I am right about the science gaffes in the story, though, am I not? - for example, a large object such as the ball entering the field entirely, instantaneously - if the device actually existed, the ball would be torn to dust as it entered gradually.
(come to think of it, doesn’t Asimov acknowledge this problem in some commentary, stating he just decided to let it slide?)
Asimov mentioned this point, in passing, in his short story “Blank!”
It’s not too scientifically rigorous, I admit. For that one, I’m not sure, (though I may be fanwanking) that it would make any difference in the scenario if the billiard ball maintained any kind of structural cohesion or was torn apart, or if the observers would even be able to tell - as long as all of its mass entered the field and enough of a strong majority of the high-speed dust particles exited travelling in the same direction. (The notion of it being a cone-shaped blast killing and wounding several other observers as well as the targeted murder man would be an interesting twist ending.) 
The one I already wondered about - there had been similar experiments in ‘reduced gravity’ before the denouement of the story, with multiple labs competing to get the lowest level, and these had clearly been about partially reducing the curvature of local space-time, not just balancing gravity out with an equal and opposite force. Did none of these researchers notice odd effects on momentum as objects entered and left a reduced-curvature field, or energy flickers? Or did those side-effects really only snap into being as curvature hit the zero mark?
Oh, I’m sorry, I misunderstood the scenario you were describing. I thought you stayed on the platform while the car moved away from you. But apparently, you mean that you move along with car 19, and become further from the paltform.
In this case, alright, I use “stationary relative to” the same way you do.
So now my question is: why would you be surprised that the time machine remains stationary relative to the Earth? The rules of how time machines are meant to work haven’t been set, of course, but, hey, most things on the Earth do remain stationary relative to it…
Ah, there we are. I conflated that and “Billiard Ball”, I guess. Still, I knew good old Uncle Isaac hadn’t ignored the issue entirely!
2 Geek demerits to me, I suppose. 
Actually, I think you are misunderstanding Shawn1767 now. An object being “stationary relative to” another object means that the two objects remain the same distance apart, no matter what else is going on. A person sitting in a car is stationary to it, and their motion relative to the earth is changes together. I think Shawn1767 means the opposite.
We’ve actually had this question before. Not as many replies, though.
It would depend on the type of time travel.
If we’re going to do time travel, in a purely science fiction way - of course - I say we do it inside of a machine that stays stationary relative to itself, and that the traveling object(s) travel from one point in time WITHIN the machine to another point in time WITHIN the machine.
Okay, a little clarification. In my first scenario about the train. I was thinking that “stationary relative to the train” meant that I was not moving and that the train was moving away. Perhaps I am not correct on that meaning. I meant that I would stay in the same place while the train moved away and the distance between us increased.
In my second scenario, I was IN the train and the space between us was not increasing. Is that what stationary relative to the train means? Then what does “moving” relative to the train mean?
Hm, I think J Cubed is right; I understood your scenario correctly the first time, not the second time.
Ok, the point is, “not moving” is also only a relative term. There’s no absolute sense in which you can say “I was not moving”. You can only say “I was not moving relative to X”. For example, “I was not moving relative to the platform and the train station and the ground around it. However, I was moving relative to the train.”
Given two objects, you can ask whether they are getting closer together or further apart or such things; however, one object in isolation cannot be described as “moving” or “not moving” in any absolute sense.
To say that I am stationary relative to X means, among other things, that my distance from X remains constant. It means that, from the perspective of someone who was stuck to X, I don’t look like I’m moving. It means that, in a frame of reference where X isn’t moving, neither am I.
If relativity is correct there is not space and time; there is just space-time a four dimensional space. Time travel would involve moving from some coordinate in space time to another so you would travel through both time and space generally – perhaps by somehow magically swapping your ability to maneuver freely along some space axis for the ability to move along the time axis.
You might look up Larry Niven’s “The Theory and Practice of Time Travel” and while your at it “The Theory and Practice of Teleportation” for one SF writers attempt to put physics into these subject.
In one of his editorials for his magazine, Asimov discussed time travel problems. An aspect of which describes the OP’s conundrum. If you’re going to be traveling back in time from today’s date (1/6/09 [or 6/1/09 for the non-Americans out there]) to, say, 7/4/1776, then you’ve got to have some kind of mechanism to track the Earth, since its in an entirely different part of its orbit at the time in which you arrive. Also, the Earth’s orbit has shifted slightly over the centuries, (not to mention rotation speeds have changed), so depending upon how far you plan on travelling in time, you’ll have to make adjustments for that, as well. Finally, the position of the solar system itself changes as our arm of the galaxy rotates around the giant black hole at its center. Go too far in time in one direction or the other, and you’ll find yourself not merely off the Earth, but in the remote blackness of space itself.
Asimov concludes his piece by saying none of that should really matter, if its a damned good story (or words to that effect).
Yes, Tuckerfan, that is what I was trying to describe.
Tuckerfan, all of that is moot if you time travel within a machine, rather than use a machine to transport you and the machine through time.
This would, of course, be limited in the time you can travel from the moment the machine was built to the moment the machine was taken offline.
Trocrisp, so, like the Guardian of Forever in Star Trek (a portal) would be different from a Delorean?
I don’t recall the specifics of the Guardian of Forever in ST, it’s been forever since I’ve seen the episode (8yearsish), but yes.
If you’re inside of a ship that travels through time, you would presumably (without accounting for Stellar-shift) come out somewhere far away from your original location.
If you walk into a room, and walk out of that same room, only whats inside of that room has changed time. Everything around it has stayed (relatively) stationary. Of course this would require a physics mastery well beyond our current ability, but if we’re going Sci Fi, lets go all the way baby!
Ok, correct me if I’m wrong here, but I was under the impression that while linear motion is relative, rotational movement is not. So, completely setting aside the issue of whether or not the earth, and the whole universe are moving linearly, we can say, with certainty, that the earth will be in a different part of its orbit, and the solar system will be in a different part of its orbit, after such-and-such a time?
Ignoring the motion of the milky way galaxy itself, it’s theoretically (and I only say theoretically because I’m not 100%, I’m 99.9%) possible to take into account all of the variables to pinpoint where the Earth, Sun and the rest of the solar system will be in any given amount of time. Simple extrapolation.