# A paradox related to relative motion [duplicate]

I know that it is a very old question but still I don't find any satisfactory solution for Achilles Paradox. Please explain me the fundamentals of Achilles paradox in terms of stage wise distance covered. Note that it is easily solvable in terms of time, but if you start analysing this event in terms of time, then there is not at all any paradox. So please explain in terms of stage wise distances only.

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## marked as duplicate by John Rennie, Stan Liou, Alfred Centauri, Danu, jinaweeMay 29 at 11:45

Why do you feel there is no paradox for time, but there is one for distance? It is much the same apparent paradox either way. –  mmesser314 May 29 at 10:21
There must be a thousand articles on Zeno's paradoxes out there in Googlespace. Can you expand your answer to explain what you do and don't understand about it. Without this detail we might as well copy and paste from one of the existing articles. –  John Rennie May 29 at 10:27
How is this related to relative motion in any way whatsoever? –  Stan Liou May 29 at 10:31
This question is just a copy/paste of another question: physics.stackexchange.com/questions/98459/the-achilles-paradox –  mpv May 29 at 10:40

Time is directly connected to distance as far as movement is concerned. Actually, movement (a change of distance) is measured with time, and there is no movement without time (or time without movement), unless one is able to travel with infinite velocity. So you can always translate any distance into respective time according to the formula:

$v=x/t$

This solves your paradox. Achilles does not travel all the sub-distances traveled by the tortoise one-by-one in consecutive units of time; he travels a number of them at a time, and that's what allows him to catch up the tortoise. They travel different numbers of distance units within the same time unit.

Take a look at this:

"During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise."

See how the word "time" is used ambiguously here? Is it the same period of time or a different one that take Achilles and tortoise to travel the sub-distances? When Achilles is finally close enough to tortoise, but still behind him, in the next time unit he will travel as much as the tortoise in the same time and then some more. In one time unit Achilles can travel the distance between them, and also overtake the tortoise. He does not need to walk with very short steps and follow the footprints of the tortoise (which is suggested by the way the paradox is formulated).

And also, you cannot talk about traveling a distance (movement) without considering time. As I said - there is no distance change without time. One requires the other.

And notice that the paradox is actually stated in terms of time. You cannot avoid it. Especially that the whole paradox is about racing :)

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