"You can never catch up"
In the paradox of Achilles and the tortoise,
we imagine the Greek hero Achilles in a footrace with the plodding reptile.
Because he is so fast a runner, Achilles graciously allows the tortoise a
head start of a hundred feet. If we suppose that each racer starts running
at some constant speed (one very fast and one very slow), then after some
Achilles will have run a hundred feet, bringing him to the tortoise's starting
point; during this time, the tortoise has "run" a (much shorter) distance,
say one foot. It will then take Achilles some further period of time to
run that distance, during which the tortoise will advance farther; and then
another period of time 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, Zeno says, swift Achilles can never
overtake the tortoise.
The refutation of this famous paradox starts with t/time, d/distance and
r/rate, or speed, wherein d = rt, r = d/t, and t = d/r.
If Achilles runs @ 10 mph, and the Tortoise runs @ 1 mph, then let both
A and T run for one hour and observe where each is located inre the other.
If both A and T start simultaneously from the same starting location, mile
marker zero, then after one hour A will be at mile marker ten and T will
be at mile marker one, and, hence, A will have outrun T and therefore will
have passed T's location.
If A at mile marker zero starts one mile behind T at mile marker one, then
after one hour A will be at mile marker ten whereas T will be at mile marker
two, and, hence, A will have outrun T and therefore will have passed T's
Similar analyses will reveal that unless T starts at mile marker 9.1, A
will always pass his location after one hour.