A Paradox of motion

"You can never catch up." "In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead." (Aristotle Physics VI:9, 239b15)

Isn't it nice to know it's not all in your head or just a feeling? I think here shall begin my look into Paradoxes and how they support so much of what we experience every day. For those of you who need a further intro into Paradoxes see this post. Now back to understanding why we "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 finite time, 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.

John I think you misstate Heisenberg slightly. Commutable states, [x,y]=0, can be defined simultaneously. While Heisenberg works here, there really is no need. Since as the distance between them gets smaller, the time it take Achilles to reach the tortoise also gets smaller, the infinite sum of distances converges to a finite amount of time. In other words, Achilles keeps on reaching the tortoise's 'starting' position faster and faster until he is there in no time!

Posted by: someguy | August 01, 2007 at 04:09 PM

Hmmm.... Assumption v=constant????

By laws physics start either implies a moment of time or a moment of action. If v=constant then Vo and To are only at equilibrium at the singular moment in time + Position in said case then yes Achilles cannot overtake the tortoise.

Heisenberg illustrated that we cannot ascertain accurately two separate physical states in the same formula within a degree a accuracy "x" where x = a finite number.

At Time N (Tn) we can either measure Vsub or Position (Psub). As Vsub is a function of T then the only measurement can be that of Psub when Vsub=Vinstant. In a function of time Vsub as well as the missing component here Vprime or acceleration implies a progressing clock at which point no accuracy can be applied to Psub!

Posted by: John Carmichael | August 25, 2005 at 04:50 PM