ORIGINAL: mr_matt
Hey Dukester, you are pretty good with math, maybe you could help me with a problem I have had for years.
A runner wants to run a certain distance - let us say 100 meters - in a finite time. But to reach the 100-meter mark, the runner must first reach the 50-meter mark, and to reach that, the runner must first run 25 meters. But to do that, he or she must first run 12.5 meters.
Since space is infinitely divisible, we can repeat these 'requirements' forever. Thus the runner has to reach an infinite number of 'midpoints' in a finite time. This is impossible, so the runner can never reach his goal. In general, anyone who wants to move from one point to another must meet these requirements, and so motion is impossible, and what we perceive as motion is merely an illusion.
Where did I go wrong?
My true math skills are pretty rusty, got to be some college age kids in here much better than me. What I lack for in young brains, though I make up for in practical applications.
This is the same type problem as the "if I'm ten feet from a wall and step half the distance to the wall each time, how many steps will it take until I reach the wall?". The theoretical answer of course is you will never reach the wall, but the practical answer is that very quickly the step length becomes smaller than your body's motor skills will allow with precision and you will indeed reach the wall simply by suffering from that affliction known as being human. Note that the more clumsy and teens who don't tie their shoes will reach the wall slightly faster that the rest of us.
In regards to your question, the same kind of thing applies from the practical aspect, if we assume the runner proceeds with constant velocity, then unless the observer can intentionally decrease their perception of the passage of time, the runner will not be observed to slow down and take an infinitely long time to reach the end. Instead, the observer will perceive the runner to reach the 50% marks at an progressively faster rate up until the upper range of their temporal perception. At that point, the runner will appear be stepping beyond the 50% mark in the minimum time unit the observer can perceive.
Duke
Edit:
One of the things I liked about the proof statement for the lack of a solution for the conveyor speed equals wheel speed argument above is the simplicity of the approach. For the equations I used:
Vc=Vw
Vw = Vp+Vc
In essence these are both equations for a line with one having an intercept of 0 and the other having an intercept of Vp. So if you were to say Vw=Y and Vc = X, you get
X=Y or flipped for Y=X for the standard format
and
Y=X+Vp
Since both lines have a slope of 1 but different intercepts, you could easily make the same proof graphically by plotting the lines. For all Vp<>0, they have no intersecting points (lines with the same slope being parallel and all), for Vp=0 they intersect at every point and there is no single unique solution.
[Ok major geometry nerdly meltown over with for the moment.
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