powell and zeno race-reasoning
10th jan 2008

This question left me all confused.. and get me really thinking. Here goes:

A race is arranged between Asafa Powell (currently the 100m world record holder) and a tortoise named Zeno.
Since Powell can run much faster than the tortoise, Zeno is given a substantial head start.
The judge fires the starting pistol and both contestants are off.
Now, the first thing Powell has to do is to get to the place from which Zeno started.
However, the tortoise, while slow, is unflagging: While Powell was busy making up his handicap, Zeno advanced a bit further.
So the next thing Powell has to do is to get to the new place Zeno occupies.
While Powell is doing this, our hero the tortoise will have gone on a little bit further still.
However small the gap is, it will take Powell some time to cross it, and in that time Zeno will have created another gap.
So however fast Powell runs, all the tortoise need to do in order not to be beaten is to keep going- to make some progress in the time it takes Powell to close the previous gap between them.
It is inevitable that Zeno the tortoise will win the race.

Do you agree with this conclusion?
Explain why or why not.

Stolen from pri's blog.
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LEON'S CONCLUSION -
Unresolvable.
(highlight the answer)

We need 3 essential-facts to establish a conclusion with either winner.
E.FACT 1: Distance of the whole race.
E.FACT 2: Speeds of Powell and Zeno.
E.FACT 3: Distance of headstart given.

Here's my lengthy explanation:

From the long paragraph from the question, we know 3 facts.
FACT 1: Zeno is given a headstart.
FACT 2: Powell's speed > Zeno's.
FACT 3: As Powell reaches Zeno's current distance, Zeno will progress even further.

Let's use some numbers here!

At time 0 second:
Zeno is given headstart, say he was 10 metre lead for now.
Powell's at 0 metre still.

At time 1 second:
Powell reaches Zeno's 10 metre distance this second(meaning Powell's speed is 10m/s).
Zeno(being a tortoise, let's just say his speed is 1m/s), inches forward 1 metre, thus in the lead still by 1 metre.
(Powell at 10m, Zeno at 11m)

At time 1.1 second:
Powell reaches Zeno's 11 metre distance(same speed).
Zeno inches forward another 0.1 metre(same speed).
(Powell at 11m, Zeno at 11.1m)

At time 1.11 second:
Powell reaches Zeno's 11.1 metre distance(same speed).
Zeno inches forward another 0.01 metre(same speed).
(Powell at 11.1m, Zeno at 11.11m)

This will go on infinitely, until the distance of the race ENDS - which up to this point, WE DON'T KNOW(E.FACT 1).

If we know the distance, say 100 metres, Powell can be moving at 10m/s, while Zeno's at 9.99... m/s, (Powell's speed still greater than Zeno's, but insignificantly), thus, Zeno will win the race because of his headstart. WE NEED TO KNOW THEIR SPEEDS(E.FACT 2).

Also, let's assume they are having a 100m race. If Zeno is given a 99.99... metres headstart, he will win wouldn't he(E.FACT 3)?

I did some brilliant time-killing. And for a short update, i went to TTS hospital to visit my dad. Apparently he's been diagnosed with an illness called "Gout", and he gotta stay over tonight for further observation. Hopefully he gets recovered soon. :)