h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Well, I am sure this conversation topic shall be termed as dry as most of my fellow men(and women) here, but still I was proud of the fact that I could still view a problem mathematically 5 years after completing my engineering ...so here goes.

Using the beautiful formulas given if the speed of the object is V and the total Distance is X,

the first half of the distance shall be (X/2), and the time taken to complete that shall be (X/2)/V or VX/2.

Similarly the next half of the remaining distance would be (X/4) and the time taken to complete that shall be (X/4)/V or VX/4.

On the same lines, the total time tken for the total distance shall be

(VX/2)+(VX/4)+(VX/8)+(VX/16)+.......to infinity

This in mathematical terminology is called an infinite geometric progression, its sum converges to a definate number and there is definate way to sum it up.This one sums to

VX( (1/2) /( 1-(1/2))
=VX ( (1/2) /(1/2) )
=VX

which is the same what you would get by a direct application of the formula given in the article.

Aha, I see every one has fallen asleep

Hey, Maths is not all that bad a subject. Sometimes it does make ur life a little better...as i just demonstrated above

Still, considering Xeno lived pre-calculus he did a pretty good job.

My math teacher explained it pretty well, in the course of defining the phrase "for all practical purposes". You line up the girls in the class on one side of the room, the boys on the other. The first move is halfway to the center of the room by both groups, then the same for the second move, third, etc. The two groups will never meet, but after a small number of moves they are close enough "for all practical purposes".

Xeno actually had 4 'paradoxes', about various things. The upshot of them all, and this is perfectly true, is that in a Universe governed by Newton's laws ('an object feeling no force travels in a straight line' and all that) motion is completely impossible. I kid you not.

It turns out you need special relativity (Einstein's thing) to explain them. And if the Greeks had really thought about them, they would have realised special relativity was the way to go way back then.

And then where would we be now?

My take on the whole situation is, Xeno argues that an very small distance over an infinitely small time means no distance over normal time. For me, where the argument breaks down is where he uses infinity... it can't be divided or multiplied or added or subtracted like a normal number, because it isn't a number.

oh, ...

First, on a non-mathematical note, Zeno's name begins, in Greek, with the same letter as 'Zeus' and 'zoology', and with a different first letter from 'xenophobe'. Just thought I'd mention.

Secondly, Almighty Rob wrote: "considering Xeno lived pre-calculus he did a pretty good job."

I would disagree. Zeno's contemporaries recognized that there was something wrong with what Zeno was saying (you don't just walk up to someone and prove that you couldn't possibly have just walked up to them), and they were able to spot his fallacy. Aristotle refutes Zeno rather clearly in Book VI of his /Physics/. Aristotle did a good job showing, without calculus, that 'infinitely divisible' and 'infinitely extended' mean different things and that Zeno's paradox relies on a confusion of these two things. There is no historical indication that Zeno himself was anything other than confused.

Well, isn't Aristotle a genius, then.

He worked out that you actually *can* move. How'd he figure that one out?

Not sure... it had something to do with drinking wine and hanging around gymnasiums...



(hey, that's not a mathematical viewpoint!)

Not that your basic math is wrong, but there is one flaw in the whole "infinate" part of all this.
Plank's constant. A plank is the shortest distance. period. There is no half plank. Admittedly, it's like 10^-15 meters
or something like that, but it is real, and talking about distances shorter than that s literal nonsense, cause they don't exist.

Just my 2 cents worth.

I'm gunna be really nitpicky.
A couple things to note:

Planck's constant is not called 'a plank', it's named after a guy named Planck. It isn't the smallest possible distance. In fact, it isn't a measure of distance at all, but angular momentum.

On the original mathematical argument made above:
If speed is V and distance is X, then time is X/V. Not XV. Other than that the geometric progression argument is dead on. And I don't know wether the greeks had access to that type of math or not, but it has nothing to do with Calculus.

They certainly did know, however, that something was wrong with Xeno's Paradox for despite the fact that it made perfect sence logically it obviously defied the real world. Luckily, as noted previously, the problem dissapears if you think of the paradox as a geometric sum. No need for relativity.

Sorry about that... I had a physics final today and my brain is cranked into problem-solving mode.

PS - I'm not actually Researcher174803... I don't know why it gave me that name. I registered as ThePatch

Well Mr Patch is absolutely right. Time=Distance/speed and so the Geometric progression shud have give X/V and not XV. I stand corrected.

Probably that explains why my professor used to tell me " You have got the brains, but you do not have concentration". Which incidently also reminds me of this anecdote, authenticity of which is not guaranteed:

It seems a lift salesman once sold the concept of a home lift to Einstein and was asked by Einstein to install one at home. After a day the salesman came back and cribbed to Einstien that he had been thrown out of the home by his Butler. Einstien git furious and called home to enquire on which he was told "but sir, your house only has one floor!!!!"

Well i am not asking you to read between the lines but if you do....

And Mr patch saved me the effort of commenting of Plank's (SIC) constant. and as always he seems to be dead on target

Mr Patch is not dead on. There is a "fundamental" length unit, based on Planck's constant, and known as a "plank". Physicists have strange senses of humour, you know.
I disagree that it is the smallest length, though, and anything less is meaningless. And it doesn't solve the problem of motion, cos how do you move from one plank to the next?
Also relativity doesn't solve it either, as it is based on differential equations, which implies that space and time are both infinitely divisible. Relativity, then, simply rephrases XZeno's paradox, it does NOT solve it.
And quantum mechanics doesn't solve motion either, cos it only says that motion happens, but does not say HOW it happens.
Any suggestions?

well it turns out that there is "Planck's Length" but only that it is not quite the "smallest unit of length". It seems it is defined as "The length scale on which the quantized nature of gravity should first become evident" and the mathematical value is 4*10^-35..so i guess as far as XENOs paradox is concerned it would not be quite relevant....

Moreover half a planck's length will still be of the order of 10^-35ish not 10^-15ish

As to Half a Dents question on what causes motion, I think I'll pass this one to someone more qualified; it does not seem that an ex-engineer, current "pointy haired" manager would be competent enough to discuss nuances of relativity

However, would love to read more on the subject if anyone cares to share any gems of wisdom.

p.s.: more on Planck's Length at:

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