Can you solve this paradox?

Rod P

You have a bow and arrow. You shoot the arrow with the bow from Point A to Point B.

But the arrow will never reach Point B, it's destination. Here's why:

Before the arrow can reach Point B, it must first travel half way (let's call that Point C).

But before the arrow can reach Point C, it must first travel half way (let's call that Point D).

But before the arrow can reach Point D, it must first travel half way (let's call that Point E).

Do you see the problem? In other words, before an arrrow can travel anywhere, it must first travel half-way, no matter how short or how small the distance. It will always be half way before it can reach the whole of any distance. This half-way can divide and sub-divide, and sub-sub-divide ad infinitum. Therefore logically, the arrow can never reach its final destination. Yet we all know that the arrow does reach Point B from Point A.

This is a paradox. How do you explain it?

Rod P.

Puternut

I just talked with Robin Hood, and he said you're full of it. See, it depends on the bow and arrow as well as the stockings you wear. If they are tight, your arrow will go farther than even point B, by passing all the other points.

Puternut

googlemagoogle

the number of halfways is indefinite. so when the arrow reaches point B, it's actually still flying.

iggy_the_fish

Wasn't it one of the greek philosophers who first started thinking this one over? I forget which one.

The answer lies in infinite sums having finite limits. In this case, you're adding 1/2, then 1/4, then 1/8 etc. If you sum this infinite series (1/{2^n}) from n=1 to infinity, you get the answer 1. If the arrow travels at roughly a constant speed from A to B, the time taken to traverse each segment in the series takes exponentially less time also. You travel smaller and smaller distances (to your mid points), but this is balanced by you traversing them in less and less time. This infinite sum has a finite limit.

Interestingly enough, if you sum the series 1/2, 1/3, 1/4, 1/5, ... you don't get a finite answer, it's infinity, even though the terms in the sum get smaller and smaller as you go on...

ig

Rod P

Puternut,

First of all, I talked with Robin Hood, and he gave me the name and location of his tailor, who I visited. I paid this guy $500.00 to make you the best stockings in the world on the same quality level as Robin Hood himself. The only difference was that they were custom made to fit you to perfection.

Now, you take the world's most powerful bow and shoot the world's most aerodynamically efficient arrow, and shoot that arrow with maximum force, so that the arrow will reach it's theoretical physical distance limit. Now let's call that distance limit "Point B". There is no such thing as shooting the arrow past this Point B. The arrow can only travel between Point A and Point B, and that's the reality here.

So now, smarty pants, how ya gonna get that arrow to Point B when before it gets there, it must first travel half-way (Point C)? And before it gets to Point C, it must first go half of that distance (to Point D). And so on...

googlemagoogle,

The number of half-ways is INFINITE, which means there are so many of them you cannot possibly count them. And because the arrow must first travel half-way of any distance between Point A and Point B in a straight line trajectory, it cannot ever get beyond the half-way point of any point along the way to its Point B destination. Therefore, it cannot logically reach Point B.

Rod P.

iggy_the_fish

Zeno's paradox it's called, although I think he used Achilles chasing a tortoise or something like that.

The answer definitely lies in the fact that you're moving through increasingly small amounts of time, which it turns out you can do in a finite time, EVEN THOUGH there are infinetly many of them.

ig.

googlemagoogle

infinite, that's the word, sorry.

the thing is, that the space between a and b becomes infinitely small.

i'm thinking of an even tougher paradox... when the arrow travels from a to b, and c is halfway, it wont even reach c, because to reach c it has to pass halfway of a to c, which is d, and it wont even reach d, as... to be continued.

iggy_the_fish

It's a good job that arrows are too stupid to realise that they can't actually fly...

ig.

Rod P

Iggy,

Yes, it is one of Zeno's several paradoxes, and your explanation isn't too bad.

The answer is very well explained at the following website:

www.mathacademy.com

Allow me to paraphrase their explanation (to fit my example of the arrow):

"What this actually does is to make all motion impossible, for before the arrow can cover half the distance it must cover half of half the distance, and before it can do that it must cover half of half the distance, and so on, so that in reality it can never move any distance at all, because doing so involves moving an infinite number of small intermediate distances first.

Now, since motion obviously is possible, the question arises, what is wrong with Zeno? What is the "flaw in the logic?"

Suppose the arrow was to travel one mile from point A to point B. Now suppose we take Zeno's Paradox at face value for the moment and agree with him, that before the arrow can travel a mile, it must first travel a half-mile. And before the arrow can travel the remaining half-mile, it must first cover half of it, that is, a quarter mile, and then an eighth-mile, then a sixteenth-mile, and then a thirty-secondth mile, and so on. Well, suppose the arrow could cover all these infinite number of small distances, how far should it have travelled? One mile! In other words:

1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 +...........

At first this may seem impossible: adding up an infinite number of positive distances should give an infinite distance for the sum. But it doesn't- in this case it gives a finite sum; indeed, all these distances add up to one (1)! A little reflection will reveal that this isn't so strange after all: if 1 can divide up a finite distance into an infinite number of small distances, then adding all those distances together should just give me back the finite distance I started with. (An infinite sum such as the one above is known in mathematics as an infinite series, and when such a sum adds up to a finite number we say that the series is summable.)

Now the resolution to Zeno's Paradox is easy. Obviously, it will take the arrow some fixed time to cross half the distance to the other side of the room, say 1/2 second. How long will it take to cross half the remaining distance? Half as long- only 1/4 second. Covering half of the remaining distance(an eighth of the total) will only take 1/8 of a second. And so on. And once the arrow has covered all the infinitely many sub-distances and added up all the time it took to traverse them? Only 1 second, and here the arrow is on the other side of the room after all.

DrMike

Except for :

Distance in not infinitly divisible. The smallest unit of distance is "Planck's length" which is about 1.6 x 10e-35 meters. Anything less than that drops below the quantum threshhold. ( i.e. does not exist )

Which introduces even stranger behaviors.

Can you solve this paradox?

Share this