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Ok, imagen an arrow, and you shoot it at a target. FORGET all the laws of gravity and motion and replace it with this:

The arrow always travels by half the distance it is to go to. Using this thinking, will the arrow ever reach the target?

ADDITION:
O, The arrow travels half the distance to the target, then it tarvels half of the distance of the distance it has already traveled. It goes by this law forever, so wil it ever reach the target???

[/Is confused]

The answer would be .5x, and x = the total distance?
It's kind of hard to understand what you're asking..really..

Obviously not.

It will eventually become incredibly close to the target but be moving so slowly, it might look as if it has hit it when it is indeed just floating in hte air next to it.

But it will never hit. =/

ADDITION:
you may be more like.... x to the 1/2 power of the distance or something if you want an actual formula...

but it eventually goes to like 1/1000000000000000000000000th of a millimeter away, as you could imagine.

Oh god no, not xeno's paradox. As its name implies, the problem posed here is a paradox, we both know that movement must be possible and know that it must be impossible, as distance becomes infinite using the halfway logic.

That's what it's called... I knew I'd heard it somewhere...

Ah, but he said gravity does not apply. Therefore, movement can slooooooooowwww....

I think. ;P

i believe this is exactly the same a an infinite geometric series with a finite sum...

You can't get to half of an angstrom?

A hydrogen atom is 0.5 angstroms

an angstrom is mass, isn't it?

or length? dunno...

But can't you divide it, it just says that it won't be that anymore? like an atom of iron can be divided but it won't be iron anymore, because it wont have those properties? I dont remember if that's an atom or angstrom. Oh well. Anyway I think you can divide the air a couple more times ;P

Ahem. I know the answer since I asked the question, and I think you guys missed something when reading the question. In this case, the infinite series of numbers equals 1, so it will hit the target. Now, it is a paradox, and I think that angstrom is also called a plank legnth?? isnt it? And you could always divide in half, there cant be an end to how much you can divide, now can there?

but wouldnt you always be 50% of the total distance remaining apart from target?

ahhh limits!!! approaches, asymptote...whatever...

Duh, com' on. I worked it out, and its on a LOT of websites too.

I thought about that problem a lot, but I worded it differently..

I don't think so. It won't theoretically, but if for one second you cut off how long the fraction can be, then yes.

Fine... we shall test it in real life *shoots arrow at target* see! *looks at target and arrow is almost, but not quite touching the target* NOOOO!!!!

The arrow would never hit. It would be at .5 repeating distance, or 5/9 in fraction form. I think.

As was mentioned previously, this is known as Zeno's/Xeno's paradox. Obviously, when you fire an arrow it will reach it's target at some point, at least if you have a hint of skill and coordination. Therefore, this paradox is false. It implies adding theoretical mathematical that do not exist and do not serve to reality. To find an accurate representation one must consider the tiem it takes for the object to move, not just the distance as that laeves an imcomplete forumla that leads to the illusion that the time will be infinite. I don't know the equations of the top of my head, but if one factors in time and speed (which is dependant on time), then as long as there is movement you will come to a finite time for the object to reach its intended target.

I'm sure Wikipedia probaby has a good article on this. I'll have a look.

Link.

Physically, yes, becuase the arrow would be too large to fit inside its theoretical distance from the point.

Mathmatically, no, becuase if "a" is the target and the arrows position is "x", a = the limit as x approaches infinity. But since x can never equal infinity, it will never reach a.

Here's another paradox like that one:

A turtle and a rabbit are about to run a race, and the rabbit wants to be fair so he gives the turtle a good head start.

T repersents the turtle and R represents the rabbit.

If the rabbit always moves twice the distance the turtle moves in any given time frame, will the rabbit ever catch up to the turtle?

EX:


So according to the above scenario, the rabbit does indeed catch and pass the turtle. But lets look at it from another point of view:



In this way, the rabbit will NEVER catch up to the turtle, becuase the turtle will always be a liiiiiiittle bit ahead, mathematically.

Both of these are perfectly logical ways of thinking about it, yet the have different answers. So which is right?

ADDITION:
And btw, BOOO to SEN formatting. I had to have my post aligned wrong in the post box AND the preview, in order for it to align right in the actual post.

if the arrow follows that law according to what you say the target is, then think that the target is a little ahead of the original target and it will hit the target

1. ____________________do-0dan_____________________<

2. _________o-0________do-0dan________________<_____

3. _________o-0________do-0dan<_____________________

o-0 = target that you made up
do-0dan = original target
_ = empty space
< = arrow

The second method is not logical because it fails to take the fact that the tortoise and rabbit are moving at different paces and the distances are finite. Obviously it's untrue because something is capable of passing another thing, it's a little harder to disprove though as it serves as an illusion to how human brains tend to grasp logic.

This'd best be done by using measureable distances. Let's say that the tortoise is moving at 5kph and the rabbit is moving at 10kph. The rabbit starts off 5 meters behind the tortoise.

Once the tortoise has moved 2 meters, the rabbit has moved 4. Now the rabbit is 3 meters behind the tortoise. If the tortoise moves 1 meter, the rabbit will move 2 meters in that same amount of time. The rabbit is now 2 meters behind the tortoise. The tortoise can then move 50 decimeters and the rabbit will move one meter. The rabbit is now 1.5 meters behind the tortoise.

The problem is that things don't move in a set of fixed frames so to speak. If we take this example to larger distances, then we see it is inaccurate and not viable for representing how the tortoise and the rabbit move respective of each other. If the rabbit starts 5 meters behind the tortoise and moves 15 meters, the tortoise will have moved 7.5 meters and therefore, the rabbit is 2.5 meters ahead of the tortoise.

To make this simple, let's say the tortoise moves 1 meter per second (it's bloody desperate for exercise) and the rabbit moves at 2 meters per second (The rabbit is still 5 meters behind the tortoise). If the tortoise moves one meter, it took one second to do so, and it took one second for the rabbit to gain a meter on it (four meters behind it). If the tortoise moves 50 decimeters, it will have taken .5 seconds for that to happen, and the same amount of time for the rabbit to gain 50 decimeters, or move one meter (now at 3.5 meters behind the tortoise). This will contue with time being broken up into smaller and smaller segments. The time measurement will become 1 second, then 1.5 seconds, then 1.75 seconds, then 1.825 seconds, end continue. Never will it exceed seconds if the added distance (and the therefore the added time) is halved.

With these speeds, it will take 5 seconds for the rabbit to catch up with the tortoise. Therefore, the above example shows that the rabbit will never catch up to the tortoise in 2 seconds, not that it will never catch up.

If we change the distance the rabbit is behind the tortoise to 1 meter, then the scenario sort of works. As the tortoise moves one meter, the rabbit moves two and they're both in the same spot. It only takes one second for the rabbit to catch up, therefore the rabbit manages to catch up despite the limitation of halved distance with halved time imposed on the scenario.

Because of this, the idea of halving the distance travelled only adds uneccesary limitations to the problem and prevents it from being correct if the time it takes for the rabbit to catch up to the tortoise is 2 or more of whatever unjit of time you are using.


Upon looking back, I noticed you used a turtle instead of a tortoise. Just pretend I used turtle instead of tortoise, it's not really relevant.

You are simply stating more evidence of the looking at it through the first scenario, which is giving the turtle and the rabbit finite velocities.

The second method IS logical, but you discredit it as fixed time frames. In fact, the steps are simply "snapshots" of time. To put it in terms of continous time, here's the same scenario in words:

The race begins and pretty soon the rabbit gets up to the place where the turtle has started from. But, of course, the turtle has moved ahead a bit to a new place, further down the course. So pretty soon the rabbit gets to that place, but again, by then the turtle has moved ahead to the next new place. When the rabbit gets there, the turtle will again have moved ahead. Therefore the rabbit can never beat the turtle because the rabbit can in fact never even catch up with the turtle, since every time he gets to where the turtle was just an instant ago, the turtle will have already left that place. Right?

This is and the arrow are just examples of Xeno's paradox which is:

If you are travelling from point A to point B, you must travel half of the distance to point B before travelling all of the distance. Now from that point you must again travel half of the remaining distance. If you continue to do so (travel half of the distance) you will never reach point B.

But this paradox is at the crux of the mysterious digital/analog relationship, but it is a strong argument that reality is digital, meaning that it has a smallest unit. This is because one DOES get to point B, so at some point we must pass an "unhalvable" point.

ADDITION:
Side note on halve limits:

It's impossible to fold a piece of paper in half more than eight times.