Invincible turtles?

[iago]

Here's a bit of a puzzle for everybody to think about.  Let's set up a hypothetical situation:

You have a bow and arrow.  There is a turtle at some distance away from you (say, 100m; it doesn't matter).  You fire the arrow at the turtle, aimed perfectly.  Now think about this:

The arrow has to pass through the half way point between you and the turle (in this case, 50m).
Then it has to pass through the halfway point between itself and the turle (in this case, 75m).
Then the halfway point between that and the turtle  (87.5m).

And so on. 

The problem is, no matter how many times you divide a number in half, you'll never get to 0. 

Therefore, the arrow will spend an infinite amount of time trying to get half way between itself and the turtle, because there's always a half way point. 

[Joe]

I've heard this one before. In theory, it works.

In practice, however, you will get down to a matter of atoms away, and if it does not move at least an atom's length, it doesn't move at all, and has stopped. But it must keep going, thunk. It hits.

[iago]

It can't move half the length of an atom?  Don't atoms have a size?

[Joe]

Well, I guess it can move half the length of an atom, but eventually you get down to nothing.

[iago]

Nothing?  Why can't it move a quarter the length of an atom? An eighth?  How could you get down to nothing if you keep dividing in half?

[MyndFyre]

Here's a bit of a puzzle for everybody to think about.  Let's set up a hypothetical situation:

You have a bow and arrow.  There is a turtle at some distance away from you (say, 100m; it doesn't matter).  You fire the arrow at the turtle, aimed perfectly.  Now think about this:

The arrow has to pass through the half way point between you and the turle (in this case, 50m).
Then it has to pass through the halfway point between itself and the turle (in this case, 75m).
Then the halfway point between that and the turtle  (87.5m).

And so on. 

The problem is, no matter how many times you divide a number in half, you'll never get to 0. 

Therefore, the arrow will spend an infinite amount of time trying to get half way between itself and the turtle, because there's always a half way point. 

Assuming constant horizontal velocity (no wind resistance), you can determine how much time has passed between each half-way point indicated, and you'll see that you don't spend an infinite amount of time.

Assuming a total distance of 100m and a speed of the arrow at 50m/s (that's a fast arrow I believe).  (The following equations are slightly off from real-life because they do not take into account gravity; if you were to factor in gravity, you would need to aim upwards, and so an initial v = 50m/s would decrease according to the vertical angle and a_g=9.8m/s²).

At halfway point 1, you'll have spent 1s to get to where the arrow is.
At halfway point 2, you'll have spent 1.5s total to get to where the arrow is, or 0.5s from halfway point 1.
At point 3, you'll have spent 1.75s total, or 0.25s from halfway point 2.
And so on.

Your distance traversed becomes infinitesimally small as you divide by 2.  However, your time travelled to reach each point also becomes infinitesimally small as you travel from one point to the next.  The relationship is modeled by the function:
 
where d is the total distance traveled, t is time elapsed, v(t) is the velocity as a function of time, and d' is the initial distance from the origin of measurement.

Your series is simply a summation of the total distance:
 
This is an infinite geometric series.  Solving by the property:
 
Your series sums to 2.  I'm not exactly sure what the 2 means...  but I'm sure it means something.  In any case, it does indicate that the series is convergent and not infinite.

In any case, it takes d/v time to traverse a distance.    If you want to get technical, it takes t=(d-d')/v(t) time.  

Congrats to MyndFyre for winning the uberl33t award for killing turtles!

[iago]

Haha yeah, you hit it on the head, and killed it (the turtle; and the question).  :-)

[Joe]

And congrats at Joe for not yet taking calculus!

EDIT -
Nice one iago. =p

[MyndFyre]

And congrats at Joe for not yet taking calculus!

You don't need to have taken calculus to understand that both the distance covered and the time taken to cover the distance become infinitesimally small at the same rate. 

[GameSnake]

lol crazy shit

[Quik]

And congrats at Joe for not yet taking calculus!

EDIT -
Nice one iago. =p

That's pre-Algebra II, especially if you take the course with minimal Trig.

[iago]

By the way, here's smart people discussing this problem (which is known as one of Zeno's Paradoxes of Motion):

http://en.wikipedia.org/wiki/Zeno's_paradoxes

"If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless." (Aristotle Physics VI:9, 239b5)

Imagine an arrow in flight. At every moment in time, the arrow is located at a specific position. If the moment is just a single instant, then the arrow does not have time to move and is at rest during that instant. Now, during the following instances, it then must also be at rest for the same reason. The arrow is always at rest and cannot move: motion is impossible.

The solution is pretty much what I would have said, but they worded it much better:

This paradox may be resolved mathematically as follows: in the limit, as the length of a moment approaches zero, the instantaneous rate of change or velocity (which is the quotient of distance over length of the moment) does not have to approach zero. This nonzero limit is the velocity of the arrow at the instant.

That's essentially the problem I laid out; however, I stole the inspiration from Terry Pratchett:

Several of them were holding tortoises on sticks. They looked a bit pathetic, like tortoise lollies.
'Anyway, it's cruel,' said the tall man. 'Poor little things. They look so sad with their little legs waggling.'
'It's logically impossible for the arrow to hit them!' The fat man threw up his hands. 'It shouldn't do it! You must be giving me the wrong type of tortoise,' he added accusingly.
'We ough to try again with faster tortoises.'
'Or slower arrows?'
'Possibly, possibly.'
Teppic was aware of a faint scuffling by his chin. There was a small tortoise scurrying past him. It had several ricochet marks on its shell.

[Joe]

Way to call us dumb. =).

Quik, I'm in Pre-Algebra 1, so eh?

[iago]

Way to call us dumb. =).

I was including myself

[Sidoh]

Doesn't quantum physics tie in here too?  IE "It's either here or there, there's no in between."

[iago]

Hmm, good question.  Although it does sound like a possibility, and it would definitely go a distance towards refuting my "half an atom's length" argument, I'm not sure if it directly applies. 

I think that the wikipedia post summed it up best.  The limit as the time slice approaches 0 doesn't have to be 0.  Which is what Mynd said, in a shorter way

[Sidoh]

Yes, you can move half an atom's length;  I agree with you on that.  There is a limit with quantum physics as to how small a distance you can move, though.

But yes, that's a more justified answer.  I think mine makes sense too, though. 

[iago]

Yes, you can move half an atom's length;  I agree with you on that.  There is a limit with quantum physics as to how small a distance you can move, though.

Once you get down to the size of an atom, however, you start facing more issues.  That's why I didn't say "it refutes", but it's a start.  There are more problems like uncertainity, like the fact that atoms can also be treated as waves, and probabilistic entities, not totally physical.  I'm not sure how motion works at that level, I'm no expert in the field. 

[Sidoh]

Once you get down to the size of an atom, however, you start facing more issues.  That's why I didn't say "it refutes", but it's a start.  There are more problems like uncertainity, like the fact that atoms can also be treated as waves, and probabilistic entities, not totally physical.  I'm not sure how motion works at that level, I'm no expert in the field. 

I don't think I've heard of treating atoms as waves, but I could just be spacing that out.  I've heard plenty of the Electron Wave theory though.

[iago]

Once you get down to the size of an atom, however, you start facing more issues.  That's why I didn't say "it refutes", but it's a start.  There are more problems like uncertainity, like the fact that atoms can also be treated as waves, and probabilistic entities, not totally physical.  I'm not sure how motion works at that level, I'm no expert in the field. 

I don't think I've heard of treating atoms as waves, but I could just be spacing that out.  I've heard plenty of the Electron Wave theory though.

It involves how the electrons orbit the aton, so you're right.  I didn't mean entire atoms, I meant bits of atom

Lots of info here, anyway:
http://www.qmw.ac.uk/~zgap118/

[rabbit]

Doesn't quantum physics tie in here too?  IE "It's either here or there, there's no in between."

"Here" is wherever the object is, and "in between" and "there" are both defined as target locations by some observer, so as soon as it is "in between", "here" is redefined to be where "in between" was, and "in between" is redefined to a new position, and does not exist (much like a point on the cartesion plane, it has location but nothing else).  "In between" is also an array, as best illustrated by this crappy image:



There are infinite "in between" points, and each is defined by changing what "here" and "there" are.  So, technically, the object is "here", "there", and an infinitude of the point "in between".

[Sidoh]

I was talking at things around the size of an electron.  I understand that in hypothetical terms, there are infinite points inbetween two points.

However, I was saying when you start moving a physical object in small enough increments, it will stop moving in lesser increments at some point.  At least that's what my minimal understanding of quantum physics explains in this example.