# Distance cut in half infinitely (shooting an arrow at tree)

(I looked for a Stupid Question section but couldn't find one anywhere - I apologize if this is the wrong section for this question.)

If you shoot an arrow at a tree, the arrow will eventually travel half the distance between your bow and the tree. From that point, you could go on infinitely saying the arrow will travel half the remaining distance to the tree. How the hell does the arrow ever hit the tree?

If my friends asked me this question, how would I respond to them in a way that makes sense? I'm basically looking for something more than "just don't think about it that way, idiot" because that's all I've got right now.

Thanks

Great forum by the way, just found it today and have spent way too much time reading already.

How about as the distance is halved each time, the remaining distance is becoming smaller and since your "halving" tends to infinity, this distance will tend to become infinitely small.

The arrow, however, as well as the tree stay the same size, so eventually your arrow will "hit" the tree (there'll be a remaining distance of dimensions smaller than that of the possible physical distance between two material objects).

On the other hand, if you're looking at it from an atomic scale, the atoms of the arrow never actually touch the atoms from the tree (Coulomb forces), but there will come a point when the repulsive forces between the atoms will become so strong that distance will no longer be halved (otherwise we'll have fusion between the arrow and the tree and both will cease to exist)...

Perhaps this is a stupid answer (let's wait and see if someone better qualified comes along, there's a hell of a lot of stupidly clever people on this forum) but perhaps it beats "Just don't think about it that way, idiot!" :rofl:

I agree. Though it might seem reasonable that one can "half" a distance infinitely between colliding objects, that very notion seems irrational to me.

There becomes a point in mathematical "halving" that further halving is so small that nothing we know of in reality can accomodate the continuance of that halving; in essence it can not exist.. therefore impact must.

I may have my explanation wrong, but I recall this issue having been settled in some way many, many years ago.

rcgldr
Homework Helper
To make this easier, assume the arrow travels at a constant speed. It takes 1/2 the time to travel 1/2 the distance, then 1/4 the time to travel 1/4 the distance. Note the speed remains constant, only the number of abstract observation points increase.

Ignoring the speed aspect and only focusing on total distance moved in this "halving process", an infinite sum can be used:

Code:
s(n) = 1/2 + 1/4 + 1/8 + 1/16 ... + 1/2^n, as n approaches infinity.

Multiply s(n) by 2:

2 s(n) = 1 + 1/2 + 1/4 + 1/8 ... + 1/2^(n-1)

Subtract s(n) from 2(sn)

2 s(n) = 1 + 1/2 + 1/4 + 1/8 ... + 1/2^(n-1)
-  s(n) =     1/2 + 1/4 + 1/8 ... + 1/2^(n-1) + 1/2^n
------------------------------------------------------
s(n) = 1 +   0 +   0 +   0 ... +         0 + 1/2^n

s(n) = 1 + 1/2^n

so the limit as n->infinity is == 1.

The some of all the distances = 1 times the total distance between arrows initial starting position and tree.

atyy
In calculus, the position as a function of time is x(t). For motion to occur, we must have dx(t)/dt = lim(∆t-->0)[(x(t+∆t)+x(t))/∆t] != 0. Is the first derivative of a function local or non-local? It is local in the sense that it exists at every time t, but it is non-local in the sense that it involves positions at two moments of time. In order to make the velocity exist at every moment of an infinitely subdivisible time, mathematicians have to add postulates such as continuity and differentiability. (Apparently things like everywhere continuous but nowhere differentiable functions exist. )

As Jeff Reid showed, by adding up all the 'halved' segments, you eventually arrive at 1.

The sum of 1/2 + 1/4 + 1/8 + 1/16 + ... is 1 even though that may seem weird. After all, you are adding positive numbers together and continue to do so for infinity. What stops you from adding just another 'halve' as soon as you reach 1? The answer is that if you actually try to calculate (or 'follow') this sum, you never reach 1. Only in the limit that you keep summing forever do you reach 1.

So, you may ask, how does the arrow ever reach the tree, when it does not even take an infinite amount of time? We are summing each of the halved distances yet we still reach the total distance in a finite time! The answer here I believe is that while the distance to travel halves, so does the time it takes the arrow to travel that distance. The time between two successive 'halved points' tends to zero equally fast than the distance tends to zero, hence you can think of them 'canceling each other out'.

This paradox is a funny one. Most people will just say that the argument is wrong. But it isn't. What makes it a paradox is that most people will not realize that the argument in its own is setting a time limit, something that is correctly pointed out by nick89.

So nothing wrong in it. The arrow will not hit the tree in the time you are allowing it to do. It will hit the tree just an infinitesimal time fraction later.

As shown, the argument/logic IS fundamentally incorrect. A LOT of physics includes competing arguments which have considerable appeal but are ultimately proven to be false. An example is Newton's concept of universal (constant) time the same everywhere for every observer.

I believe it was an ancient Greek who "proved" one could never cross a stream by noting it consisted of an "infinite" number of points and would therefore take an infinite time to cross....the calculus of Newton showed how this argument was invalid.

Jeff's post is a nice approach....

An alternative is to note that at approximately constant speed, it takes no longer to cover the second half of the distance than it did the first...just double the 1/2 distance time of flight....

HallsofIvy
Homework Helper
This is, of course, "Zeno's Paradox" (that's the Greek that Naty1 referred to). The resolution is that it there is no reason to think that it requires an infinite time to cross an infinite number of points.

Zenon paradox if I'm not wrong.

If you divide the flight in halves, the time to cover it will also be halves. The sum of halves is infinetly near the unit but will not reach the unit. True for the distance, true for time. So the arrow will not reach the distance ins the allowed time.

Arrow flying at 10m/s. Initial distance:160m

In 16s, the arrow will hit the tree, but the argument allows just 8s+4s+2s+1s+1/2s+1/4s+... which is extremely close to 16, but in fact it is 15.9999s.
The distance series: 5+2.5+1.25+.... or just below 10.

What is wrong in the paradox is the abusive adverb 'never' that some people use. But the initial question was 'How the hell does the arrow ever hit the tree?' . The answer is just letting the arrow an additional infinitesimal amount of time.

Very interesting replies! Thank you all for taking the time to explore this question with me. I shall return to my friends with a much better answer to the question. Or at least I will be able to direct them to this informative thread. :)

russ_watters
Mentor
Looks to me like the OP's wording mixes two of the paradoxes together, but in a way that creates the flaw that Jeff pointed out: http://en.wikipedia.org/wiki/Dichotomy_paradox

Zeno's arrow paradox isn't about halving the distance, it is about defining motion in discrete points (you can't be where you are and moving at the same time). The dichotomy paradox is about distance, but doesn't deal in time.

I'm not a big fan of Zeno's paradoxes - to me they seem to be relatively simple misunderstandings of how math works/what it is for. Given when they were written, he's forgiven, but I don't think they hold any value today.

rcgldr
Homework Helper
A bouncing ball approximates a geometric series. It could bounce an infinite number of times in a finite amount of time and while moving a finite distance in the process. A billiard ball or ping pong ball bouncing on a hard surface allows a person to hear the increasing rate of bounces as a high frequency. In the case of a real ball, eventuallly the bounce height becomes less than the deformation, and the ball stops bouncing, although it's center of mass will continue to oscillate (I don't know the math at this point).

Do a web search for bouncing ball geometric series and you'll find a few hits. I didn't bother trying to find the "best one".

gmax137
If my friends asked me this question, how would I respond to them in a way that makes sense? I'm basically looking for something more than "just don't think about it that way, idiot" because that's all I've got right now.

Tell your friends to aim at twice the distance to the tree. That way, when the arrow is released, first it goes half the distance, and hits the tree. End of story. What's the problem?

Lol I like that one. I'll use that answer first to shut them up for a second while I try to remember everything from this thread. Then when they get that "aha" moment, I'll be ready.

The advice to “not think of it that way” is in fact valid. Unless you are using some special type of “geometric progression” bow to launch the arrow, it will move towards the tree according to uniform parabolic motion. That is, the path is dependent only on the initial velocity and angle that it leaves the bow as well as the gravitational acceleration of the earth.
However, even if you insist on thinking of the path of the arrow in terms of a geometric progression, mathematics still provides a definitive answer to the problem. If you call the distance to the tree unity, and the arrow moves according to the progression:
1/2 + 1/4 + 1/8 + 1/16 . . . .
The Sum of this convergent series is given by: a / (1 – r) where a = the first term and r is the common ration. In this case, a = 1/2 and r also = 1/2
The Sum then is 1/2 / (1 – 1/2) which is 1, the distance to the tree!

I read this about 9 years ago as a first grader, and its made me think ever since. Anyways, as schroder said about going up 1/2 +1/4 + 1/8 it would be ininity time until you get to 1. So why does it not take infinity time to get to the tree? Well i was thinking, this may make no sense at all. but does it reach the tree because time doesnt really exist outside of our minds so infinity time can exist making the arrow reach the tree? Someone wanna help me out with this? has this allready been thouht of, or is this just compleltey stupid, ive never tooken physics or anything its just me thinking freely :)

You are implying that the velocity of the arrow decreases dependant on some function of distance. These paradoxes are not really paradoxes at all - only math that has been taken out of context.

I read this about 9 years ago as a first grader, and its made me think ever since. Anyways, as schroder said about going up 1/2 +1/4 + 1/8 it would be ininity time until you get to 1. So why does it not take infinity time to get to the tree? Well i was thinking, this may make no sense at all. but does it reach the tree because time doesnt really exist outside of our minds so infinity time can exist making the arrow reach the tree? Someone wanna help me out with this? has this allready been thouht of, or is this just compleltey stupid, ive never tooken physics or anything its just me thinking freely :)

I suspect that time exists outside of our minds but will kinda let that go...

You seem to be making the assumption that some specific time, like say 1/100 of a second, is needed to compute the next distance of the arrow to the tree for each halving of distance. But obvioussly the arrow does not pause for these computations, so they must, if they exist at all, must be instantaneous. Thus they, in total, occupy no time and the arrow hits the tree.

I read this about 9 years ago as a first grader, and its made me think ever since. Anyways, as schroder said about going up 1/2 +1/4 + 1/8 it would be ininity time until you get to 1. So why does it not take infinity time to get to the tree? Well i was thinking, this may make no sense at all. but does it reach the tree because time doesnt really exist outside of our minds so infinity time can exist making the arrow reach the tree? Someone wanna help me out with this? has this allready been thouht of, or is this just compleltey stupid, ive never tooken physics or anything its just me thinking freely :)

My thoughts are this:

Space/time is not linear. As such, there is no reason to presume that a collapsing distance must collapse in a linear fashion. If it did, the arrow could never reach the tree.
But the arrow does hit the tree.
From this easily reproducible observation, we must conclude that space/time is curved.
Because of this curvature "halving" is not actually "half" and impacts may thus occur.

You are halving distance, but you are halving time intervals as well. So the arrow won't hit the tre on the time given. Which is true. But the arrow will in fact hit the tree as time does not stop.
Simple as this.

Drakkith
Staff Emeritus
My thoughts are this:

Space/time is not linear. As such, there is no reason to presume that a collapsing distance must collapse in a linear fashion. If it did, the arrow could never reach the tree.
But the arrow does hit the tree.
From this easily reproducible observation, we must conclude that space/time is curved.
Because of this curvature "halving" is not actually "half" and impacts may thus occur.

Why would you need to divide space and time at all? Traveling at X velocity, an arrow will travel Y distance in Z time. Simple as that. Halving the time halves the distance traveled, sure. But we don't need to start breaking things into halves and quarters. There's no reason to.

Would I be right in saying this goes back the planck length, eg. the smallest length possible, if you could cut the space between the archer and the tree into infinite divisions then the arrow could never reach the tree, s=d/t but if s=$$\infty$$/time would never give you an answer and therefore the arrow would never reach the tree.

So basically you keep cutting in half until you reach the planck length which then makes the distance up to one

hm scientifically i agree with Naty1, his argument was the best in my opinion plus he seemed to say everything that i wanted to. if you wanted to take the simple problem solving approach Gmax's answer was awesome.
Then again what do i know i'm just an 8th grade girl who loves to study physics.
-Ivy-