# Arrow chasing a soldier

1. Nov 23, 2013

Do anyone know the "Arrow chasing a soldier" paradox?I don't know the actual name,so couldn't google it.
Someone throws an arrow at a soldier.The soldier runs away to flee.The arrow actually travels faster than the soldier,so it should hit him.The paradox states that as the distance halves and halves,the arrow actually does not touch him.I want an explanation on this.
Note:I can google this, only if I know the name.

2. Nov 23, 2013

### Enigman

You are looking for an hybrid restatement of Achille's turtle in terms of Arrow paradox...
Achilles and turtle paradox:
In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

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

Last edited: Nov 23, 2013
3. Nov 23, 2013

### phinds

You might also be interested in Zeno's Paradox

4. Nov 23, 2013

### Enigman

The arrow, achilles and turtle and Dichotomy paradoxes are all Zeno's paradoxes.
Dichotomy paradox is more or less a restatement of Achilles and turtle and both deal with space, while Arrow paradox deals with time.

5. Nov 23, 2013

Hmm...I was thinking if a PF member could give a non-Calculus answer to this.The web answers seems disturbing and difficult to understand.

6. Nov 23, 2013

### Staff: Mentor

7. Nov 23, 2013

### Enigman

I don't like the statement of Achilles and torty paradox but math's sound.

Last edited by a moderator: Sep 25, 2014
8. Nov 23, 2013

I don't see any non-Calculus here.It's full of calculus(I don' know Calculus yet.)

9. Nov 23, 2013

### Enigman

See the vid. its pretty simplistic.
Random factoid: Zenosparadox is a cheat code in Age of Mythology: The Titans expansion. It spawns all heroes of both campaigns at your primary town center. Okay I'm off to learn Alice Chess...
E

10. Nov 23, 2013

### Staff: Mentor

If the arrow's position is $p_A=v_A t$ and the target's position is $p_T=v_T t + d_0$ then the arrow hits the target when $p_A=p_T$ or by simple algebra $t=d_0/(v_A-v_T)$.

Without calculus it is clear that the arrow reaches the target in a finite amount of time despite the fact that there are an infinite number of points along its trajectory. Actually calculating the infinite sum as an infinite sum requires calculus, but we know from algebra what the answer must be even if we cannot calculate it directly without calculus.

If you want something more than the algebraic answer then I think you need go ahead and learn calculus.

11. Nov 23, 2013

### Enigman

Correction.

12. Nov 23, 2013

### HallsofIvy

Staff Emeritus
"Zeno's paradox" is based on the assumption that it is not possible to pass through an infinite number of points in a finite time. That is incorrect.

13. Nov 23, 2013

Hmm.For example,If someone travels a distance of 1m,the 1m can be divided into infinite pieces.Right?

14. Nov 23, 2013

### phinds

Mathematically, yes. Whether or not space is actually quantized is an open question, but in either case it doesn't matter in terms of how fast/far you can move

15. Nov 23, 2013

### Mandelbroth

Well, you are asking us to not use calculus to answer a calculus question, or at least a question that is best answered in terms of the machinery of calculus. If you want a good answer, calculus is the way to go.

16. Nov 23, 2013

Haahha.The most precious cheats to me are:
JUNK FOOD NIGHT
ATM OF EREBUS
TROJAN HORSE FOR SALE
PANDORAS BOX
L33T SUPA H4X0R

These cheats are enough to win even in titan mode.
oh:I forgot, O CANADA for emergencies. lol

17. Nov 23, 2013

Well,Thank you all.I've now understood the Paradox well.

18. Nov 23, 2013

### Enigman

[off topic hijack]I can handle titan mode campaigns... random map not so much...
The trick to campaign is never do what your objective says.The campaigns are built based on triggers, so don't complete the starting objectives just act towards the final objective. Eg.
*For the first scenario Kraken attacks the docks objective kill it. Don't, let it live while you beef up your army and resources. The next wave won't come till its dead.
*Osiris box scenario you just get out of prison. Objective look at the osiris box- don't as soon as you do timer starts to the next shuffle...just get triggers out of the way and you have thrown a wrench in the AI.
Oh and I made a mistake Zenosparadox is for random god powers, mixed up with atlantisreborn
[/off topic hijack]

If you have mathematica you could check this wolfram demo out:
In case you don't, you could download the free CDF player
(wolfram demos are a pretty decent waste of time so its worth it)

Last edited: Nov 23, 2013
19. Nov 26, 2013

### Tobias Funke

Zeno's paradoxes seem like something that we only have second-hand accounts of, which are then repeated in distorted or incomplete versions in freshman calculus books.

My understanding is that the paradoxes, if we give the benefit of the doubt and call them that, are only paradoxes when taken together; Zeno was trying to argue that both infinitely divisible space and discrete space are impossible, which is why he came up with multiple scenarios. If Achilles can never catch the tortoise assuming infinitely divisible space, that's no paradox if space is actually discrete, is it?

I never liked or could take seriously the "solutions" in those calculus books. They're just showing off and kind of missing the point, since if you're going to solve it using techniques the Greeks would clearly reject outright, you may as well make it easier on yourself, walk up to the wall and touch it, and say QED. Claiming that you need calculus to show that a faster runner will eventually overtake a slower one is just going to reinforce the common belief that mathematicians are crazy people who prove (column proofs of course...) obvious things.

20. Nov 26, 2013

### phinds

Tobias, I mean no disrespect but I think you are the one that has it wrong. It doesn't matter whether space is continuous or discrete, the basic argument of the "paradox" has the same problem and yes the calculus solutions DO show the proper argument whether "common people" like it or not.

Positing that space is discrete would be unproven speculation.