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BNC
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Optimal arrow length
« on: Aug 24th, 2003, 1:36am » |
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Prove that for every given bow, there exist an optimal arrow length, for which the arrow will have the maximal flight distance. (assume all other arrow factors are not changed).
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Icarus
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Re: Optimal arrow length
« Reply #1 on: Aug 24th, 2003, 11:35am » |
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And may we assume as part of our model that flight distance varies continuously with arrow length. Otherwise proof is impossible.
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BNC
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Re: Optimal arrow length
« Reply #2 on: Aug 24th, 2003, 12:31pm » |
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Why, ofcourse. But do you know of any physical phnomenon (i.e., not mathematical) that is non-continous in the "day-to-day" scale?
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SWF
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Re: Optimal arrow length
« Reply #3 on: Aug 24th, 2003, 12:39pm » |
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on Aug 24th, 2003, 12:31pm, BNC wrote:| do you know of any physical phnomenon (i.e., not mathematical) that is non-continous in the "day-to-day" scale? |
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The amount of force that can be applied to the end of an arrow. Try and pull a 10 meter long arrow all the way back in a bow and something is likely to break.
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| « Last Edit: Aug 24th, 2003, 12:40pm by SWF » |
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BNC
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Re: Optimal arrow length
« Reply #4 on: Aug 24th, 2003, 12:48pm » |
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on Aug 24th, 2003, 12:39pm, SWF wrote:
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<<hidden text removed>>
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But then I would argue the process is irreversible, but is still continuous.
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Icarus
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Re: Optimal arrow length
« Reply #5 on: Aug 24th, 2003, 5:16pm » |
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on Aug 24th, 2003, 12:31pm, BNC wrote:| Why, of course. But do you know of any physical phenomenon (i.e., not mathematical) that is non-continuous in the "day-to-day" scale? |
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No mathematical description of the real world is anything more than an assumption. I figured it was best to clear up this point now, instead of having someone raise it later.
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wowbagger
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Re: Optimal arrow length
« Reply #6 on: Aug 25th, 2003, 1:35am » |
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on Aug 24th, 2003, 12:31pm, BNC wrote:| Why, ofcourse. But do you know of any physical phnomenon (i.e., not mathematical) that is non-continous in the "day-to-day" scale? |
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How about the electric field of a charged (hollow) sphere?
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James Fingas
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Re: Optimal arrow length
« Reply #7 on: Aug 25th, 2003, 9:05am » |
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on Aug 24th, 2003, 12:31pm, BNC wrote:| Why, ofcourse. But do you know of any physical phnomenon (i.e., not mathematical) that is non-continous in the "day-to-day" scale? |
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Whether or not your cottage burned down as a result of playing with matches?
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Sameer
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Re: Optimal arrow length
« Reply #8 on: Aug 25th, 2003, 9:54am » |
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You are still in the primitive era and you don't know how to make a bow and an arrow? 
Also you are now tired of bow-arrow games and use guns instead?
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BNC
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Re: Optimal arrow length
« Reply #9 on: Aug 26th, 2003, 12:06am » |
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on Aug 25th, 2003, 1:35am, wowbagger wrote:
How about the electric field of a charged (hollow) sphere? |
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It's been a while since I had to solve such problems. However, I dare to guess that for a mathematical description of a sphere, with zero width walls, and perfecly smoth surface, the answer is non-continous. But for a real-life physical sphere, it would be continuous non-the-less.
But, I may wrong. As I said, it's been a while...
on Aug 25th, 2003, 9:05am, James Fingas wrote:
Whether or not your cottage burned down as a result of playing with matches? |
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Any continuous phenomenon may be reduced to a non-continuous case by "brute-force discretization". Take for example my x-coordinate location (in an arbitrary axes system) as I wander through my home. I think we will all agree that, disregarding the off-chance that I will tunnel from room to room, the process is indeed continuous. Now I can define “have I gone through the x=2m mark” – and that would be non-continuous. I claim that the phenomenon described is continuous, just the answer to the question, as a yes/no question, must be discreet.
In a similar manner, I think that the process of “burning down the house while playing with matches” is continuous. I may burn anything from part of the single match I started with, and up to the entire neighborhood, with, basically, any number of step in the way. If we will define the measurement as the damage done, you will get an answer that is as continuous as allowed by the unit you choose.
Now, naturally, by selecting the proper discreet-by-nature answer, you can get a discreet answer (e.g., “how may chairs are in your home?”) – but that’s not the point!
In any case, I think we have diverged from the original question…
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| « Last Edit: Aug 26th, 2003, 12:16am by BNC » |
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James Fingas
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Re: Optimal arrow length
« Reply #10 on: Aug 26th, 2003, 10:13am » |
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My point was that for a process with sufficient positive feedback, a continuous input can be made into a discontinuous output.
Once the cottage starts burning hot enough, you won't be able to stop it. So it goes from 'mostly not burnt' to 'gutted' with a small change in initial conditions. This is not the best example of this, either...
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Icarus
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Re: Optimal arrow length
« Reply #11 on: Aug 26th, 2003, 2:25pm » |
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on Aug 26th, 2003, 12:06am, BNC wrote:In any case, I think we have diverged from the original question…
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Yes! I'm sorry that a question which was only intended to head off objections that would be raised to the solution without it, has instead led to a whole series of objections before any solution is posted!
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towr
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Re: Optimal arrow length
« Reply #12 on: Aug 26th, 2003, 3:10pm » |
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on Aug 24th, 2003, 1:36am, BNC wrote:| Prove that for every given bow, there exist an optimal arrow length, for which the arrow will have the maximal flight distance. (assume all other arrow factors are not changed). |
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Assuming all other factors like lift, drag, thickness etc stay the same, the only thing that changes with length is mass.
So the only important things that change is the impulse and thus momentum. Energy should stay the same, since the forces in the bow are the same, and the distance it's tentioned over. So lighter and shorter is better.
Of course to effectively aim an arrow it needs to be longer than the maximum the bow is tretched, so you can steady it both on the bowstring and against the bow.
So the optimal length would be the shortest arrow that still allows that.
If mass also doesn't change than density must change (something must give after all). That'd have repurcussions for the effect by friction, so shorter would again be better..
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| « Last Edit: Aug 26th, 2003, 3:13pm by towr » |
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James Fingas
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Re: Optimal arrow length
« Reply #13 on: Aug 27th, 2003, 8:31am » |
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on Aug 26th, 2003, 3:10pm, towr wrote:
Assuming all other factors like lift, drag, thickness etc stay the same, the only thing that changes with length is mass.
So the only important things that change is the impulse and thus momentum. Energy should stay the same, since the forces in the bow are the same, and the distance it's tentioned over. So lighter and shorter is better.
Of course to effectively aim an arrow it needs to be longer than the maximum the bow is tretched, so you can steady it both on the bowstring and against the bow.
So the optimal length would be the shortest arrow that still allows that.
If mass also doesn't change than density must change (something must give after all). That'd have repurcussions for the effect by friction, so shorter would again be better..
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I disagree. The reason there is an optimal arrow length is a tradeoff between two effects:
1) Drag is largely constant with arrow length, but increases drastically with speed (proportional to speed cubed, I think). So a faster, lighter arrow may not go as far as a slower, heavier one.
2) There is a finite amount of energy you can put into pulling the string back (so a heavier arrow will go slower).
Somewhere in the middle is an optimal length, where the arrow goes slow enough to not dissipate all its energy quickly, but is still going fast enough to travel a long distance.
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Sameer
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Re: Optimal arrow length
« Reply #14 on: Aug 27th, 2003, 3:11pm » |
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If I remember my differential equations correctly, and if u mean air resistance by "drag" then it is proportional to the velocity i.e. F = -bv where b would be resitance coefficient.
I am still confused and think that towr's and James' arguements are correct. Consider a length of arrow then the maximum energy you can give to the arrow is only that much string you can pull to cover the length of the arrow. So ultimately there is a length where the break point of string may reach and the bow will break. So to have an arrow beyond that lenght is meaningless. Once your lenght is fixed w.r.t the string energy now comes the point of determining mass and also consider drag. I guess then it would be a problem of solving a differential equation with the E (energy) as constant and acceleration, velocity and mass as our parameters  
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BNC
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Re: Optimal arrow length
« Reply #15 on: Aug 27th, 2003, 4:00pm » |
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on Aug 27th, 2003, 3:11pm, Sameer wrote:
.... So ultimately there is a length where the break point of string may reach and the bow will break. So to have an arrow beyond that lenght is meaningless. ... |
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You could pull the the string only as far as it safely goes...without breaking...not "utilizing" the entire length of the arrow. That doesn't make the longer arrow meaningless (but maybe sub-optimal)
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| « Last Edit: Aug 27th, 2003, 4:08pm by BNC » |
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Kozo Morimoto
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Re: Optimal arrow length
« Reply #16 on: Aug 27th, 2003, 10:09pm » |
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According to the specs of the riddle "assume all other arrow factors are not changed", we must operate under the assumption that everything is done in a vaccuum. So the only thing the length would change would be rotational or tumbling effect if not fired perfectly?
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wowbagger
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Re: Optimal arrow length
« Reply #17 on: Aug 28th, 2003, 1:47am » |
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on Aug 27th, 2003, 10:09pm, Kozo Morimoto wrote:| According to the specs of the riddle "assume all other arrow factors are not changed", we must operate under the assumption that everything is done in a vaccuum. |
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I don't think so. It doesn't say anything about a vacuum - nor about any fluid in which the arrow moves, I know.
BNC will surely clarify this point.
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BNC
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Re: Optimal arrow length
« Reply #18 on: Aug 28th, 2003, 1:52am » |
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No, vacuum assumption is not required here.
A point of clarification: the riddle asks for a proof that the optimal length exists, not to find that length.
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Sameer
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Re: Optimal arrow length
« Reply #19 on: Aug 28th, 2003, 6:19am » |
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on Aug 27th, 2003, 4:00pm, BNC wrote:
You could pull the the string only as far as it safely goes...without breaking...not "utilizing" the entire length of the arrow. That doesn't make the longer arrow meaningless (but maybe sub-optimal)
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Well I wanted to say sub-optimal (language  ), my point being since it would be only possible to stretch string to its breaking point, an optimal length w.r.t. the string would be only the length of arrow that is equal to the stretch.
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PUPPY
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Re: Optimal arrow length
« Reply #20 on: Aug 30th, 2003, 4:49pm » |
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i don't know
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James Fingas
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Re: Optimal arrow length
« Reply #21 on: Sep 2nd, 2003, 12:47pm » |
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Rereading the question, it doesn't rule out zero-length or infinite-length arrows. Using this information, I can prove that there is no optimal arrow length.
The "optimal arrow length" exists if and only if there is a maximum on the graph of arrow flight distance versus arrow length.
Note that the arrow flight distance is the distance from the person firing the arrow (initially holding the arrow's tail) and the place where the arrow head lies when the arrow comes to rest after it is released. The arrow flight distance is therefore at least the arrow length.
As the arrow length goes to infinity, the flight distance goes to infinity. Therefore there can be no optimal arrow length.
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towr
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Re: Optimal arrow length
« Reply #22 on: Sep 2nd, 2003, 1:48pm » |
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sure there is, the world is round, and the arrow will bend under gravity, or break under it. Either way at some point it is longer than optimal.
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BNC
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Re: Optimal arrow length
« Reply #23 on: Sep 3rd, 2003, 12:16am » |
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on Sep 2nd, 2003, 12:47pm, James Fingas wrote:
Note that the arrow flight distance is the distance from the person firing the arrow (initially holding the arrow's tail) and the place where the arrow head lies when the arrow comes to rest after it is released. The arrow flight distance is therefore at least the arrow length.
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I don't know. I would define the flight distance as the distance between the arrow head before fire, and the location at the end of the flight (remember -- its an arrow, not a spear). Hence, infinite arrow would have zero flight length.
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aero_guy
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Re: Optimal arrow length
« Reply #24 on: Sep 3rd, 2003, 7:57am » |
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I think the problem of actually finiding the distance is very tricky as when the length of the arrow changes, the optimal angle of flight changes as well, but proof of an optimum may be easier. Lets look at the equations:
Drag is proprtional to a bunch of constants, the surface area of the arrow, and the velocity squared (much simplification here), so:
D=c1LV2
mass is proportional to the length
m=c2L
the energy transmitted to the arrow is assumed a constant (as long as the arrow can be pulled back that far which is another assumption)
E=.5mVinit2
so we get
Vinit=c3/sqrt(L)
I may come back to this and take a look, but and obvious method for the proof of an optimum length is:
zero length gives us inifinite initial velocity but also infinite squared drag. Doing the math you will see that the limit brings the distance traveled to zero. Infinite length gives us infinite mass and zero initial velocity, therefore it doesn't move. The system as presented is continuous. There exist positive values of distance. Therefore there exists some maximum (though not necesarily a single point) on the range of zero to infinite arrow length.
oh wait, these equations say that the initial drag will be a constant (the other terms drop out at t=0).
OK, modify the above. The minimum is not based on infinite drag, but on minimum pull. As below a certain length E will begin to go down based upon how far you can pull back. At L=0, E=0. Ok, that is better.
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