wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> easy >> Optimal arrow length
        
(Message started by: BNC on Aug 24th, 2003, 1:36am)
Title: Optimal arrow length
Post by BNC on Aug 24th, 2003, 1:36am
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).
Title: Re: Optimal arrow length
Post by Icarus on Aug 24th, 2003, 11:35am
And may we assume as part of our model that flight distance varies continuously with arrow length. Otherwise proof is impossible.
Title: Re: Optimal arrow length
Post by BNC on Aug 24th, 2003, 12:31pm
Why, ofcourse. But do you know of any physical phnomenon (i.e., not mathematical) that is non-continous in the "day-to-day" scale?
Title: Re: Optimal arrow length
Post by SWF on Aug 24th, 2003, 12:39pm

on 08/24/03 at 12:31:40, BNC wrote:
do you know of any physical phnomenon (i.e., not mathematical) that is non-continous in the "day-to-day" scale?

(hidden text)
[hide]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.[/hide]
Title: Re: Optimal arrow length
Post by BNC on Aug 24th, 2003, 12:48pm

on 08/24/03 at 12:39:03, SWF wrote:
(hidden text)
<<hidden text removed>>


But then I would argue the process is irreversible, but is still continuous.
Title: Re: Optimal arrow length
Post by Icarus on Aug 24th, 2003, 5:16pm

on 08/24/03 at 12:31:40, 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?


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.
Title: Re: Optimal arrow length
Post by wowbagger on Aug 25th, 2003, 1:35am

on 08/24/03 at 12:31:40, 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?

How about the electric field of a charged (hollow) sphere?
Title: Re: Optimal arrow length
Post by James Fingas on Aug 25th, 2003, 9:05am

on 08/24/03 at 12:31:40, 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?


Whether or not your cottage burned down as a result of playing with matches?
Title: Re: Optimal arrow length
Post by Sameer on Aug 25th, 2003, 9:54am
You are still in the primitive era and you don't know how to make a bow and an arrow?  ;D
Also you are now tired of bow-arrow games and use guns instead?
Title: Re: Optimal arrow length
Post by BNC on Aug 26th, 2003, 12:06am

on 08/25/03 at 01:35:23, wowbagger wrote:
How about the electric field of a charged (hollow) sphere?

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 08/25/03 at 09:05:58, James Fingas wrote:
Whether or not your cottage burned down as a result of playing with matches?

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…
Title: Re: Optimal arrow length
Post by James Fingas on Aug 26th, 2003, 10:13am
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...
Title: Re: Optimal arrow length
Post by Icarus on Aug 26th, 2003, 2:25pm

on 08/26/03 at 00:06:50, BNC wrote:
In any case, I think we have diverged from the original question…


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!
Title: Re: Optimal arrow length
Post by towr on Aug 26th, 2003, 3:10pm

on 08/24/03 at 01:36:57, 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).

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..
Title: Re: Optimal arrow length
Post by James Fingas on Aug 27th, 2003, 8:31am

on 08/26/03 at 15:10:52, 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..


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.
Title: Re: Optimal arrow length
Post by Sameer on Aug 27th, 2003, 3:11pm
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  ??? ::)
Title: Re: Optimal arrow length
Post by BNC on Aug 27th, 2003, 4:00pm

on 08/27/03 at 15:11:08, 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. ...


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 [hide](but maybe sub-optimal)[/hide]
Title: Re: Optimal arrow length
Post by Kozo Morimoto on Aug 27th, 2003, 10:09pm
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?
Title: Re: Optimal arrow length
Post by wowbagger on Aug 28th, 2003, 1:47am

on 08/27/03 at 22:09:15, 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.

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.
Title: Re: Optimal arrow length
Post by BNC on Aug 28th, 2003, 1:52am
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.
Title: Re: Optimal arrow length
Post by Sameer on Aug 28th, 2003, 6:19am

on 08/27/03 at 16:00:49, 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 [hide](but maybe sub-optimal)[/hide]

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.
Title: Re: Optimal arrow length
Post by PUPPY on Aug 30th, 2003, 4:49pm
i don't know
Title: Re: Optimal arrow length
Post by James Fingas on Sep 2nd, 2003, 12:47pm
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.
Title: Re: Optimal arrow length
Post by towr on Sep 2nd, 2003, 1:48pm
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.
Title: Re: Optimal arrow length
Post by BNC on Sep 3rd, 2003, 12:16am

on 09/02/03 at 12:47:03, 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.


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.
Title: Re: Optimal arrow length
Post by aero_guy on Sep 3rd, 2003, 7:57am
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.
Title: Re: Optimal arrow length
Post by aero_guy on Sep 3rd, 2003, 8:01am
Thus, the optimum length will likely occur with L less that or equal to L for max pull.  In all likelihood it will be very near or equal to this value depending on how E decreases in this neighborhood.
Title: Re: Optimal arrow length
Post by Icarus on Sep 3rd, 2003, 3:21pm
Of course, the problem does not ask you to find the optimal length, merely to prove that it exists. Which you have now done!

The reason this is "easy", is because all you needed to do was realize that after some length, travel distance decreases and remains below some level obtained earlier. Combine this with the fact that distance varies continuously with length, apply the "continuous on a finite closed interval implies a max value" theorem, and you are finished.
Title: Re: Optimal arrow length
Post by THUDandBLUNDER on Sep 3rd, 2003, 6:36pm

Quote:
...apply the "continuous on a finite closed interval implies a max value" theorem

Rolle's theorem, I think.
Title: Re: Optimal arrow length
Post by Icarus on Sep 3rd, 2003, 7:15pm
No, its not Rolles' theorem. It's generally called the "Extrema theorem" (since it also predicts a minimum). It is a special case of a general (and simple) un-named topological result:

The continuous image of a compact set is also compact.

Combine this with theorem that a subset of [bbr] is compact iff it is closed and bounded (such as a finite closed interval) and the fact that any closed set contains all its limit points (definition of closed), and you have the extrema theorem.

Rolles' theorem applies the extrema theorem to a differentiable function:

Rolles' Theorem: if [smiley=f.gif] is differentiable on an interval ([smiley=a.gif], [smiley=b.gif]) and continuous on the right at [smiley=a.gif] and on the left at [smiley=b.gif], and if [smiley=f.gif]([smiley=a.gif]) = [smiley=f.gif]([smiley=b.gif]), then there is a [smiley=c.gif] [in] ([smiley=a.gif], [smiley=b.gif]) such that [smiley=f.gif] '([smiley=c.gif]) = 0.
Title: Re: Optimal arrow length
Post by Icarus on Sep 3rd, 2003, 7:54pm
Concerning discontinuous phenomena: Just this week I came up with a situation in my job that involves a situation that is nearly discontinuous if not actually so:

Consider this: You have two vertical bars and a crossbar between them connected by sliders. The slider on one vertical bar is at the end of the crossbar, and the crossbar is able to pivot freely. The crossbar passes through the other slider and is able to both pivot and slide through it. Between the two vertical bars are two circular posts. As the slider on the left is raised, the crossbar rests first on the left post, until the moment it comes in contact with the 2nd post. At this point it lifts off the first post and starts pivoting about the 2nd post.

If you raise the left slider at a constant speed, the speed of the right slider will be discontinuous, and vice versa.

At an extremely small scale, I believe this is actually a continuous action. But for all pratical purposes, it is discontinuous.


Powered by YaBB 1 Gold - SP 1.4!
Forum software copyright © 2000-2004 Yet another Bulletin Board