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Thread: Paintball Spin Physics - Getting to the final Answer

  1. #121
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    Well... I'm no scientist, but i would have to agree with bjjb99 to some extent about the Reynolds number in relation to the aerodyamics of the PB in question.

    It was my impression that the discussion was relating to the Paintball that, out of a series of shots would, for no explainable reason change directions in mid flight. Now I ask is did this same change of direction occur with the nylon balls as well as the paintballs in the test firing?
    If so then the answer would point towards a issue of aerodynamics to a sphere in flight. If that were so I would rest my case.

    It is only my obervations on the paintball field that lead me to this conclusion. that all paintballs are not created equal, no ball is without its own characteristics as I said earlier...To what final conclusion this thread takes toward the determation of "x" a person must discount all other factors...THEN account for the varible characterists of each ball fired.

    But then thats what we call science...

    Play on....

  2. #122
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    hmm

    any reynolds affect would be more pronounced on a paintball
    than say a black powder ball. This is because of the surface
    area/weight ratio, or desity.

    That makes things interesting, I know that for distance shooting
    we load boat tail bullets which have the back edge of the bullet
    tapered to decrease turbulence behind the projectile.

    .............
    well, I thought I would have free time this week, guess not.

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  3. #123
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    Originally posted by bjjb99
    I am leery about extrapolating behavior seen at these Reynolds number values up into the regime occupied by a paintball moving at 280 fps. If my constants and Reynolds number forumla are reasonably accurate, then an Re = 300 paintball would be moving at around 1.6 fps. That's two full orders of magnitude we're extrapolating over here, and it makes me nervous.
    I read that the effect continued well past those numbers. I think it's just easier to photograph at the slower speeds. I'll try to find the information and post it (with links).

    Okay, I found it. Everything I have posted has been about flows past a cylinder. Here is what they say about spheres. It appears that the cross over point from minor to major influence is at approximately the reynolds numbers paintballs fly at.

    Flow Past a Sphere

    Wakes behind spheres are observed to be steady for Reynolds numbers below 300-400. Above this limit (which also depends on the surface finish) vortices break off and are periodically released to form vortex loops that are connected like in a chain.

    At Re above 6000 the vortex shedding is very periodic, with Strouhal number ranging from 0.125 to 0.20, the largest figure being a limit at high Reynolds numbers (Achenbach, 1974). Similar wakes can be observed behind particles falling in water. Effects of the surface geometry have been studied for the evaluation of the aerodynamic performances of sports balls (Metha, 1985)

    Here is the link that includes the above text:

    http://www.aerodyn.org/Unsteady/unsteady.html
    Last edited by hitech; 01-06-2003 at 02:59 PM.

  4. #124
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    *chuckle* In the time between my reading your initial statement about finding and posting the link and your actually posting the link, I happened across the very same link!

    I'll agree that the Karman vortex effect does occur at high Reynolds numbers. As part of the link I found while you were posting it, they mention something called the Strouhal number. I dug into this a bit, and it seems that the vortex shedding frequency can be determined from this number, the object's diameter, and the object's velocity. Here's only one of several links I found. This one in particular has Strouhal numbers in the Re ~ 10^5 range, so it worked out nicely with our paintball studies.

    http://www.enseeiht.fr/hmf/travaux/C...s/strouhal.htm

    For a 68 caliber paintball moving at 280 fps, I get a vortex shedding frequency of about 940 hertz. I don't know if that means one vortex every 1/940th of a second, or one vortex pair per 1/940th of a second. This would mean that we have a vortex (or vortex pair) every 4 inches or so as the ball heads downrange. Given that the vortex shedding is "very periodic" at high Reynolds numbers, wouldn't this result in a purely oscillatory motion instead of what Tom is seeing, namely a strong divergence in one direction near the end of a ball's travel? Even near the end of its flight, a paintball that has enough energy to break on a target still has a pretty darn high Reynolds number.

    On another note

    Since vortex shedding alters the pressure distribution of the air surrounding it, do you think this shedding would be audible? Do you think that might be the characteristic "whoosh" sound you hear when a paintball flys past your head? On a lark, I played a 940 hertz pure tone using Mathematica, and it sounds fairly similar to the fundamental tone you hear when a paintball goes by... something to think about, I suppose.

    BJJB

  5. #125
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    Originally posted by bjjb99
    Given that the vortex shedding is "very periodic" at high Reynolds numbers, wouldn't this result in a purely oscillatory motion instead of what Tom is seeing, namely a strong divergence in one direction near the end of a ball's travel? Even near the end of its flight, a paintball that has enough energy to break on a target still has a pretty darn high Reynolds number.
    I'm thinking that very periodic could easily mean significantly more vortices are shed from one area of the sphere that the other. It would follow that the sphere would then change course. This would alter the angle of attack and maybe that could further alter things. As you have probably guessed, I don't know. However, it sure seems feasible to me.

    Originally posted by bjjb99
    For a 68 caliber paintball moving at 280 fps, I get a vortex shedding frequency of about 940 hertz...This would mean that we have a vortex (or vortex pair) every 4 inches or so as the ball heads downrange. Since vortex shedding alters the pressure distribution of the air surrounding it, do you think this shedding would be audible?...On a lark, I played a 940 hertz pure tone using Mathematica, and it sounds fairly similar to the fundamental tone you hear when a paintball goes by your head...
    Very interesting. Something has to be causing the vibrations that cause that noise.

  6. #126
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    YES!!!!! YESSSS!!!! I LOVE YOU GUYS!!!!

    We have arrived!! I didn't think it would ever happen that players would take the time to investigate the science behind paintball flight to its logical conclusion but the magic of AO comes through.

    YES vortex shedding happens in the R numbers we are talking about.

    YES we can verify this because paintballs make a buzzing sound when they go by.

    YES this happens with perfectly round nylon balls too.

    Now for the last step. BJ mentioned that well maybe its just oscillating so it really doesnt have an effect on accuracy. Here is the last thing to look up. Its called "The drunken mans walk". Its a statistical model that fits what we are talking about here to a tee. Go get it!!

    A very excited,

    AGD

  7. #127
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    Wow, I just read this thread, from page 3 (yeah, I skimmed a bit) and this is just incredible. Only on AO could you see this kind of thing. Absolutly amazing.

    It's awesome that we've found out (apparantly TK's always known) about this "reynolds number" but more importantly (to me anyway ) how do we combat this? or Can we?

    Is the future of paintball going to see differantly shaped paintballs?

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  8. #128
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    Originally posted by AGD
    YES!!!!! YESSSS!!!! I LOVE YOU GUYS!!!!
    Awwww... shucks.

    Originally posted by AGD


    YES vortex shedding happens in the R numbers we are talking about.

    YES we can verify this because paintballs make a buzzing sound when they go by.

    YES this happens with perfectly round nylon balls too.

    Now for the last step. BJ mentioned that well maybe its just oscillating so it really doesnt have an effect on accuracy. Here is the last thing to look up. Its called "The drunken mans walk". Its a statistical model that fits what we are talking about here to a tee. Go get it!!
    I'm already familiar with random walks of many sorts, and I would agree that a series of random "kicks" perpendicular to the ball's trajectory would result in a random walk sort of behavior.

    In a normal random walk, if the steps are of uniform length but each in a random direction, then the average distance between the start and end location is directly related to the square root of the number of steps taken. I cannot remember off the top of my head what happens if the steps are of both a random length and a random direction... gonna have to think on that one a bit.

    For a paintball, the steps change the ball's velocity perpendicular to its trajectory... more along the lines of a random walk with random stepsize and direction.

    However, to conclude that vortex shedding and a random walk are reasonable culprits for the observed inaccuracies of paintballs, we need to establish two things first:

    1. The vortex shedding must be random in nature. The references I've seen so far seem to indicate that the shedding is periodic (i.e. repetitive and thus nonrandom, at least the way I interpret the term "periodic") at high Reynolds numbers.

    2. The shed vortices must impart enough momentum to the ball in a direction perpendicular to the ball's trajectory to affect its flight by the amount observed at the end of the ball's travel.

    Once we establish these two and can do a bit of modeling, I think we might have a nearly complete explanation of what's happening with respect to paintball accuracy.

    Incidentally, if vortex shedding is the cause, then the shed vortices can only impart so much momentum to the ball during its flight. If the paintball is more massive (thicker/denser fills, anyone?), then that momentum would not change the ball's lateral velocity as much. Similarly, heavier balls are less sensitive air resistance and would tend to travel farther (I think... it's early and I'm operating on about 8 hours sleep for two days).

    Tom, have you witnessed lighter balls having a statistically larger shot grouping than heavier balls fired at the same initial velocity?

    BJJB

  9. #129
    whats the hypothesis, and conclusion? I am realy not in the mood to read 7 pages of posts.

  10. #130
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    Here is the definition of a random walk (I'm assuming that the "Drunken Man's Walk" is a random walk):

    A random process consisting of a sequence of discrete steps of fixed length. The random thermal perturbations in a liquid are responsible for a random walk phenomenon known as Brownian motion, and the collisions of molecules in a gas are a random walk responsible for diffusion. Random walks have interesting mathematical properties that vary greatly depending on the dimension in which the walk occurs and whether it is confined to a lattice.

    Here is a link that elaborates on this more:

    http://mathworld.wolfram.com/RandomW...mensional.html

    It seems like a very simple concept to me. Each vortex shed would "pull" the paintball in the opposite direction that it was shed in. If they shed in random directions then the paintballs flight should follow a random walk. Tom, you never managed to photograph evidence of the vortices shedding from the paintballs, did you?

  11. #131
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    Confedman-

    what this means is.. we have been looking for what this "X" force is that is making hte balls go all walkabout on us.

    We find that when the Reynolds number in in this certain range or higher, vortexes are shed off of parts of the ball in that neat little chain loop fashion seen in the pics. Each one of these sheddings would pull the ball in on the opposite side. The sheddings pullign the ball here and there make for that nice unpredictable movement we all see.

    Now we just must define it better.

    Unfortunately I don't see any real world fix for this. It is a natural occouring thing.

    I think I got all that right. That was sorta a recap for me.

  12. #132
    hmm does the walk about worst or better with thicker fill?(im guessing better). we should try like a powder ball, but not too fine since some people would have a horrible time breathing if they got hit in the mouth(throught the mask). how about a fill thats like a damp flour? muddy but not dusty.

  13. #133
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    I don't think it has anything to do with the fill of the paintball. It's simply a naturally occuring phenomenon that happens when you've got a non-spinning object that's traveling though the air at high speeds/reynolds numbers.

  14. #134
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    Ok look at these applets. They demonstrate where the ball will end up (usually off center) at the end of the random walk.

    http://math.furman.edu/~dcs/java/rw.html

    Lets imagine that the applet above allows you to "fire" a ball down range. If we are shooting a ball 80 feet, and we assume that the vortex gets shed and takes a "Step" every 4 inches (as proposed above) then we would get about 250 steps. This is too few to see anything in the applet but lets input 500 steps and then run and clear the program a bunch of times to see the effect.

    Now notice that if you didnít know any better (and couldn't see the small movements) the balls would look like they were CURVING away from center. Also notice that some go right down the middle. Every once and a while you get a real "flier" that curves way off. Does this look familiar??

    Here is another applet that records were each ball "hits" at the end of the walk. Every hit gets recorded as a red bar that builds vertically. The red bars are distributed to the left and right of center. This is only one dimensional but you should be able to get the idea. Hit start and let it run for a while.

    http://stp.clarku.edu/simulations/on...onal-walk.html

    Once you get enough "hits" your graph will look like a bell curve. This means that MOST of your shots will be pretty close to center and you will get fewer and fewer hits the farther you get from center.

    Stack up enough hits and use the gaussian button on the right to see how well your shot pattern fits the bell curve. Fits pretty good after a bunch of trials right?

    Now here is the leap. The fact that it fits a bell curve means that the results are RANDOM!! The gaussian fit is a test for RANDOMNESS. Why is this so important?? After all its what you were expecting right? Itís important because if itís random, then NOTHING IS AFFECTING IT!!! If it were the bolt, or the barrel or anything else that could improve accuracy it would not make a bell curve. Well actually the turbulence is affecting it but in a completely random way.

    So lets think about it, the ball gets pushed around by the bolt, the air blast, the barrel and whatever else is in the way until it leaves the barrel. Now its alone by itself in the free air and instead of clear sailing to the target it gets viciously attacked by the ďXĒ factor Shredding Vortices. Hundreds of times it a couple of seconds it gets pummeled mercilessly back and forth through the air. After a thorough beating it ends up on target randomly distant from the center.

    Now I ask you, after this thrashing, this pinball machine of air flow, this twisting vortex tail of tormented accuracy, how does the barrel, bolt or air blast make your paintball more accurate??

    AGD

  15. #135
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    The barrel can make your shot more accurate by keeping the velocity more consistant with the proper paint/barrel match.....
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  16. #136
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    Right, keeping velocities consisitant is the only thing that we paintballers can do in order to make our guns more accurate.

    the graph that's on applet one shows where the ball would land, assuming that velocity is consistant, and that nothing else is effecting the ball's flight.

    now having inconsisitant velocities is alltogether pointing the ball in a differant direction, wheter it be higher (higher velocity) or lower (lower velocities). It's like moving the graph itself around, and then re-drawing your random points.

  17. #137
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    My opinion is that the gun can't effect the paintball's accuracy besides consistency. Like joey said, the higher and lower velocities would move the target graph of the landing point. So that means it would take more steps or less steps in the random walk applet providing that it still gets pummeled by airflow at the same rate. That means the longer it flies the more chance of random movement and the shorter it flies the less chance of movement. The only problem with the shorter distance is... well... shorter distance.

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  18. #138
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    Originally posted by joeyjoe367
    I don't think it has anything to do with the fill of the paintball. It's simply a naturally occuring phenomenon that happens when you've got a non-spinning object that's traveling though the air at high speeds/reynolds numbers.
    I think you're missing the intent of my statement. It's not that the fill somehow changes the way the air travels around the ball, it's that a denser fill (and thus a heavier ball) is more resistant to the forces resulting from the way air travels around it. Since I don't know how much brand-to-brand (or batch-to-batch) variance there is in paintball fill density, I can't conclude that this is a signficant contributor to accuracy.

    BJJB

  19. #139
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    Ah, I see. a denser fill/heavier ball would be less effected by the vortex shredding since it carries more momentum than a lighter ball.

    I believe that there are regulations concerning the weight of a paintball. A heavier ball packs a bigger punch at 300 fps than a *EDIT*lighter*EDIT* ball, so safety concerns are there.

    Unless you lower the speed at which you fire the ball. You might actually not lose any range since the heavier ball loses velocity slower than a lighter ball.

    Hmm... How about a weighted 10 gram nylon ball? That could test our theory... I would be more than willing to bet that it would be more accurate.

  20. #140
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    Last time I checked (years and years ago now...) paintballs averaged between 50 and 54 grains. I tend to use 52 grains as my average when doing calculations. They may weigh differently now?...

    manike

  21. #141
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    I think the average is about 3.2 grams.....

    Some paintballs weigh A LOT less. RPS Lightning weighs 2.5 grams

    Here are some weights I have found so far:

    Inferno--3.27g
    JT--3.19g
    Flash--3.23g
    Direct Hit--3.26g
    PMI--3.21g
    Marbalizer--3.25g
    Lightning--2.56g (Now RPS Mercury)
    Big Ball--3.18g
    All Star--3.23g
    All Star Yellow--3.14g
    Evil Marbs--3.22g
    Great American Premium--3.17g
    Last edited by ES13Raven; 01-08-2003 at 12:39 PM.

  22. #142
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    Originally posted by AGD
    Ok look at these applets. They demonstrate where the ball will end up (usually off center) at the end of the random walk.

    http://math.furman.edu/~dcs/java/rw.html
    I could not get this one to work through the various firewalls and such... all I got was a page with some text on it but no java applet, so I wrote my own random walk using Matlab. It takes a series of one unit steps in random directions and computes the endpoint of the travel.

    Originally posted by AGD
    Here is another applet that records were each ball "hits" at the end of the walk. Every hit gets recorded as a red bar that builds vertically. The red bars are distributed to the left and right of center. This is only one dimensional but you should be able to get the idea. Hit start and let it run for a while.

    http://stp.clarku.edu/simulations/on...onal-walk.html
    You can't use this applet to describe the distribution of paintball hits. The behavior of a fixed stepsize position-based random walk in one dimension is not the same as a fixed stepsize position-based random walk with random direction in two dimensions. The "distance from center" distributions are nowhere near identical.

    In the one dimensional case, the mean distance from center at the end of travel is equal to zero, and the standard deviation is equal to the square root of the number of steps taken. The distribution of distances from multiple trials fits a gaussian nicely.

    In the two dimensional case things are quite different. Now the mean distance from center at the end of travel is equal to the square root of the number of steps taken. The standard deviation is darn near half the mean value. The distribution's shape is by no means gaussian anymore, and actually more resembles a Poisson distribution.

    Additionally, a two dimensional position-based random walk is not identical to what's happening during a paintball's flight. At best it is a mapping of the ball's lateral velocity components during flight. This mapping must then be converted to the actual ball's position, which is a cumulative process.

    After a bit of number crunching, it looks like the Poisson-like distribution is retained through the conversion of random walk based velocities into actual distances from the origin, but the magnitudes of the distances are much greater than for a plain positional random walk. Instead of the mean distance being around the square root of 250 (around 15.8 times the step size), the mean distance is now upwards of 1000 times the step size. In addition, the relationship between the mean distance and the number of steps taken now seems to be more quadratic in nature instead of the familiar "square root of number of steps taken".

    So you're right in saying that a random series of "kicks" to the paintball during its flight will result in a randomization of the shot pattern. However the shot pattern you get from the effects of these "kicks" is not gaussian in nature, assuming the kicks are all of equal magnitude. Instead, it should be more Poisson-like than gaussian. This has the interesting effect of making the probability of hitting close to your intended point of aim very small, and giving the shot grouping a donut shape (if the Poisson distribution is sharp enough).

    With respect to whether the "kicks" are randomly oriented, I still have reservations. I've yet to see a reference that explains why shed vortices would be periodic in time but random in orientation when they come off the back of the paintball. All I've seen so far is that at high Reynolds numbers the vortex shedding is periodic, and I still interpret that to mean "cyclic with respect to time and space".

    BJJB

    Incidentally, I created a short-lived Von Karman vortex street in my bathtub a couple of nights ago, using nothing more than my arm. Fascinating stuff...

  23. #143
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    Hmmmmmm

    Certainly, the pressure variables from a random shedding vortex is one element of the actual flight of a paintball but I think that to conclude that it is the only element involved in the size of the shot pattern on target, is really a stretch. If the hardware played no role in the event, it must then follow that any device or combination of hardware fired at the same velocities with the same batch of paint should show essentially identical shot groupings (in terms of the size of the shot group), but that is not the type of results that I've seen.
    It seems to me that the calculations of the results of a single effect on a rigid sphere should not be held as the last word in predicting the flight path of a flexible and not so round projectile like a paintball. Since a paintball in flight can present both round and elliptical shapes to the airstream and at varying "angles of attack", it apears that the pressure variables in the vortex become much less random. The hardware can affect the status of the ball (relative to shape and rotation) when it enters the air stream; which in turn will influence the results or magnitude of other forces that act on the ball in flight.
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  24. #144
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    Originally posted by bjjb99
    I've yet to see a reference that explains why shed vortices would be periodic in time but random in orientation when they come off the back of the paintball. All I've seen so far is that at high Reynolds numbers the vortex shedding is periodic, and I still interpret that to mean "cyclic with respect to time and space".
    I have been looking for the same evidence/explanation. However, it seems more likely that the shedding frequency is periodic and the orientation is random, at least random with respect to the previous vortex's orientation. After all, what would the orientation be? It seems that any slight difference in the air flow around the sphere would influence the shed orientation, especially with our not quite round spheres. It may not be truly random, but it doesn't seem that there is any way to predict it (the orientation).

    I haven't played around with any of the applets, but I may. Or, I might write my own. The code for the walk is extremely simple, I just need to come up with a simple way to represent the results.

    Man this is fun stuff. Now I know why I took physics in college just for fun.

  25. #145
    ben_JD Guest

    I am not a Deep Blue member

    If my posting this here is inappropriate, please erase.

    My question may seem trite, but are the vortices shed by a paintball strong enough to influence a second paintball following along a similar path?

    i.e. do vortices shed by prior paintballs further disrupt subsequent paintballs in a long string of shots (indepedent of the subsequent paintball's own vortex shedding issues)?

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    Re: I am not a Deep Blue member

    Originally posted by ben_JD
    If my posting this here is inappropriate, please erase.

    My question may seem trite, but are the vortices shed by a paintball strong enough to influence a second paintball following along a similar path?

    i.e. do vortices shed by prior paintballs further disrupt subsequent paintballs in a long string of shots (indepedent of the subsequent paintball's own vortex shedding issues)?
    Good question. I do know that vortices can hang around for a surprisingly long time. And I do know that wing tip vortices from aircraft can disrupt the airflow over a subsequent aircraft's wing. So I suppose it is at least possible for a paintball's vortices to affect subsequent shots.

    For now, however, I think I'm going to concern myself with how the vortices affect the paintball that's doing the shedding. Can anyone point me towards a reference describing the magnitude of momentum change imparted by a shed vortex? Preferably one with big writing and lots of pretty pictures...

    BJJB

  27. #147
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    Originally posted by hitech


    I have been looking for the same evidence/explanation. However, it seems more likely that the shedding frequency is periodic and the orientation is random, at least random with respect to the previous vortex's orientation. After all, what would the orientation be? It seems that any slight difference in the air flow around the sphere would influence the shed orientation, especially with our not quite round spheres. It may not be truly random, but it doesn't seem that there is any way to predict it (the orientation).

    How about "two consecutive orientations are radially symmetric about the paintball's trajectory"? I'm not saying that we should be able to predict where the shedding occurs; rather, I'm suggesting that the shedding oscillates between a couple of distinct locations on the ball. Once the oscillation is started, it may well be tough to kick it out of such a back-and-forth mode... sort of a metastable oscillation state, if you will.


    Originally posted by hitech


    I haven't played around with any of the applets, but I may. Or, I might write my own. The code for the walk is extremely simple, I just need to come up with a simple way to represent the results.

    What exactly are you trying to represent? A good scatter plot from multiple random walk trials is a decent way of showing where each path ends up. A histogram of "final distance from origin" data will show the distribution reasonably well. And of course there's the good old mean and standard deviation results to look at too.

    Do you have access to Matlab? If so I've got random walk code already written, along with a transformation from acceleration/velocity to positional data. You can plot pretty much any variable against any other as long as the vectors/arrays are the same size.

    Originally posted by hitech

    Man this is fun stuff. Now I know why I took physics in college just for fun.
    Heh. Guess I have fun for a living then.

    BJJB

  28. #148
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    At 300 fps, firing 10 balls per second, the paintballs are going to be 30 feet apart in the air.

    What we need to do is figure out if/how the vortices in the air will spinoff and possibly cause our next shot to vary off course. The thing is, at 10 balls per second at 300 fps, yes they may be 30 feet apart which seems like alot but they are travelling at that same speed so the balls will be 0.1 seconds behind each other. Will the air turbulence from the vortices shedding hang around for 0.1 seconds? because if so, that ball behind the first is going to come through that are where the first ball shed the vortice and hit that turbulence exactly 0.1 seconds later going 300fps, and possibly shedding vertices of it's own.

    I might have to make some 3d animation of this process in real-time if i have a chance... that would really convince people.

  29. #149
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    BJJB,
    Now I see what you are getting at. Yes, it would be nice to find something that states the vortices shed randomly. I could see them oscillating and the orientation spinning, possibly back and forth.

    Not I do not have Matlab, I don't even know what it is. I was thinking of writing a simple VB program. Having the program walk is easy, showing it on the screen is what I have to figure out. Simply listing the ending point with direction and distance requires math formulas I don't know. I may have taken physics in college, but I've never used it since.

    I'm sure it's not as much fun when it's work.

    Here is a link on the force generated. Is it any good?

    http://www.eas.asu.edu/~squires/sphere_summary.html

  30. #150
    Join Date
    Dec 2001
    Location
    Northern VA
    Posts
    316
    Hitech,
    Matlab is a data processing and visualization programming language. It was designed to allow rapid scripting of processing algorithms. The scripts are processed interpretively instead of being compiled, like the old interpretive BASIC programs of yesteryear. Matlab operates on matrices very quickly, allowing the user to skip having to write a bunch of nested loops.

    Distance from origin is simply the Pythagorean theorem:

    distance_from_origin = sqrt(X_endpoint^2 + Y_endpoint^2)

    Direction would be a trigonometric exercise if you want angles instead of a vector. I used a pre-canned function called atan2... it's a four-quadrant arctangent function, covering the full 360 degrees of a circle instead of just a portion of it.

    The link you listed had a nice plot of force coefficient versus some formula that I couldn't identify. I think the greatest contribution I obtained from that link was a new suite of search terms to plug into search engines.

    BJJB

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