SpudFiles

Alright, we've all heard the ever present new-guy range claims, like, for a 2" barrel ball-valve pneumatic, "i shot 800 meters w/ a potayto!11!". Most of us simply ignore these comments, or post an exasperated response as to how that can't be true. Here's my method of proving ridiculous range claims false. Anybody can argue against a GGDT readout, maybe their valve opened faster, maybe the air expanded faster, yada yada. But, by proving that even under the ABSOLUTE BEST conditions, their launcher couldn't make that range, then you can definitively say they are lying, unless they're shooting some sort of ammo that generates lift somehow.

Anyways, here's the method. I post only so that the physics and math buffs can point out errors (if some exist) and suggest a more optimistic model.

So, basically, it goes a little something like this:

The first things to assume, for very optimal conditions and ease of calculation are:

• Instead of balancing ambient pressure and taking into account the column of air after the projectile in the barrel, just assume the cannon is firing into a vacuum. Use the absolute pressure for these calculations, (i.e. if the operating pressure is 202 kpa, then the absolute pressure is 101 kpa).
• Assume there is no friction in the barrel.
• Assume you are using a 100% efficient valve, even better than a burst disk. Also, assume your propulsive gas has no limits to how fast it can expand. Basically, it will expand infinitely fast unless something gets in its way.
• Assume that all work done by the gases is transferred directly into kinetic energy of the object. No energy is lost.
• External ballistics also occur in a vacuum.

These parameters are very optimistic, but if you prove that even under these conditions, the cannon cannot make the range claimed, then the claim is disproved.

First, you need to figure out the work done by the projectile.

Using a composite function of Boyle's Law and simple geometric formulas for volume, you can derive an equation that yields pressure as a function of distance. Multiplication of the barrel area divided by ten will give the force as a function of distance if the pressure is in terms of kPa (1 kilopascal = 1 newton per 10 cm²). For ease of calculation, this is left undistributed into the function. The function is then simplified as much as possible.

Finally, you integrate the function from 0 to d to determine the work done on the projectile. Since all work is transferred into kinetic energy of the projectile, divide this energy by half the mass, and then take the square root to determine the muzzle velocity.

Now, it is a simple range calculation in vacuum to determine whether their claim is completely bogus or maybe has some merit to it...

So, here's a sample problem:

Initial Pressure: 800 kPa
Chamber volume: 1,000 mL
Barrel diameter: [4/sqrt(pi)] cm ≈ 2.26 cm
Barrel length: 100 cm
Projectile mass: 1 kg

Boyle's Law

Boyle's Law Rearranged

Composition into P as function of d

Simplified...

Multiplication by surface area divided by ten (simplified to 2/5)

Integral of force with respect to d (work)

Factoring out of constant to get integrand in the form of u'/u

Evaluated and simplified integral

Transcendentals approximated to rationals...

Equation for kinetic energy...

Posulated Approximate Muzzle Velocity

And, from here, you can calculate the range in a vacuum easily through standard high school physics.

Didn't he mention something about keeping this to "ease of calculation"?

Well, I think you just won the boobie prize of being the official forum mythbuster.

Actually, if you take these conditions, in a vacuum, the launcher will have a range of about 3 and a half miles.

This absolute maximum range isn't much use for disproving optimistic claims, since I have never heard a claim much more generous than "over a mile".
Even a modest 100 m/s prediction is good for a kilometre of range if drag is ignored.

Drag, that killer of range, just can't be ignored for these purposes.

I reckon that a dispute over GGDT or range calculation results would have to be pretty considerable in the first place you to justifiably say that X yards of range is impossible.

99% of the time, I can't see where my projectiles fired into the distance land - most of the time, I can't even see them at any point in their flight. But from calculations, even with optimistic predictions, I know they can't be landing more than maybe 400 yards away at most.

My argument is that at their generous ranges, they simply wouldn't be able to see the projectile to know where it landed.

You can see a .22 bullet against the sky in the right conditions so you should be able to see anything you are likely to fire from a spud gun.

Hawk eye is right...if you get the sunlight at the right angle. For us its off the back porch in late afternoon. The light reflects off the bullet and you see it fly down range. Kind of cool.
The easy way to check would to have some brave soul go an stand out there and tell you where it landed. We've done it a few times. They used a four wheeler to duck behind and were able to track the object coming at them. Most the time it shot over them and landed several yards away.

Its kind of hard to read the equations... I'm very interested and saves me from posting in this. I'm hoping to do a math report for college on finding the optimal weight of an object with a given velocity and different drag co efficients. Basically I want to graph and compare the trajectories for heavier and lighter objects that have all the same variables except mass.
Does that make any sense and can any one help? I don't want to hijack the thread so PM me if you would...

@Jared: Well, in short, the heavier objects will decelerate less for a given drag force, so a heavier projectile is always going to fly further for a given velocity and drag, with no upper limit, although the returns will be diminishing.

I don't know how far you are into your project, but a better question would be: "For a given initial kinetic energy, and set drag conditions, what is the best projectile weight?"
That way, a heavier projectile will be launched slower, and thus lose some of it's advantage.
That question would actually have a resolvable (non-infinite) optimum weight.

I won't disrupt the thread more, so if you want to discuss it further we'll do it by PM.

@Hawkeye: Well, you're probably enjoying better conditions than I am. Normally I find things I fire just can't be seen.
I don't often fire on really nice days with good sunlight - it's the UK, we're lucky if we see a blue sky once in a year.

Still, things tend to disappear so fast they're next to invisible - others might find different, but I tend to work at high velocity.

However, I ask you - have you ever seen a speeding .22 round 100 or 200 yards away? Seeing things near isn't too hard, but when they get far away, you're going to have a hard time.
Some of the range claims I've heard would require similar eagle-eyed-ness.

Ragnarok wrote:@Jared: Well, in short, the heavier objects will decelerate less for a given drag force, so a heavier projectile is always going to fly further for a given velocity and drag, with no upper limit, although the returns will be diminishing.

On a related informative note....

In the ballistics world (which includes everything from your .22 LR bullet to re-entry vehicles) there's a value referred to as the "Ballistic Coefficient."

The Bc is defined as the ratio of mass to drag forces. The higher the Bc, the better a projectile is at maintaining it's velocity.

More interesting, two objects having a similar Bc will fly the identical trajectory (assuming similar starting velocities). One could be the size of a house while the other is the size of a bug.... Doesn't matter. Same Bc means same trajectory (of course, getting two objects that far separated in size to have the same Bc would be a trick in and of itself!).

Forgive my ignorance but what is the d int he third step? I see later on it is Work but I don't see how you can substitute it in there in the 3rd step.


Math looks good though, aside from that everything adds up.

D_Hall wrote:In the ballistics world (which includes everything from your .22 LR bullet to re-entry vehicles) there's a value referred to as the "Ballistic Coefficient."

Yes, although I couldn't consider myself a total expert at aerodynamics, I have been reading around the subject recently.

I was "assigned" (although I didn't have much choice in the matter) the task of re-writing and improving Joannaardway's old range calculator, so I've been scouring different internet sites and textbooks for all the scraps of information I can find.
Currently the folders full of this information run at 55.5 Megabytes, along with all the work and a couple of scanned scrawlings I've put in there with it - and the spreadsheet itself, a spare and a couple of others on related matters.

I've steered clear of using Bc specifically, because this requires the user to calculate a Bc themselves, instead, the program calculates it itself from Cd, Mass and projectile area (I think that's it, but no doubt as tired as I am, I've missed something... or maybe not.)
The other reason is that Bc is usually only commonly published for firearms bullets, and I wanted to separate the program from one made for regular firearms.

What I did steal though was the data from the drag models that the firearms industry has developed to help with transonic and supersonic effects, even if they are a little rare in the spudding world.
Finding the models for the unusual shapes spudders use like cylinders and blunt nosed projectiles was a regular nightmare - but, nightmares have bad dreams about me...

Anyway, I need to stop rambling, and go and lie on that flat soft thing I keep in my bedroom for stubbing my toes on.

Potatobusters.....

Rokmonkey wrote: Forgive my ignorance but what is the d int he third step? I see later on it is Work but I don't see how you can substitute it in there in the 3rd step.

The d is distance down the barrel. 4 (the cross-section area of the barrel) times d equals volume in the barrel at distance d, and add that to 1,000 gets you total new volume.

Thanks, makes sense now.

Ragnarok wrote:I was "assigned" (although I didn't have much choice in the matter) the task of re-writing and improving Joannaardway's old range calculator, so I've been scouring different internet sites and textbooks for all the scraps of information I can find.

If you've got any questions, feel free to ask... In a former life I used to make my living writing flight simulations for missile/projectile systems (ie, very complicated range calculators ). That's actually a bit of why the ballistics stuff in GGDT is so crude.... GGDT was (to me) an academic excercise. There was/is no challenge in writing a range calculator for me so I've no interest in doing it.

In any event, feel free to ask.

Currently the folders full of this information run at 55.5 Megabytes, along with all the work and a couple of scanned scrawlings I've put in there with it - and the spreadsheet itself, a spare and a couple of others on related matters.

You may want to look into a program called "DATCOM". It is a program that was written by the Air Force to do aerodynamic predictions and is available online/free these days as it is considered obsolete (but is still a great tool for the hobbyist).

Finding the models for the unusual shapes spudders use like cylinders and blunt nosed projectiles was a regular nightmare - but, nightmares have bad dreams about me...

I wouldn't kill myself worrying about it... Unless you're counting on rifled barrels such projectiles are going to tumble (and then all hell breaks loose!).

Can't forget to calculate what phase the moon is in or the latitude of the shot. :-p Sorry, couldn't resist. My biggest problem is finding a field big enough to test my launcher with. If I lived in Eastern, not Western Washington I could have a desert, but around here, shooting over the water at a boat might be my best bet. Maybe Boeing's parking lot... I'm pushing 300yards with the 23 gram round at 300psi. Diminishing returns around 350psi. I think the nerfy round is deforming at that psi and at that weight. Going to try a heavier mix.

However, I ask you - have you ever seen a speeding .22 round 100 or 200 yards away? Seeing things near isn't too hard, but when they get far away, you're going to have a hard time.
Some of the range claims I've heard would require similar eagle-eyed-ness.

Your right we don't see them from long. Its just for the split second that the sun hits them just right.... 8)
But if you sick a scope on a paint ball gun or a low power BB gun you can often see the trajectory of the projectile. It used a scoped on my brothers BB gun to teach him about wind age...

I would think shooting over water would be the easier way to figure out how far it goes. As long as it floated...even if it sank if you had some kind of dye to mark where it hits the water...

Edit: I haven't quite begun the project yet but its in the process of being started I'll pm you when I need the helps thanks.... 8)