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External Ballistics Primer for Paintball


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#1 UV Halo

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Posted 07 June 2010 - 11:50 PM

External ballistics is the part of the science of ballistics that deals with the behavior of a non-powered projectile in flight.


"non-powered projectile" sounds like a paintball to me.

Background:
I've been playing since 1989/90 and in that time, I've always wanted something that would allow me to make more accurate shots from a further distance away. I was one of those guys, back in the very early 90s who experienced first hand, the ridiculous amount of hype surrounding paintball products in all price ranges, in every source of information about paintball. Coincidentally, I've also had a very long standing recreational interest in physics.

Myths: Some of these myths could be attributed to marketing techniques employed by most manufacturers in those early years. Some other myths are attributable to the improper (or lack of) understanding in physics. Take for example the statement:. "45 degrees gives the longest range". I've seen this myth in ads, web forums, web videos by respected players and even articles in respected paintball magazines.

Early Research and Study: I have faith (maybe misplaced) that over the years, a lot of companies did research on paintball ballistics. However, none of them really shared what they learned. Eventually, Tom Kaye published a lot of his experimental data and fostered a lot of scientific discussion via the Deep Blue forum on Automag Owners(AO). Regrettably, not being an Automag owner, I was oblivious to their discussions at the time.

Gary Dyrkacz ultimately built upon a lot of the discussion on AO and his efforts led to the "Paintball Trajectory Calculator". He presents a very technical and detailed discussion of the Newtonian physics and fluid dynamics (aerodynamics, more specifically) that influence the trajectory of a paintball. Unfortunately, he found himself lacking one key piece of information: the drag profile for a 'mostly spherical, smooth gelatin surfaced, object with a single seam'. Instead, for his calculations he used drag data derived from the testing of smooth spheres. I'm not writing off his calculator as a failure by any means. Given the limitations he faced, his calculator performs very well even by today's standards.

In my own research, I learned about the science/mathematics of external ballistics. I've come to think of it as a mathematical shorthand that allows for tweaking the key variables that influence a projectile's trajectory. At first glance, a lot of individuals dismiss external ballistics as only pertaining to firearm projectiles, velocities, etc. However, I found that airgunners (the guys who shoot high-end pellet guns for hunting small game, and marksmanship competition) also use the mathematics of external ballistics to predict the performance of their projectiles. I conducted a lot of my own research in this area because I was interested in resurrecting the "Safety Paintball". I learned everything I could only to find that it's still under patent. In any case, knowing the math isn't enough. You need the data to run through it. Generally speaking, there are three methods of collecting this data:

  • The weakest method of is to measure how much a projectile drops if fired at a given velocity and distance. The problem we have all seen with this is that paintballs tend to spread (due to inaccuracy) and this can impact your ability to measure the true drop vs, a shot hitting low because of spread.
  • The absolute best method is a doppler radar tracking and measuring the speed of a projectile as it moves through its entire flight path. Yeah, fat chance of that ever happening in this next to cottage industry.
  • Dual Chronograph measurements are the 'standard' method in firearm and airgun communities. The idea here is to measure the speed of a projectile as it moves through two optical chronographs some distance apart. The difference in speed measured between the two chronographs allows for the calculation of the drag the paintball was subjected to. However, nobody I knew owned one optical chronograph, let alone two.


A Breakthrough:
Cockerpunk and Bryce Larson got it in their heads to do a "ranged" chronograph test to simply see how much paintballs actually slow down between two points so, they could try to create a curve that would describe how fast a paintball would be going at any point in the curve.. What they didn't seem to realize was that they were conducting a test that would allow one to calculate the drag, if they knew how to apply external ballistics. This is where I came in.

Intro to the math: A key piece of data describing a projectile's performance is the Ballistic Coefficient:

In ballistics, the ballistic coefficient (BC) of a body is a measure of its ability to overcome air resistance in flight.


So, what is the ballistic coefficient? Well, it is a measure of how aerodynamic a projectile is both in terms of it's shape (i.e. pointy nosed, round nosed, flat tail, boat tail, etc) and it's mass vs. size. Mathematically, it is expressed as:

BC = SD / i

BC: Ballistic Coefficient (how well does it resist drag) You can obtain this value through solving with SD and i, or you can use data gathered through dual chrono testing to determine it with the aid of a BC calculator like the one I used here.

SD: Sectional Density (how heavy is it vs how wide it is)

SD = M/A
M: Mass of the projectile in pounds
A: Square of the Diameter in inches. Yes, I know that this is not the same as the area of a circle. This is a common shortcut in external ballistics used for comparison purposes. This can be gotten away with because the ratio of D^2 to PiR^2 will always be the same.

i: Form Factor (how pointy is it, how is the back end shaped, etc): Drag Coefficient / Drag Coefficient of the G1 model bullet. WTF? Yeah, in external ballistics, everyone is comparing their round to the G1 model bullet which is a projectile that has a Ballistic Coefficient of 1. Think of the "i" as a common point of reference. You express how your projectile performs relative to the standard model and, calculation software can then predict how your round will perform when given other variables (like mass, velocity, etc).

So, doing all the math has allowed me to feed data to Chairgun Pro (ballistics software optimized for airguns) and make fairly accurate predictions. Unfortunately a lot of this data has been lost by the server so, I'll be recreating it as time permits.

Note: in later posts in this thread you will see me making references to projectiles weighed in grains, 7000grains = 1 pound

Edited by UV Halo, 07 June 2010 - 11:51 PM.


#2 UV Halo

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Posted 08 June 2010 - 12:32 AM

Applying the math to a .683, 3g DXS Silver ball that Bryce Larson and Cockerpunk shot in their dual chrono test. Note initially, I did not publish this data because I was hoping to use it to my advantage on the paintball field. Unlike previously, I'm only using the 50ft data as the trajectory was likely straighter (less elevation)

Testing conditions: 1k ft altitude, 70*f

Average velocity at Muzzle: 290.52

Average velocity at 50': 205.71

Note: You could use one single shot. I use averages and not just one shot so that I can get a hypothetical average of the ball orientations.

Calculated BC: .0060

"TrueBC" (adjusted for Temp and elevation): .0057


I don't need to solve any equations to start making predictions for .683, 3g projectiles.

I'm going to solver for the form factor here as it will come in handy when making predictions for .50cal rounds and hypothetical 'large bore' rounds.

.0057 = (0.0066lbs/.683^2) / i

.0057 = .0141 / i

i = 2.4737


Data and Graphics derived from the exploitation (I use Chairgun Pro, software specifically designed for airgun ballistic calculations):

Note: All Velocities are 300FPS.

Maximum Range: 94 Yards
Angle for Max Range: 26 Degrees
Terminal Energy: .453FtLb


Velocity Vs Distance
Posted Image

Drop Vs Distance
Posted Image

Energy Vs Distance
Posted Image

Time Vs Distance
Posted Image

Distance Vs Deflection Caused by 3MPH Wind
(a wind strong enough to drift smoke but, not strong enough to move a wind vane, you would barely feel it).
Posted Image

EDIT: Fixed pics

Edited by UV Halo, 19 January 2013 - 03:15 PM.


#3 UV Halo

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Posted 08 June 2010 - 08:19 AM

Applying the math to a .50, 1.21g G.I. Milsim ball that Bryce Larson weighed. Can anyone PM me (or create a separate thread) data derived from a dual chrono .50 test?

BC = SD/i
BC = (.0027/.25) / 2.4737
BC= .0044

This BC value is below the lower limit for my ballistic calculator. Until I get dual chrono data for G.I. Milsim rounds, I will use the lowest BC I can (.0050).

Maximum range: 86 Yards
Angle for Max Range: 25 Degrees
Terminal Energy: .160 FtLb



Velocity Vs Distance
Posted Image

Drop Vs Distance
Posted Image

Energy Vs Distance

Posted Image

Time Vs Distance
Posted Image

Distance Vs Deflection Caused by 3MPH Wind (a wind strong enough to drift smoke but, not strong enough to move a wind vane, you would barely feel it).
Posted Image

Edited by UV Halo, 19 January 2013 - 04:19 PM.


#4 UV Halo

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Posted 08 June 2010 - 08:23 AM

Applying the math to a theoretical .710, 3.4g Paintball reportedly shot by Bob Long in the early days.

.710 Paintball
Volume = .1874ci
Mass = .1874ci * .0396 = .0074lbs (51.8gr)
SD = .0074 / .71^2 = .0147
BC = .0147 / 2.4737 = .0059

Maximum range: 96 yards
Angle for Max range: 26 Degrees
Terminal Energy: .526FtLb


Velocity Vs Distance
Posted Image

Drop Vs Distance
Posted Image

Energy Vs Distance
Posted Image

Time Vs Distance
Posted Image

Distance Vs Deflection Caused by 3MPH Wind (a wind strong enough to drift smoke but, not strong enough to move a wind vane, you would barely feel it).
Posted Image

Edited by UV Halo, 15 April 2012 - 02:26 PM.


#5 UV Halo

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Posted 08 June 2010 - 09:50 AM

Applying the math to a .683, 3.11g Tiberius Arms First Strike Round that Punkworks shot in a dual chrono test (I'll update this comment with a link to the data soon).

Testing conditions: 1k ft altitude, 70*f

I calculated the Ballistic Coefficient demonstrated for each shot and the average, corrected for standard environmental conditions (zero altitude, 70F) is:

.0162

I'm going to solve for the form factor here as it will come in handy when making predictions for other virtual First Strike rounds. Numbers rounded for readability. Final Number accurate to four decimal places.

.0162 = (0.0069lbs/.683^2) / i

.0162 = .0.0147 / i

i = .9070

Data and Graphics derived from exploitation (I use Chairgun Pro, software specifically designed for airgun ballistic calculations):

Note: All initial velocities are 300FPS.

Maximum Range: 178 Yards
Angle for Max Range: 32 Degrees
Terminal Energy: 1.388FtLb


Velocity Vs Distance
Posted Image

Drop Vs Distance
Posted Image

Energy Vs Distance
Posted Image

Time Vs Distance
Posted Image

Distance Vs Deflection Caused by 3MPH Wind (a wind strong enough to drift smoke but, not strong enough to move a wind vane, you would barely feel it).
Posted Image

Edited by UV Halo, 19 January 2013 - 03:51 PM.


#6 The_Economist

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Posted 08 June 2010 - 10:45 AM

This right here is why I love Punkworks. Nice work, UV Halo.


#7 UV Halo

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Posted 13 February 2011 - 07:20 PM

Added First Strike Calculations based on Dual Chrono data and, my weight measurements.

#8 betasniper

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Posted 15 February 2011 - 08:33 PM

So wait. The FS round is better shaped than a G1 bullet?
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#9 UV Halo

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Posted 15 February 2011 - 09:34 PM

So wait. The FS round is better shaped than a G1 bullet?


From Wiki:

The BC gives the ratio of ballistic efficiency compared to the standard G1 projectile, which is a 1 pound (454 g), 1 inch (25.4 mm) diameter bullet with a flat base, a length of 3 inches (76.2 mm), and a 2 inch (50.8 mm) radius tangential curve for the point....The G1 model standard projectile has a BC of 1.


So, no. The BC of a FS is only a fraction of how good the G1 Model Bullet.

#10 Steephill

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Posted 15 February 2011 - 09:49 PM

Just looking at all this math hurts my brain.

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#11 spankopotomous=paintball

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Posted 15 February 2011 - 09:55 PM

sooo... whatever has an i value closest to 1 is the best?

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#12 betasniper

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Posted 15 February 2011 - 11:38 PM

Ah, the form factor for a G1 is .5191 ok. That makes more sense.
So a FS is 57% of a G1 performance wise while a paintball is 20% of G1. Got it.
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#13 UV Halo

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Posted 15 February 2011 - 11:44 PM

Just looking at all this math hurts my brain.


If it makes you feel any better, Math is not my strong suit- I only made it through Geometry in High School (not Algebra 2). On the other hand, I love science and paintball.

sooo... whatever has an i value closest to 1 is the best?


No, whatever has the BC value closest to 1.0 (or better) is the best. The i value is just a component of BC. If you want to isolate SD (by keeping the diameter and the mass the same), lower i values are better.

For example:
Standard G1 Bullet:
BC = SD/i
1 = [1lb/(1"^2)]/i
i = 1

A better shaped theoretical bullet:
1.10 = [1lb/(1"^2)]/i
i(1.10) = 1lb/(1"^2)
i = .909090...

Ah, the form factor for a G1 is .5191 ok. That makes more sense.
So a FS is 57% of a G1 performance wise while a paintball is 20% of G1. Got it.


Nope. Check my math above. on Wiki, the .5191 is the drag coefficient for the G1 model bullet. Do not confuse drag coefficient with Ballistic Coefficient (BC) or, Form Factor (i)

Edited by UV Halo, 15 February 2011 - 11:48 PM.


#14 betasniper

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Posted 16 February 2011 - 12:24 AM

Oh, Now I see. You are right. I'm tired. I'll read more tomorrow.
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#15 wagz86

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Posted 20 May 2011 - 12:44 PM

Would it be possible to see the fs rounds chart (the drop vs distance one) but the isolated and expanded view of say 0 to -100 drop scale and the 0 to 50 yards range. That section is what I'm most interested in and the numbers for it seem to be very compressed because of the large scale of the chart

Ps. I guess to make it easier. What I'm specifically looking for is how long do the fs rounds maintain a flat trajectory before dropping off. Thanks

Edited by wagz86, 20 May 2011 - 12:47 PM.


#16 wagz86

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Posted 23 May 2011 - 11:47 AM

Ok wait. I just dled the pics of the charts and expanded them myself. According to these charts the fs round maintain velocity better than normal .68 rnds. Yet the .68 round flys further before dropping to say. -100. So they fly further but lose velocity faster. Am I missing something

#17 UV Halo

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Posted 26 May 2011 - 11:58 PM

You need to re-look at the drop vs distance charts. The .68PB only makes it about 47yds before dropping -100". The FSR on the other hand goes a little more than 60yds forwards before dropping 100"

#18 wagz86

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Posted 27 May 2011 - 12:03 AM

Ok right I see that now. The mistake I made is the scale on the two charts is a tad different. Now what about my first question. Im having trouble see how far the fs round and normal .68 rounds travel before beginning to drop.

#19 UV Halo

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Posted 27 May 2011 - 10:26 PM

Ok right I see that now. The mistake I made is the scale on the two charts is a tad different. Now what about my first question. Im having trouble see how far the fs round and normal .68 rounds travel before beginning to drop.


Literally, not very. They start dropping as soon as they leave the barrel. So, the answer depends on if you want to count fractions of an inch, or several inches, and at those scales, there won't be much of a difference.

Regular PBs drop 10" at around 18yds. FS on the other hand, make it about 20yds. This difference in distance gets larger at longer distances because the PB slows down, therefore it doesn't travel as far horizontally, while it continues to drop just as fast as the FS round.

#20 wagz86

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Posted 27 May 2011 - 10:38 PM

Ahh. Ok that makes sense then why u formatted ur charts that way. Sorry I have a really hard time keeping up with all this math. Put it into the form of a word problem lol. Anyways I get it now thank u very much

#21 UV Halo

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Posted 15 April 2012 - 02:13 PM

Added Theoretical .71 Paintballs (same density) shot by Bob Long.

#22 bball

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Posted 23 May 2012 - 01:06 PM

:) glad I found this

#23 UV Halo

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Posted 25 May 2012 - 10:36 AM

:) glad I found this


I'm glad you found it helpful!! Thanks!

#24 Hitech

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Posted 24 September 2012 - 04:32 PM

Applying the math to a .683, 3.11g Tiberius Arms First Strike Round that Punkworks shot in a dual chrono test (I'll update this comment with a link to the data soon).
Drop Vs Distance
Posted Image


Thanks for this thread and in particular the First Strike data. Any chance of getting the above chart for only say about the first 200 or do inches of drop. Something that maybe has a granularity of one inch?

Thanks. Great thread, wish I would have stumbled on it much earlier.

#25 UV Halo

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Posted 11 October 2012 - 01:42 PM

I apologize for the delay- I somehow missed this question. I'll get on it tonight.

#26 UV Halo

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Posted 11 October 2012 - 06:09 PM

Posted Image

Unfortunately, that's the best I can do at present. Each Horizontal line = 1"

Edited by UV Halo, 11 October 2012 - 06:11 PM.


#27 Hitech

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Posted 11 October 2012 - 06:22 PM

Unfortunately, that's the best I can do at present. Each Horizontal line = 1"


That's great! It's hard to find somewhere I can site in my scope for the First Strike rounds. With this chart I can get the scope close, probably close enough (left to right is already correct). I should be able to set it right where I want it. Posted Image







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