External Ballistics, The First Strike Round, and Paintballs

[uv_halo]

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 early 1990 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 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.

Early Research and Study: I have faith (maybe misplaced) that over the years, at least a few 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" (page currently down). 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". 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.
• 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 a 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 of Punkworks 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 this one.
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 it is, how the back end is 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). If you have the BC (derived from dual chrono test), and the SD (measured and calculated), you can solve for the form factor with i = SD/BC.

So, doing all the math has allowed me to feed data to Chairgun Pro (ballistics software optimized for airguns) and make fairly accurate predictions. The following posts below will apply these techniques.

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

[uv_halo]

Applying the math to a .683, 3g DXS Silver ball that Bryce Larson and Cockerpunk shot in their dual chrono test. 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 at above conditions

"TrueBC" (adjusted for 70deg f, Sea Level): .0057

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

I'm going to solve for the form factor (i) 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 initial velocities are 300FPS.

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

Velocity Vs Distance


Drop Vs Distance


Energy Vs Distance


Time Vs Distance


Distance Vs Deflection Caused by 3MPH Wind

[uv_halo]

Applying the math to a .683, 3.11g Tiberius Arms First Strike Round that Punkworks shot in a dual chrono test with a smoothbore barrel (it appears that this data has been lost, I hope to do a re-test in the future).

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


Drop Vs Distance


Energy Vs Distance


Time Vs Distance


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).

[uv_halo]

Applying the math to a .50, 1.21g G.I. Milsim ball that Bryce Larson weighed. This is an example of how one could take a known form factor, and determine the BC without using a dual chrono. However, I would like a dual chrono test at some point.

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). Know that the actual performance is most likely worse than depicted below.

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

Velocity Vs Distance


Drop Vs Distance


Energy Vs Distance


Time Vs Distance


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).