From Cd to FF to SD to BC and Df too... maybe I finally get it??? LOL

[Poorman Plinker]

I think I finally get it!!! I have used as many of Mr. Sterne's replies to my posts to formulate this. (Thanks Bob!)

Is this close to the common pellet ballistic terminology and thinking used today?

Once you have the Drag Coefficient (Cd) you can determine the Form Facor (FF) and the Drag Force (Df). Once you determine the Sectional Density (SD) and you have the FF you can determine the Ballistic Coefficient (BC). You can calculate the subject pellet coeffecient of drag (Cd) if you have two velocities and a known distance between them, plus the projectile weight and caliber and the atmospheric conditions under which the test was conducted. The Cd of the subject pellet is the deceleration of the subject pellet at a specific speed range.

The Drag Coefficient of a G model (Cg) is based on shape and standard ballistic models and varies slightly at subsonic Mach Numbers. The standard drag function for each standard bullet is unique. What this means is that at sea level standard atmospheric (STP) conditions the standard drag deceleration of the standard reference bullet is the product of the velocity of the bullet multiplied by the value of the standard drag function at that velocity. A standard drag function (G1,G7, etc.) can often be found as a  pair of numbers on a table of the specific speed of the bullet in the air and the drag deceleration of the standard bullet at that bullet speed.

In the G1 drag curve, the Cd drops markedly under Mach 0.3 then slowly declines until around Mach 0.8, The G1 drag curve is based on an artillary shell shape with  a caliber of 1.0", a weight of 1.0 lb, length of 3.28", a 2" radius tangential nose curve and a flat base. The G7 drag curve is based on a rifle bullet shape.

The G7 Cg is more parallel to most subsonic projectile shapes depending on its nose and length. We can assume that comparing a round nose pellet to the the G7 Cg is at a constant for velocities under Mach 0.8 and that is indeed the case for a sphere. The Cg of nine standard projectiles are well known and 2 models are commonly programed in BC calculators over a large range of Mach numbers.

One could certainly argue that artillary shells and rifle bullets do not give a representative comparison to pellet performance. However, the G models will have to do until someone creates a P1 model and the software to calculate the drag differences. In any case it is important to know which model is used when comparing data from different sources.

Most G1 and G7 Cd/Mach charts show that¸ until Mach .8, round nose pellets have improving or parallel Cd comparison numbers respectively. Because of the differences the FF of subsonic pellets will vary markedly against a G1 model and stay fairly consistent against a G7 model.  In general the G1 model yields comparatively high BC values and is often used by the sporting ammunition industry.

For a cylinder (wadcutter)  the Cd starts to increase at velocities as low as Mach 0.6 (670 FPS). For round nose pellets the Cd starts to increase at about 850 FPS. Once you delve into higher Mach numbers and transonic speeds the Cd is no longer a constant. When you approach Transonic velocities (typically over Mach 0.8 or 900 fps) air becomes more highly compressible and Cd rises  markedly in both the G1 and the G7 model.

Form Factor (FF) has a specific meaning in ballistics. It is a measure of the drag of a projectile relative to whatever standard  it is compared. The FF is a number found from the equation: FF = Cd (study projectile) / Cg (G model used). FF is determined by dividing by the Cg of the drag model you select. As mentioned above the G1 (artillary shell shape) models are highly variable and the G7 (rifle bullet shape) models are more consistent with subsonic pellet Cd performance.

How much a projectile deviates from the applied reference projectile is mathematically expressed by the form factor. Form Factor represents the specific drag deceleration of the pellet divided by the standard reference bullet speed for a ratio to the drag deceleration of the standard bullet. If your pellet has a lower Drag Coefficient (Cd) at a given velocity, then its FF will be less than 1.0 at that velocity. If the Cd is greater than the Cg model, then the FF will be more than 1.0. If the Cd is more than 1.0 the BC will be better than a model projectile of the same SD. If the FF of the particular standard projectile you are comparing your projectile to is 1.0 then the BC equals the SD.

Sectional Density represents the lbs./sq.in. with which the bullet resists acceleration in the barrel or deceleration in the air or flesh. The higher the SD, the better a pellet maintains its velocity and energy but the harder it is to accelerate in the first place. A pellet with high SD requires more pressure or barrel length to have equal FPE to a pellet with low SD.

The Sectional Density (SD) is determined by using uses the diameter (groove, land, pellet cross-section or nominal caliber) and the weight of the pellet in grains in the following formula: SD = Gr. Wt / 7000 / d^2 or SD = Gr. Wt./ (7000*d^2) - which is easier for hand calculators. The 7000 is a divisor that converts grains to lbs. The results are in lb/sq.in. and valid for any shape.

BC = SD / FF where BC is the Ballistics Coefficient, SD is the Sectional Density and FF is Form Factor. However, it is only valid for the velocity and conditions at which the form factor/drag coefficient test was conducted and in relationship to the drag model selected. Using a different drag model results in a different BC. A projectile might have a BC (G1) of 0.500, and yet a BC (G7) of 0.200 .

The Drag Force (Df) also includes the air density (Ad), along with the velocity (v), and cross sectional area (a), related by the equation: D = Cd x Ad x v^2 x a / 2. If you double the diameter of a pellet from .177 to .357 even though they may have identical shape and Cd the larger will have four times the drag force. For subsonic velocities, if you double the velocity, you will also have four times the drag force as air resistance increases by the square of the velocity. Because the  Cd is a constant, the subsonic drag would only roughly double (1 / 0.7)^2 = 2.04; the actual increase depends on the shape of the projectile, and it's corresponding Cd vs Velocity curve.

[rsterne]

Sorry, but I disagree with some of your conclusions, particularly that it may be better to use the G7 model for pellets.... It is for a very low drag boattail design, about as different from a pellet as you can get.... nearly the opposite end of the spectrum, in fact.... A sphere (GS) would be a better choice than the G7....

When calculating the Cd, you actually calculate the deceleration first from the difference in velocities, then the drag force (for which you must know the weight and caliber), and eventually end up with the Cd over the range of velocity between the two speeds used.... To determine the FF, you must decide which drag model to use, and the G1 model is the closest we have, unless using ChairGun, where the GA model is a bit better than the G1.... The G7 isn't even close....

SD calculations as used in ballistics are not in lb/in^2.... because the PI/4 factor is not included in the calculation.... They have no "proper" units, although psi is a reasonable way to think about how the number works in resisting deceleration....

HTHs....

Bob

[Poorman Plinker]

Please Mr. Sterne,
Do not apologize for helping me get the facts straight... I am very happy for your help, notes and corrections.

I have looked at a number of Cd/Mach charts to draw my conclusion... I agree Gs would be better, if you could find an online program that says it uses it LOL. All in all the Gs, G7 and Ga are all pretty flat in the subsonic range. But the G1 is all over the place. Given the choice of 2 models I say G7 over G1. I agree the Ga (Pellet1) is designed for airguns so would be best.

If I am understanding what you are saying about establishing Cd...
Calculate deceleration first, v1-v2/d or something like that
then Df=  D = Cd x Ad x v^2 x a / 2. but that requires Cd first???
Then use the Df/D to determine Cd
then Cd/Cg = FF
I will try posting some formulae for review...To be honest I continue  to be baffled about resolving them.

Thanks for the tip on the difference between ballistic and physics SD.

[rsterne]

If you use G7, your BC values for pellets will be ridiculously low.... For example, instead of a value of BC (G1) = 0.036 for the 18.1 gr. JSB Heavy at 850 fps, the BC (G7) would be only 0.019.... That would cause supreme confusion to anyone without a complete understanding of what you have done, and what they are doing.... The default drag model is always G1, and while it may be the shape of an artillery shell, it is very close to many rifle bullets.... The SD looks after the scaling factor automatically.... The difference between G1 and GA in the Subsonic is very small, but G1 is much closer to GA than G7 is....

The deceleration is the delta V divided by the time to travel the distance between the Chronos.... a = (V1-V2) / t.... The time t is the distance between the Chronos divided by the average velocity V = (V1-V2) / 2.... The Drag Force can be calculated once you know the deceleration, from F = ma.... You know the mass (don't use weight) and you know the acceleration (deceleration), so Df = m x a.... In order to calculate the Cd, you rearrange the drag formula Df = Cd x Ad x V^2 x Fa /2 .... It becomes 1 / Cd = ( Ad x V^2 x Fa ) / ( 2 x Df ).... which you then invert to Cd = (2 x Df) / (Ad x V^2 x Fa) .... All the quantities on the right side are known, Df you calculated from F = ma, and you know the air density (Ad), average velocity (V), and frontal area (Fa).... so you can calculate the Cd.... It is extremely important to make sure you use consistent units, of course....

HTHs....

Bob

[Poorman Plinker]

Mr. Sterne,

I will take your notes and study them for a bit to get a grip. Posted in the Guild before I saw this so ignore or respond as you choose.

I saw a Cd/mach chart somewhere that I will review again today. I agree that G1 and G7 create different FF. I just think the 7 is flat in subsonic like GA and real pellets.  The challenge is that using any on-line BC program could use either and comparing data can be challenging if not down right erronious.

[Doug Wall]

All of this theory and math gets to me after a while. I use the simple formula (Ch)+(ai)+(rG)+(un). It usually gets me pretty good info.   

[rsterne]

I just think the 7 is flat in subsonic like GA and real pellets.

The actual Cd of pellets is nowhere near flat in the Subsonic, it has a distinct minimum, the position of which varies from Mach 0.5-0.6 for wadcutters, to as high as Mach 0.75 for heavy, round-nosed pellets.... The G1 model has that minimum to some degree, and the GA model has it accentuated.... The G7 is indeed flat (as is the RA4), which is incompatible with pellets.... Here is the area of interest for velocities, with the appropriate drag models shown....



Note that the G1 Model is the closest to the GA, with the GA model having more of a "dip" to the Cd curve between 600-700 fps.... This is consistent with most experimental data I have seen for pellets.... There is no question in my mind that if the GA model is not available, and your choice is between G1, G7, or GS.... that using the G1 model for pellets is your best choice not only Subsonic, but throughout the velocity range.... The RA4 model, below 1000 fps, is a "constant drag" model and very close to the original ChairGun model.... which produced TERRIBLE correlation with pellets in early versions of that program when you strayed above 900 fps in particular.... The current GA model is the best we have for pellets at the moment.... but IMO it needs tweaking, particularly in the Transonic and Supersonic range.... I also believe that the Cd used in the GA model is too low below Mach 0.5.... the actual "dip" in the Cd curve is even more exaggerated than shown....

Bob

[Poorman Plinker]

THANKS Mr Sterne,
Duely noted and I completely agree based on the graph you have presented. As always there a conflicting views and data posted elsewhere on-line. I am going to follow your lead on Cd curve too as I trust your research is more directly related to pellet performance.
 

[anti-squirrel]

My main gripe with everything is mixing units.  I laud your efforts, and appreciate the toil you're exerting here. 

However- and this is just my own opinion-for the majority of shooting performed by many airgunners (the 50-meter-and-under stuff), a lot of this has little bearing unless involved in low-power airguns as per 12FPE guns used for headshotting squirrels/rats/mice in the UK or serious competition.  For those investing in such, how many people are going to break out all the math before they sight in, learn their arc, and start wacking pests or paper?

I'm a good example of the low-power pester.  I can do the math, but I find it easier to know my ballistic arc at the nominal ranges I shoot because in the end, 1/8" inch (considering ballistics and power levels affecting POI) "off my aim point" still means a dead animal.  And the run-of-the-mill ballistic calculators work "Well enough".

It is only in the rarefied air of long range hunting/pesting/target matches using full-power guns that I think getting heavy into the formula will really net any benefit.  I could be very wrong, but with with so many different projectile shapes and weights even among regular domes in a given caliber, the average airgunner will likely look at all this information, shake his of her head, then go outside and shoot with little regard for formula and proceed to shoot stuff.

Please understand- for an intellectual pursuit, this information is worth reading.  Thus, I thank you for your effort.