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All Springer/NP/PCP Air Gun Discussion General => Air Gun Gate => Topic started by: Poorman Plinker on September 03, 2017, 11:48:12 AM
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Thanks to the assistance of various members of various forums (Especially Bob Sterne) I think this list of formulae is pretty close to complete and current
TABLE OF BALLISTICS FORMULAE
This is an excerpt from Airgun Reference Book III: Pellet Ballistics. These formulae are adapted from firearm ballistics to help the pellet ballistics enthusiast understand the theoretical factors effecting pellet performance. They could be applied with a calculator however, the results are strictly dependent on the variables and formulae applied.
Like many of the formulas published in this book, these can be substituted with calculators that are available online. The use of various calculators may produce various results as they may not use the same formulas. Unfortunately, they typically do not cite the exact formulas used for their calculations.
These formulae are provided to help the ballistics enthusiast with a theoretical understanding of the factors effecting pellet performance. They could be applied with a calculator if no internet access exists. However, the results are dependent on the variables and formulae applied.
The use of calculator programs usually does not give the user the opportunity to verify the formulae used. Because of this, it is difficult to compare results from different programs. In general, most programs use the least number of variables. The more variables that are used the more accurate the results. Determining drag by using pellet diameter only is not going to be as accurate as using the nose shape, length, caliber and air density. These formulae are for the investigator to better understand the factors that effect performance, not to replace or compare with other ballistic formulas or programs.
Foot Pounds Energy FPE is calculated from the Newtonian equation: E=1/2mv2. Or, E = W in pounds times vsquared divided by 2g. Pellet weight is expressed in grains so 2g is multiplied by 7000 (grains per pound). Ultimately:
FPE = (gr x v^2)/450240.
Where:
FPE = Energy (foot pounds)
m = Mass
v = Velocity (FPS)
gr = Pellet weight (grain)
Conveniently, the FPE at 673 FPS is approximately equal the pellet grain weight, the FPE is double the grain weight at 950 FPS and one half the grain weight at 474.5 FPS.
Frontal / Bore Area in square inches equals Pi times the radius squared:
a= Pi x r^2
Where:
a = Area (sq inch)
r = Radius (inches)
Pi = 3.14159
a .177= .0246 sq.in,
a .217 = .0369 sq.in,
a .254 = .0490 sq.in,
Circumference is equal to the area divided by the diameter divided by 4:
c = Pi x d
Where:
c = Circumference (inches)
Pi = 3.14159
d = Diameter (inches)
.177 c =.556"
.217 c =.680"
.250 c =.784"
Area to Circumference relationships are used to explain why larger calibers have greater FPE than smaller calibers using the same power plant. The formula expresses the potential propulsion area versus the potential drag area. A larger factor represents a larger amount of push versus pull. The factors are determined by dividing the sectional area by the circumference:
K = a/c
Where:
K= Area/circumference factor (dimensionless)
a = Area (sq inch)
c = Circumference (inches)
.177 a =.025 sq.in: c =.556" (factor of 0.044).
.217 a =.037 sq.in: c =.680" (factor of 0.055)
.254 a =.049 sq.in: c =.784" (factor of 0.062).
Ogive equals the height of the pellet's point plus the radius divided by the diameter of the cylinder:
o= (h+r)/d
Where:
o = Ogive (dimensionless)
h = Height (inches)
r = Radius (inches)
d = Diameter (inches)
Round 1.00
Domed 0.90
Pointed 1.25
Hollowpoint 0.75
Flat 0.50
Meplat equals one minus half of the Flat Area divided by Pellet Diameter,
Mp= 1 - (flat/d) /2
Where:
Mp = Meplat (dimensionless)
flat = Flat diameter (inch)
d = Pellet diameter (inch)
Round 1 - (0.00 / 2) = 1.0
Domed 1 - (0.05 / 2) = .975
Pointed 1 - (0.02 / 2) = .99
Hollowpoint 1 - (0.50 / 2) = .75
Flat 1 - (1.00 / 2) = .50
A value of one could be attributed to round nose (flt/d), of 0.5 to a Wadcutter (flt/d) and the flat area (flt/d) attributed to Hollow Points (approximately 0.75). Estimated pellet meplat designations could be adjusted to the P1 ideal pellet shape.
Adjusted Area could be used when studying pellets of the same caliber with different waist diameters. If the pellet is cylindrical then an unadjusted cross-sectional area could be used to determine sectional density and drag potential. The frontal cross-section alone does not give an accurate picture of the shape or volume of a pellet.
Ar = r max + r min / 2;
Aa = Pi x (Adj.r^2)
Where:
Ar = Adjusted radius (inch)
r max = Maximum radius (inch)
r min = Minimum radius (inch)
Aa = Adjusted area (sq inch)
Pi = 3.14159
Nose Coefficient (n) as a result of measurement
Estimated Ogive times Meplat = (n) proximity to standard 1.0:
Round 1.00 x 1.00 = 1.00
Domed 0.90 x 0..975 = .8775
Pointed 1.25 x 0.99 = 1.2375 (1-.2375=.7625)
Hollowpoint 0.75 x 0.75 = 0.5625
Flat 0.50 x 0.50 = 0.25
Drag Surface Area is determined by adding the side area to the frontal area
Ds = Adjusted side surface + frontal area
Ds = 2(Pi x Ar x h)+(Pi x r^2)
Where:
Ds = Drag surface area sq inch
Pi = 3.14159
r = Radius (inch) or
Ar = Adjusted radius (inch)
h = Height (inches)
Pellet Coefficient
Cp = n x Ds
Where:
Cp = Pellet shape coefficient (dimensionless)
n = Nose coefficient
Ds = Drag surface area (sq inch)
Volume of pellet. If Ar is used to represent (as a decimal) used as a percentage it may be useful in determining volume more accurately than just using the caliber.
V=pi x Ar x h
A drag coefficient can also be calculated mathematicall
bore area = (caliber/2 squared) times pi
Sectional Density is a number that describes the relation between a pellet's cross section and the pellet's mass. This widely used formula uses weight and diameter and assumes gravity:
SD = W / d ^2
SD = gr / 7000 x d^2
Where:
SD = Sectional density(dimensionless)
W = Weight (pounds)
gr = Weight (grains)
d = Diameter (inches) (land, groove, pellet cross-section or nominal caliber)
The 7.8 grain .177 pellet SD is (7.8 / (7000*0.177*0.177) = 0.0356.
The 14.4 grain .22 pellet SD is (14.4 / (7000*0.217*0.217) = 0.0437.
The 25.4 grain .25 pellet SD is (25.4 / (7000*0.25*0.25) = 0.0581.
MASS equals the weight in pounds divided by the force of gravity:
m = W/g
m = gr / (7000 x 32.1737)
m = gr / 225218
Where:
m = Mass
W = Weight (lbs)
gr = Weight (grain)
g = Force of gravity on the object (32.1737 FPS^2).
Bore Volume determines the maximum potential amount of propellant that will be available to be applied to the base of the pellet.
BV = (Pi x r^2)L
Where:
BV = Bore volume (cubic inches)
Pi = 3.14159
r = Radius (inches)
L = Length of barrel (inches)
.177 - 10 inch = .25 cubic inches (cu. in.), 18 inch = .44 ci, 24 inch = .59 ci.
.22 - 10 inch = .37 cubic inches (cu.in.), 18 inch = .67 ci, 24 inch = .89 ci.
You can calculate the maximum possible FPE of any PCP. In addition to the bore volume determining the amount of air (propellant) used, the other factor is the pressure of that gas. The following formula:
Max. FPE = BV / 12 x Pressure (PSI).
Where:
FPE = Force available to accelerate the projectile (ftlbs.).
BV = Bore volume (cu.in.)
12 = Converts inches to feet
Energy (ft.lb) = Distance (ft.) x Force (lbs.)
Recoil law of conservation of momentum states that the magnitudes of their momenta must be equal:
mv1 + mv2 = 0
m1 x v1 + m2 x v2 = 0
Where:
m1 = Airgun/shooter mass
v1 = Velocity of airgun/shooter
m2 = Pellet mass
v2 = Velocity of pellet
Despite the high velocity of the pellet, the small pellet-mass to shooter-mass ratio results in a low recoil velocity although the force and momentum are equal.
Corrected Muzzle Velocity . Pellet speed in feet per second is typically measured at a distance from the muzzle and is corrected to the actual muzzle. Corrected muzzle velocity results when:
MzV = D x K+IS
Where:
MzV = Corrected muzzle velocity results (FPS)
D = Distance from the muzzle to the chronograph midpoint (feet)
K = Air density constant (0.64)
IS = Instrumental speed (FPS)
FPE equals one-half the mass times the velocity squared:
FPE = m x v^2
FPE = W x v^2 / 2 x g
FPE = gr x v^2 / 2 x 32.1737 x 7000
FPE = gr x v^2 / 450240
Where
m = mass
W = Weight (pounds)
gr = Weight in grains
v = Velocity (feet per second)
g = Gravitational force (32.1737 FPS^2)
7000 = number of grains in a pound.
An 8-grain pellet at 800 FPS, the energy would be 8gr times the square of the number 800 (640,000). And, 8gr times 640,000 is 5,120,000. Divide that number by the constant 450240 and you get 11.371712. But, 11.37 foot-pounds will do.
Revolutions per-second of a barrel with a rifle twist 1 turn in 16 inches is said to have a twist rate of 16. The MzV (muzzle velocity) times 12 divided by twist rate equals the rate of rotation per second:
RR = MzV x 12 / tr
Where:
RR = Rotational rate per second
tr = Twist rate
MzV = Muzzle velocity (FPS)
12 = Number of inches in a foot
With twist rate of 16, a projectile traveling at 450 FPS rotates at 337.5 revolutions per-second. To determine revolutions per minute simply multiply by 60 (60 seconds per minute). For the 450 FPS pellet in the 16 twist barrel is rotating at 20250 rpm.
EFFECIENCY HPA and PCP
This formula calculates the volume of air per shot FPE. This formula can be used for HPA and PCP airguns:
VE = (VR x (FP-RP) / 15) / (E x S)
Where:
VE = Volume of air per shot energy
VR = Volume of Reservoir (cu.in.)
FP = Fill Pressure (PSI)
RP = Refill Pressure level (PSI)
E = Cumulative average FPE per shot
S = Number of Shots
15 = PSI per one atmosphere
A stock Discovery shoots as follows:
(8.2 x (2000-1200) / 15) / (22.8 x 25) = (8.2 x 800 / 15) / (570) = 0.77 Cubic Inch Per Shot
Pump Airguns can use this formula to calculate the volume of air per shot FPE:
VE = ((VP x N) / E) x K
Where:
VE = Volume of air per shot energy
VP = Volume of pump (cu.in.)
N = Number of pumps
E = FPE per shot
K = Constant (57%)
Using a Crosman 2289, the swept volume is about 18cc, and the valve about 1.6 cc. If you call it 1.1 cu.in (at one atmosphere) per stroke with 14.3 gr. JSB Express pellets in a 14" barrel:
10 pumps = 541 fps = 9.30 FPE
(1.1 x 10) / 9.30) x 57% = .67 cubic inch per shot
The efficiency factor for a pumper adjusts for pumping losses. The 57% was typical for relatively high numbers of pumps with the stock pump on the Crosman 1377/2289. It is likely too low for the 39X series, or the the 13XX/2289's at low numbers of pumps, or if a rigid Flat Topped Piston is installed.
CO2 The amount of CO2 contained in a 12-gram powerlette can be calculated into cubic inches at Standard Temperature and Pressure. There are .454 kilograms of CO2 in 8.743 cubic feet of gas. The .454 kilograms equals 454 grams. One cubic foot equals 1728 cubic inches. Or, 454 grams equals 15108 cubic inches. Dividing both by 37.83 results in 12g = 399.37 ci. Therefore, if you get 400 FPE, that works out to an efficiency of 1.00 FPE/ci.
VE = (VR / E x N) / K
Where:
VE = Volume of air per shot energy
VR = Volume of Reservoir (cu.in.)
E = Cumulative average of all shots (FPE)
N = Number of Shots
K = Constant (4)
In a normal Crosman 2240 configuration, the CO2 gun shots above 390 FPS produced 31 shots that averaged 431 FPS (5.9 FPE).
(399.37 / 5.9 x 31) / 4 = (399.37 / 183) /4= 2.18 /4 = .55
The average cubic inch per FPE result in HPA is .70. The CO2 result of .50 is 27% less than .70. When the FPE of a .22 caliber 14.3 grain pellets was shot in a 24" barrel with both CO2 and HPA using roughly the same PSI the resulting FPE differed by 27%. The factor of 4 is used to make the results relate to the HPA formula.
Air Resistance is directly proportional to the cube of a pellet's speed. Double the speed and the air resistance increases eight times. We could express resistance as product of velocity and use a formula such as:
Ar = v^3
Where:
v = Muzzle velocity in FPS
Ar = Air resistance in FPS
If we compare the resistance value of a pellet, traveling at 500 FPS vs. a pellet traveling at 700 FPS we see 700 FPS pellet has a higher resistance value (174% more).
500 FPS^3 = 125000000
700 FPS^3 = 343000000
The above formula is only for comparative purposes and only gives a very general idea about the effect,
Air Density is a way of expressing the mass of air per unit of volume. Air density is an important value to consider in aerodynamics. The following steps will explain how to calculate dry air density:
Ad = p/(K x T)
Where:
Ad = Air density (kg/m^3)
p = Air pressure in Pascals (Pa)
K = Gas constant for dry air (287.05)
T = Temperature (Kelvin)
Here is an example :Use the ISA standard values for conditions at sea level. P = 101325 Pa and T = 15 deg C. The air density is calculated to be:
Ad = (101325) / (287.05 * (15 + 273.15)) = 1.2250 kg/m3
Density of Humid Air can be calculated as the sum of the densities of the two gases, dry air and water vapor, in proportion with their partial pressures. The moist air density formula:
HA= pd/K x T + pv/Kv x T
Where:
HA = Humid Air Density (kg/m^3)
pd = Partial pressure of dry air (Pa)
K = Gas constant dry air (287.05)
T = Temperature (Kelvin)
pv = Partial pressure of water vapor (Pa)
Kv = Gas constant for water vapor (461.495 jules)
Drag Force (Df) is the resistance offered by the medium against an object moving through it. Drag force = mass times the deceleration divided by the time over which it was measured.
Df = m x dc
Df = Force of resistance (ft.lbs.)
m = mass (pounds) (gr / 225218)
dc = deceleration (v1-v2) / t (ft/sec^2)
t = D / (v1-v2/2) (seconds)
D = Distance between chronos (feet)
or
Df = m(v1-v2/t1-t2)
Where:
Df = Force of resistance (ft.lbs.)
m = Mass (to lb.)
v1-v2 = Reduction in velocity (FPS)
t1-t2 = Time over which the reduction was measured (seconds)
or
Df = Cd x Ad x V^2 x Fa /2
Drag Coefficient (Cd) represents the aspects of an object that create drag force (Df) when the object moves through a fluid medium. It is possible to borrow and modify a formula from bullet form factor to determine a pellet coefficient of drag (Cd). For example:
Cd = (2 x Df) / (Ad x v^2 x a)
where
Cd = Drag coefficient (dimentionless)
Df = Drag force (ft.lbs.)
Ad = Air density (lb/ft^3 / 32.1737)
v = Average velocity v1-v2/2 (FPS)
a = Reference area (feet^2) (in^ / 144)
Form factors are often normalized, that is, the value ranges from zero to one. A form factor equal to one indicates that the subject coeficient (Cd) is equal to the model coefficient (Cg). The form factor is determined mathametically by:
FF = Cd / Cg
Where:
FF = Form factor (dimensionless)
Cd= Subject drag coefficient at specific velocity (dimensionless)
Cg= Model drag coefficient at specific velocity (dimensionless)
The Mach number allows us to define flight speeds in which the air compressibility effects vary. The Mach number (M) is the ratio of the speed of an object moving through a fluid (v) to the speed of sound in that fluid (c). As a ratio of two speeds, it is dimensionless
M = v x c
Where:
M = Mach number
v = relative velocity of air and object
c = speed of sound under study conditions
Ballistic Coefficient (BC) of a body describes its ability to overcome air resistance and wind deflection in flight. BC is inversely proportional to drag. A high BC number indicates low drag The idea behind the development of the Ballistic Coefficient (BC) was a dimensionless scaling factor that defines the performance of a projectile relative to a G standard.
BC = SD/FF
Where:
BC (small arms) = Ballistic coefficient (dimentionless)
SD = Sectional density (dimentionless)
FF = Form Factor (dimentionless)
Time of flight in air is a little difficult to determine unless you have a ballistic computer program to do this for you or you have access to two chronographs or appropriate ballistics tables.
T = 2D/(v1 + v2)
Where:
T = Time of flight (seconds)
D = Distance (feet).
v1 = Muzzle velocity (FPS)
v2 = Remaining velocity at distance (FPS)
Time of flight in vacuum is easier to determine and only requires muzzle velocity. The pellet velocity remains constant in a vacuum since there is no air to slow it down. The formula would be :
Tv = D/v1
Where:
Tv = Time of flight in vacuum (seconds)
D = Disance (feet)
v1 = Muzzle velocity (FPS)
Wind deflection can be calculated mathematically if the information is available. The most common formula is represented by:
Wd = (Ws x 12) x (T-Tv) x F
Where:
Wd = Pellet displacement (inches)
Ws = Crosswind speed (FPS), (MPH x 1.467 = FPS)
T = Time of flight. (2 x ft / MzV+684)
Tv = Time of flight in vacuum (ft / MzV)
F = Sine of the angle at which the wind approaches the pellet
Let us use a 10 mph crosswind and the target at 50 yards. Here are the results.
.177 ; 7.8 gr; 50 yd BC 0.020 Drift = 5.41"
.22; 14.4 gr; 50 yd BC= 0.0246 Drift = 5.29"
Estimate the impact velocity of your airgun pellet we can start with a few assumptions. Assuming you are using a standard weight pellet the muzzle velocity could be 150 FPS less than the light weight pellet used to develop the advertised velocity.
ev = (av-150) - K x D
Where:
ev = Estimated velocity at range (FPS)
av = Advertized muzzle velocity (FPS)
D = Distance to target (yards)
K = Constant related to MzV
MzV at or below 800 FPS = 2.5
MzV between 800 and 900 FPS = 5.0
MzV above 900 FPS = 7.5
Momentum equals mass times velocity. Momentum is the tendency of an object in motion to stay in motion. In ballistics, momentum is usually measured in slug feet per second (SFPS) or weight feet per second.
SFPS = m x v
SFPS = gr x FPS / 7000 x g
SFPS = gr x FPS / 225218
Where:
SFPS=Momentum (slug weight)
m = Mass (to lbs)
v= Velocity (FPS)
gr = Grain weight
g = Gravitational pull ft/sec^2 (32.1737)
The slug is the unit of mass, where the pound is the unit of force. The pound is a unit of weight and weight is defined as the force of gravity on an object. Since the acceleration of gravity in common units is 32.16 ft/s^2, it follows that the weight of one slug is 32.16 pounds.
Momentum/game weight theory you take the pellet weight in grain times the FPS and divide by 1000 to determine a reasonable game weight at 10 yards.
GW = gr x MzV/1000
Where:
GW = Acceptable game weight 10 at yards (pounds)
gr = Pellet weight (grains)
MzV = Muzzle velocity (FPS)
Penetration of parafin medium can be estimated based on caliber, weight and velocity (reverse enginered from results presented on ChairGun). We can use a simple formula to estimate the theoretical penetrations based on caliber, weight and velocity:
Pn = gr x FPS/K
Where:
Pn = Penetration (inches)
gr = Pellet weight (grains)
FPS = Impact velocity (FPS)
K = Caliber factor
Caliber factor .177 = 2200
Caliber factor .22 = 3200
Caliber factor .25 = 4200
There are only five variables for penetration, if the bullet doesn't distort (expand) or tumble, weight, diameter, velocity, and the shape (ogive and mepllat). For a fixed shape, penetration is proportional to the product weight, diameter and velocity. While this is not a coefficient formula it would give a reference number that could be used to compare two different pellets of the same or different calibers for penetration potential.
Pn = gr x d x v x n
Where:
Pn = Penetration
gr = pellet weight (grains)
d = pellet diameter
v = Impact velocity (FPS)
n = Nose shape factor (ogive & meplat)
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I see some errors in the above, and I'm not about to try and find them all.... eg. Form Factors can be any number, they do not have to be less than 1.... In the last section, regarding penetration, you are multiplying by the diameter instead of dividing by the area.... for equal shapes, Penetration is SD x V.... The terms Ogive and Meplat have specific definitions in Ballistics, changing them is confusing....
Bob