Approximate ballistics formulas for spherical pellets in free flight

E.J.Allen

Department of Mathematics and Statistics,Texas Tech University,Lubbock,TX 79409-1042,USA

1.Introduction

In the present investigation,approximate ballistics formulas are derived for spherical projectiles of shotguns or muzzleloaders using realistic drag coefficients.To derive the ballistics formulas,the differential equations are simplified using dimensionless variables for velocity and distance and transforming the problem into one for velocity versus distance rather than for velocity versus time.The drag coefficient is accurately approximated by a continuous piecewise linear function that depends solely on Mach number.Using this drag coefficient,the ballistics equations are solved exactly resulting in analytical expressions for velocity and flight time versus distance.This work is useful as,currently,accurate ballistics formulas for shotguns and muzzleloaders are not known analytically and ballistics curves must be calculated computationally.An important assumption is that the pellets are not interacting during flight.Therefore,the formulas are most appropriate for large shot sizes and open chokes.The presentation is complete and selfcontained and the formulas can be readily implemented.

Three forces are important in shot pellet dynamics.A major force due to the drag of the air acts in the opposite direction of the motion of the pellet.Two minor forces involve the downward force of gravity and a sideways force due to a crosswind that acts perpendicular to pellet travel.(The component of the wind in the direction of pellet travel is small compared with the pellet's muzzle velocity and is generally neglected.)The drag on the pellet is much larger than gravity or crosswind forces and,as a result,the flight dynamics with drag can be first calculated and then corrected for gravitational drop and wind drift[19,26].

Newton's second law for a pellet undergoing drag has the form

wheremis the pellet mass,ais the acceleration,andFis the drag force on the shot pellet(See Table 1 for notation).The drag forceFincreases with velocity and with pellet diameter.The drag on single spherical objects in air has been studied experimentally and theoretically in several investigations,e.g.,[3,9,10,14,18,21,23,24].The drag forceFdepends on velocity,Reynolds number,and the Mach number.For spherical pellets,the drag force is equal to[4]

Table 1Notation used in the paper(length units are all cm to simplify formulas).

whereDis pellet diameter,ρa(bǔ)is air density,vis pellet velocity,andC（M，Re）is the drag coefficient.The drag coefficient,C（M，Re）,is a measure of the pellet's resistance in air and depends in a complicated way on the Reynolds numberReand the Mach numberMfora compressible fluid such as air[3,18,21,22].The Mach number is equal to the pellet velocity divided by the speed of sound in air and the Reynolds number is equal to the product of the pellet diameter,air density,and pellet velocity divided by the air viscosity.The Mach and Reynolds numbers are considered in greater detail in the next section.In particular,though,under constant atmospheric conditions and a given pellet diameter,the Mach and Reynolds numbers are proportional to pellet velocity.

By equations(1)and(2),the pellet velocity satisfies the differential equation

Equation(3)is the principal differential equation for the exterior ballistics of spherical pellets.The solution of equation(3)gives the pellet velocity with flight time and,by integration,the trajectory distance with flight time.Then,corrections due to gravity or crosswind drift can be applied.If atmospheric conditions are constant and the pellet size and composition are fixed,thenReandMare proportional to pellet velocity and equation(3)can be written as

wherekis a constant andGis a function which depends on pellet velocityv.

Due to the complicated nature of the drag coefficient as a function of Mach number and Reynolds number,equation(3)or(4)cannot be solved exactly.However,in order to understand the nature of pellet ballistics,simple approximations have been made for the drag coefficient so that analytical solutions to(3)or(4)can be determined and studied.For example,it is sometimes assumed thatG（v）=cviwherecis a constant andi=0 ori=1[1,11].The resulting solutions yield insight into the ballistics dynamics but are not highly accurate for the entire range of pellet velocities in a typical trajectory.

If the drag coefficient is known in a functional and/or tabulated form,then(3)can be accurately solved in a computational manner.Some computational methods successively solve the differential equation in small time steps for pellet velocity and position.A classic ballistics program was developed by E.D.Lowry and used by J.Taylor to generate numerous shotshell ballistics tables[26].Several ballistics programs currently available online include:“Shotgun Load Comparison Calculator Program”of Westslope Magazine[30], “Shotgun Simulator”of Blackbart Software[5],“Shogun Ballistics”of Connecticut Muzzleloaders[12],and “Hornady Ballistic Calculators”of Team Hornady[17].One disadvantage of these computer programs is that the drag coefficients used in the programs are,in general,not clearly described.

In the present investigation,the ballistics equations for velocity and flight time are solved exactly using an accurate approximation to the drag coefficient.In order to accomplish this,equation(3)is first redefined as a problem involving dimensionless velocity as a function of dimensionless distance rather than velocity as a function of time.The simpler forms of the equation and its solution motivate this change.Next,by considering drag coefficients for different Reynolds numbers and for pellets of different sizes,it is explained how the drag coefficient can be accurately approximated by a continuous piecewise linear function that depends solely on Mach number.Using this drag coefficient,the ballistics equations are solved exactly resulting in analytical formulas for pellet velocity and flight time with trajectory distance.The derived ballistics formulas are tested and several examples are described.In addition,for completeness,relevant notes are given on corrections for gravitational drop and wind drift,on the estimation of certain physical parameters of air,and on the effects of shot clouds and string length.

2.Derivation of formulas for pellet velocity and flight time

In this section,the flight dynamics of spherical projectiles from shotguns or muzzleloaders is shown to satisfy a simple differential equation involving Mach number and a dimensionless,scaled,distance variable.It is explained how the drag coefficientC（M，Re）can be accurately approximated by a continuous piecewise linear function of Mach speed for spherical projectiles from shotguns or muzzleloaders.Using the approximate drag coefficient,the differential equation is then solved exactly to find formulas for pellet velocity and flight time with trajectory distance.For clarity,some of the definitions stated in the Introduction are more thoroughly described.In the fifth section,it is shown that the derived formulas accurately describe the ballistics dynamics of spherical pellets.

Using several important unit-free numbers,equation(3)can be put in a simple general dimensionless form.LetMbe the Mach number,the ratio of the pellet velocity to the velocity of sound in air.Specifically,Mis defined as

For given atmospheric conditions,the speed of sound in air,vs,is constant soMis proportional to pellet velocityv.The Mach number can therefore be considered to be a dimensionless velocity.As Mach 1 corresponds to about 1150 ft/s,shot pellets and round balls have velocities of interest between about Mach 0.2 and Mach 2.0.

Similar to the Mach number,letRebe the Reynolds number,defined as

whereDis the shot pellet diameter in cm,ρa(bǔ)is the density of air in gm/cm3,andμais the viscosity of air in gm/(cm sec).The units of diameter,density,viscosity,and velocity all cancel making Reynolds number a dimensionless quantity similar to the Mach number.AsD,ρa(bǔ)and μaare constant for a particular problem,Reis proportional to pellet velocityv.Therefore,Recan also be interpreted as a dimensionless velocity.Reynolds number increases with velocity and with pellet size.Indeed,shotgun pellet velocities have Reynolds numbers effectively between 10,000 and 300,000 but round balls for muzzleloaders have Reynolds numbers as high as 600,000.For example,extreme values of the Reynolds number are obtained by a small pellet of low velocity and a large pellet of high velocity.At sea level in dryair at 27°C,a 9-shot size pellet with the low velocity of 250 ft/s hasRe=10，000,whereas 000 buckshot with the high velocity of 1700 ft/s hasRe=300，000.Furthermore,a 54 caliber round ball at a high velocity of 2250 ft/s has a Reynolds number of 570,000.Notice also that most round balls in common use,e.g.,32,36,38,45,50,and 54 caliber,have diameters less than 9/16 inch in diameter.However,a 58 caliber round ball has a diameter of 0.570-0.575 inch which is slightly greater than 9/16=0.5625 inch.For a fixed pellet diameter and under given atmospheric conditions,Reynolds numberReis directly proportional to Mach numberMand bothReandMare proportional to pellet velocity.

Finally,similar to Mach number and Reynolds number,a dimensionless numberzproportional to distance is defined as

wherexis the trajectory distance,Dis the shot pellet diameter,ρpis the pellet density,andkz=Dρp/ρa(bǔ)is a scaling factor with the same units as distancex,i.e.,cm.Again the units cancel and,in this case,zcan be thought of as a dimensionless distance.

The drag forceFdepends on velocity,Reynolds number,and the Mach number.For spherical pellets,the drag force is given by(2)whereC（M，Re）is the drag coefficient which depends onReandM.The drag coefficient is the most difficult quantity to deal with in exterior ballistics and,as a result,requires the most explanation.Geometrically,C（M，Re）forms a three-dimensional surface above the （M，Re）plane made up of points consisting of the Reynolds number and the Mach number[21].From a given point on the（M，Re）plane,the numberC（M，Re）is the height to this three dimensional surface.The equationRe=cforca constant or the equationRe=sMfor a given value ofsform two other planes in three dimensions that are perpendicular to the （M，Re）plane.Intersecting the three-dimensional surface with either of these new planes gives a two-dimensional,S-shaped curve.This two dimensional curve is the curve of the drag coefficient versus Mach number for a given value ofcors.The three-dimensional surface,intersecting planes,and S-shaped curves of intersection are illustrated in Fig.1.In this illustration,the dark triangular region is the planeRe=sM,the dark rectangular region is the planeRe=c,the meshed region is the three-dimensional surface of drag coefficients,and the two S-shaped intersecting curves are the curves of drag coefficient versus Mach number whenRe=cor whenRe=sM.

Fig.1.Illustration of the plane Re=constant(dark rectangle)and the plane M=sRe(dark triangle)intersecting the drag coefficient surface forming S-shaped curves of drag coefficient versus Mach number.

Examples of the S-shaped curves of drag coefficient versus Mach number are shown in Fig.2.In the left graph,recommended drag coefficients[18]forRe=10，000 are plotted as open circles.In the right graph,drag coefficient data[9]for 9/16-inch spheres are plotted as closed circles.In this case,Re=Dρa(bǔ)v/μa=sMwheres=Dρa(bǔ)vs/μa≈3× 105is fixed sinceD=9/16 inch and ρa(bǔ)，vs,and μaare constants.Miller and Bailey[21]show that the drag coefficient slowly increases as Reynolds number increases for any fixed value of Mach number.In addition,the drag coefficient increases with pellet size as the Reynolds number is proportional to pellet diameter.As shot pellets and round balls effectively have Reynolds numbers greater than 10,000 and diameters generally less than or equal to 9/16 inch,it is reasonable to assume that most shot pellets and round balls have drag coefficient values between these two curves in Fig.2.In other words,for shot pellets and round balls,the drag coefficient is generally bounded above by the drag coefficient of 9/16 inch spheres and bounded below by the drag coefficient of pellets with Reynolds number 10,000.

In the present investigation,the two S-shaped drag coefficient curves in Fig.2 are approximated by continuous piecewise linear curves made up of three straight-line segments.Each piecewise linear curve is completely determined by the values of the drag coefficient at the four endpoints of the three line segments.These four endpoints are taken at Mach speeds ofM1=2.0,M2=1.2,M3=0.7,andM4=0.2.LetC1（M）represent the drag coefficient curve forRe=10，000 andC2（M）represent the drag coefficient curve for 9/16 inch spheres.TheC1（M）andC2（M）curves are specified in the first two columns of Table 2 by the values at the four endpoints.The average of these two curves is referred to asC（M）and is specified in the third column of Table 2.The piecewise linearC1（M）andC2（M）curves are shown in Fig.2 as solid lines.

These two piecewise linear curves fit the data well;the average relative error between each piecewise linear curve and the data points is 2.0%for the left curve and 2.1%for the right curve.The average curveC（M）is shown in Fig.3 along with the drag coefficient points taken from Refs.[9]and[18].As can be seen,the average curve fits well all of the drag coefficient points and,for simplicity,is used throughout this paper to represent the drag coefficient for both shot pellets and round balls.

Using the values at the four endpoints given in Table 2,the average drag coefficientC（M）is completely determined on the interval 0.2＜M＜2.0 and is given explicitly by the continuous piecewise linear function:

Fig.2.Drag coefficient versus Mach number for Reynolds number 10,000(left)and for 9/16 inch spheres(right)(data points are from Refs.[9]and[18]).

Table 2Drag coefficients at four Mach numbers.

Returning now to the ballistics equation(3),the relation

Fig.3.Average drag coefficient versus Mach number(open points from Ref.[18]for Reynolds number 10,000 and closed points from Ref.[9]for 9/16 inch spheres).

is substituted into(3)as similarly applied in Ref.[11].Next,using the dimensionless numbersMandzand the relation πD3ρp=6m,equation(3)simplifies to

In equation(9),Mach numberMis a function of dimensionless distancezrather than a function of timetas in(3).The advantage of(9)over(3)is the much simpler form of the solution,especially,for a piecewise-lineardrag coefficient.Finally,substituting the approximate drag coefficient(8)into(9)gives

where the drag coefficientCdepends only onM.The initial value of the pellet's Mach number isM0=v0/vswhere v0is the muzzle velocity.This simplified formula basically implies thatMonly depends onz.In other words,the solution of this scaled equation determines the ballistics dynamics under any given weather conditions and pellet size and density for pellets in free flight.

Equation(10)can be solved exactly or approximately.An approximate solution,with relative error less than 2% for 0.2 ≤M（z）≤ 2.0, is given by the rational functionM（z）≈2/（1+a1z+a2z2+a3z3+a4z4） wherea1=0.861482，a2=0.158335，a3=-0.035699, anda4=0.006225.This approximation can then be used to calculate pellet velocities and flight times with respect to distance.However,in the present investigation,equation(10)is solved exactly.As the drag coefficient is defined separately in(8)on three different Mach number intervals,the form of the solution depends on the magnitude of the initial Mach numberM0=v0/vs.Let Case 1 correspond toM0＞1.2,Case 2 correspond to 0.7＜M0＜1.2,and Case 3 correspond toM0＜0.7.These three cases differ by the number of Mach number intervals experienced by the pellet as it slows down.In Case 1 all three intervals are seen,in Case 2 only two intervals are seen,while in the third case only one interval is seen.For example,ifM0=1.1 then the pellet starts in the interval(0.7,1.2)but as the pellet slows down,its Mach number decreases and it eventually enters into the slowest interval(0.0,0.7).In this example,the pellet is of Case 2 and experiences two Mach number intervals.

2.1.Solution for Case 1,M0＞1.2

WithC（M）given by equation(8),equation(10)is solved exactly forM0＞1.2 and the pellet velocity at trajectory distancexis equal to:

The flight times are given by

where

The valuesx1andt1are the distance and time when the pellet's speed is equal to Mach 1.2 whilex2andt2are the distance and time when the pellet's speed is equal to Mach 0.7.Notice that formulas(11)-(12)forxandtonly depend on the values of vs,M0=v0/vs,andkz=Dρp/ρa(bǔ).Thus,the formulas only depend on five values:speed of sound,vs,muzzle velocity,v0,pellet diameter,D,pellet density,ρp,and density of air,ρa(bǔ).

2.2.Solution for Case 2,0.7＜M0＜1.2

WithC（M）given by equation(8),equation(10)is solved exactly for 0.7＜M0＜1.2 and the pellet velocity at trajectory distancexis equal to

where

In addition,the flight time is equal to:

where

The valuesx1andt1are the distance and time when the pellet's speed has slowed to Mach 0.7.Notice again that formulas(13)-(14)forxandtonly depend on five parameter values:speed of sound,vs,muzzle velocity,v0,pellet diameter,D,pellet density,ρp,and density of air,ρa(bǔ).

2.3.Solution for Case 3,M0＜0.7

WithC（M）given by equation(8),equation(10)is solved exactly forM0＜0.7 and the pellet velocity and flight time at trajectory distancexare equal to:

Two explicit solutions forvandtusing equations(11)-(16)are illustrated in the examples section.First,however,drop and drift corrections are discussed,estimation of environmental parameters is explained,and several tests are performed on the derived ballistics formulas.

3.Drop and drift corrections

The first significant correction to the shot pellet trajectory is the drop due to gravity.This downward drop in inches is equal to

whereg=386.1 inches/sec2is the gravitational acceleration andtis the flight time in seconds.

The second correction is the drift due to a crosswind.Let vwbe the lateral wind velocity in mph,tbe the flight time in sec,xthe flight distance in cm,and v0the muzzle velocity in cm/sec.It is assumed that the crosswind velocity is small compared with the pellet velocity during flight.(For example,a wind speed of 10 mph=14.7 ft/s is small compared with the speed of a pellet.)Didion's well-known formula then gives the approximation

for the lateral drift in units of inches noting that 1.0 mph=17.6 in/sec.A geometric argument for Didion's formula is given in Ref.[19].One way to see Didion's formula is by considering the components of drag force and velocity in the two perpendicular directions:the direction of the crosswind and the initial direction of the pellet.Geometrically,the ratio of the components of the drag force in these two directions is equal to the ratio of the components of the velocity in these directions.This observation leads to the differential equationdvL/（vL-vw）=dva/vawith solution vL=vw（1-va/v0）where vLis the lateral speed of the pellet and vais the speed of the pellet along the pellet's initial direction.If the wind speed is small compared with the pellet's speed,vais approximately equal to the speed of the pellet with no wind,i.e.va≈v.Didion's formula(18)then follows.

4.Air density,viscosity,and speed of sound in air

Given the air temperatureTa,air pressurePa,and relative humidityRH,all of which are standard meteorological information,the air pressure ρa(bǔ),the air viscosity μa,and the speed of sound in air can be estimated.The viscosity of air depends on temperature but is practically independent of pressure and varies only slightly with humidity.For example,at 20°C,air viscosity decreases less than 1%from dry air to fully saturated air.Sutherland's formula[8,25]gives air viscosity as a function of temperature

whereTais air temperature in°C.

Air density ρa(bǔ)depends,however,on temperature,pressure,and relative humidity.A formula for air density based on the ideal gas law and the concept of virtual temperature is given by Ref.[7]

whereRg=84.763（inches Hg cm3）/（°Cgm）is the gas constant,Pais the air pressure in inches Hg,andPVis the vapor pressure in inches Hg.The vapor pressure depends on the relative humidity using,for example,the Antoine equation[31]

whereRHis the relative humidity as a fraction between 0 and 1,Tais in°C,andPVhas units of inches of mercury.

The speed of sound in air,vs,depends on the air temperature,Ta,but only slightly on the humidity.It can be calculated using the formula from Ref.[16]

wherePVis the vapor pressure calculated using equation(21).Here,vshas units of cm/sec.

5.Tests

Equation(11)through(16),derived using the average piecewise linear drag coefficient defined by equation(8),were checked against ballistic data of references[6,11,15,29].In the calculations,as atmospheric conditions were generally not specified with the ballistics data,International Standard Atmosphere[27]at sea level was assumed in the calculations.Specifically,dry air was assumed at a pressure of 29.9 in Hg and a temperature of 15°C.

In the first test,the velocity at 100 yards was compared for several different caliber round balls having a muzzle velocity of 1450 ft/s.Velocity data were obtained from Ref.[15]and are displayed in Fig.4 for 32,36,38,45,50,and 54 caliber round balls.(Several of the reported 100-yard velocity values of reference[15]were interpolated to have a muzzle velocity of 1450 ft/s.For example,for the 50 caliber bullet,muzzle velocities of 1355 ft/s and 1493 ft/s gave velocities at 100 yards of 879 ft/s and 911 ft/s,respectively.Linearly interpolating these values gives an estimated 100-yard velocity of 901 ft/s for a muzzle velocity of 1450 ft/s)Next,velocities at 100 yards for different sizes of round balls were calculated using equation(11)for a muzzle velocity of 1450 ft/s.The solid line in Fig.4 gives the calculated velocities at 100 yards.Good agreement is seen between the calculated values and the test data.For a higher round ball muzzle velocity of about 2000 ft/s,however,calculated velocities at 100 yards using equation(11)are consistently about 1%-2%higher than the test results.For example,36,45,and 50 caliber round ball test velocities at 100 yards were reported as 949 ft/s,1065 ft/s,and 1131 ft/s for muzzle velocities of 2015,2021,and 2012 ft/s[15].The calculated velocities at 100 yards using equation(16)were,respectively,966 ft/s,1091 ft/s,and 1145 ft/s.

Fig.4.Round ball velocity at 100 yards for various calibers with muzzle velocity of 1450 ft/s.

In another test,for a 50 caliber round ball with diameter 0.495 inches and muzzle velocity 2004 ft/s,the round ball velocities at 50 yards and 100 yards were reported as 1523 ft/s and 1137 ft/s,respectively[6].The calculated velocities for this test using equation(11)were 1502 ft/s and 1147,respectively,for 50 yards and 100 yards and were within 1.5%of the reported values.

For smaller sphere sizes,comparisons were made for experimental data for a 36 gm load of BB lead shot with a muzzle velocity of 400 m/sec and for a 28 gm load of size 4 steel shot with a muzzle velocity of 322 m/sec.The experimental data were reported by Compton[11],who extensively studied shot cloud dynamics.The results show fairly good agreement between calculated times and velocities as indicated in Fig.5.The better agreement for the larger,denser pellets may be due to less interactions among the pellets during the initial few yards of flight[11].

One last test was performed,in this case,for 6-shot lead pellets with a muzzle velocity of 1375 ft/s.In Fig.6,calculated values are compared with tabulated values given in Ref.[29]whose“values were based on a carefully executed and well-monitored set of ballistic measurements”.Plots in Fig.6 shows close agreement between the calculated and reference values.

6.Examples

Two examples are described to illustrate use of equations(11)-(16).In these examples,the parameters vs,M0,andkzare first calculated and then substituted into the equations.

6.1.Example 1

In the first example,lead pellets of shot size 6 have muzzle velocity 1350 ft/s.The temperature is 86 F,the pressure is 29.5 inHg,the relative humidity is 40%,and the crosswind is 15 mph.Of interest is the velocity and flight time of the pellets at 30 yards and 60 yards.Also,the drop and the drift are to be estimated.

Forx=30 yards=2743.2 cm,the estimated flight time and velocity are 0.0892 s and v=811.85 ft/sec.Also,by equations(17)and(18),drop=1.54 inches and the drift=5.95 inches.Forx=60 yards=5486.4 cm,the estimated flight time and velocity aret=0.2229 sec and v=565.82 ft/sec.For 60 yards,the calculated drop=9.59 inches and the drift=23.65 inches.Plots of the velocity and flight time using equations(23)and(24)are given up to 80 yards in Fig.7.

Fig.5.Pellet velocities and flight times for lead and steel pellets with diameters 0.425 cm and 0.302 cm,respectively.

Fig.6.Pellet velocities and times for lead pellets with diameters 0.277 cm.

6.2.Example 2

For this example,a 50 caliber lead round ball(0.495 inch diameter)is shot with a muzzle velocity of 2004 ft/s.The velocity and flight time are required at 50 yards,100 yards and 200 yards.Atmospheric conditions are 20°C,30 inHg,and 10%relative humidity.The lateral wind speed is 5 mph.

For this problem, the muzzle velocity v0=2004×30.48=61082 cm/sec,the pellet density ρp=11.34 gm/cm3,pellet diameterD=1.2573 cm,air pressurePa=30 inHg,relative humidityRH=0.1,air temperatureTa=20°C,and lateral wind speed vw=5 mph.By equation(21),the vapor pressurePV=0.0690 inHg and by equation(20),the air density ρa(bǔ)=0.0012063 gm/cm3.The speed of sound using equation(22)is vs=34340 cm/sec.Next,scale constantkzis calculated using equation(7)askz=Dρp/ρa(bǔ)=11820 cm.AsM0=v0/vs=1.77874＞1.2,Case 1 is required with three parts to the flight time and pellet velocity.Substitutingkz,M0,and vsinto the equations in Case1,x1=6361.2，x2=18792，t1=0.1277, andt2=0.5394. Next,substituting these values into equations(11)and(12)gives

and

Fig.7.Pellet velocity and flight time for the first example.

Forx=50 yards=4572 cm,the estimated flight time and velocity are 0.0866 s and 1508.9 ft/s.Also,by equations(17)and(18),drop=1.45 inches and the drift=1.03 inches.Forx=100 yards=9144 cm,the estimated flight time and velocity aret=0.2009 sec and v=1157.5 ft/sec.For 100 yards,the calculated drop=7.79 inches and the drift=4.51 inches.Forx=200 yards=18288 cm,the estimated flight time and velocity aret=0.5186 sec and v=801.4 ft/sec.For 200 yards,the calculated drop=51.92 inches and the drift=19.29 inches.Plots of the velocity and flight time using equations(25)and(26)are given out to 200 yards in Fig.8.

7.Shot clouds and string length

For convenience and completeness,some results on shot cloud dynamics and string length are briefly summarized in this section for comparison with the assumption made in the present investigation on free flight of the pellets.

7.1.Shot cloud

The undispersed pellets in the shot column experience more complicated drag forces than free,single pellets[10,14,23,24].Pellets traveling directly behind other pellets may have lower drag forces while adjacent-traveling pellets may have higher drag forces.In addition,deformations occurring due to pellet-pellet and pelletbarrel interactions result in nonspherical pellets with higher drag coefficients.Consequently,pellet velocities in the shot cloud gradually differ longitudinally in space.As the pellets disperse away from the shot column,the pellets become free from interactions with other pellets and move as single pellets.It appears that if a pellet is over 3.5 diameters away from other pellets,it may experience little interaction with other pellets[11,24]and may be considered to be moving as a free,single pellet.To discuss the effect of dispersal on the pellet dynamics,it is useful to estimate the distance at which pellets begin to travel freely.

To investigate this,the dispersal of shot pellets from cylinder,modified,and full chokes was computationally simulated.As commonly classified,it was assumed that 40%,60%,and 70%,of the pellets land in a 30-inch circle at 40 yards,respectively,for cylinder,modified,and full chokes[2].In addition,as generally assumed[11,20,28],the dispersal distribution was taken to be Gaussian so the density of the pellets passing distancexis given by

Fig.8.Pellet velocity and flight time for the second example.

whereu=x/40,Nis the total number of pellets,and ρ（r，x）is the number of pellets per unit area at a radial distance ofrinches and a trajectory distance ofxyards.(Parameteruis proportional toxto model the linear spreading pattern of the pellets with trajectory distance as observed experimentally[13].)For cylinder,modified,and full chokes,σ is equal to 14.84 inches,11.08 inches,and 9.666 inches,respectively.At each distancez=10，15，20，25 yards and for each choke,100 shots were simulated and each pellet was counted as either single or not single.A pellet was classified as a single pellet if the pellet was at least 3.5 diameters from every other pellet.In each shot there was one ounce of 4-shot or 8-shot pellets whereNis equal to 135 for 1 ounce of 2-shot and 410 for 1 ounce of 8-shot.The results of the computations are summarized in Table 3 and Table 4.Notice that over 3/4 of the pellets are traveling singly at 15 yards even for full choke and 8-shot pellets.Also,the velocity of pellets that travel freely from the muzzle out to 15 yards are still moving at speeds in excess of 80%of the speed of the shot column even if the shot column experiences no drag whatsoever.The computational results indicate that it appears generally reasonable to estimate the dynamics of shotgun ballistics using free-sphere ballistics especially for open chokes or large shot sizes.One effect of pellets gradually dispersing away from a dense shot cloud is the production of a longitudinal distribution of the pellets which is more pronounced for tighter chokes and smaller pellets.This is discussed next.

Table 3Percentages of pellets that are traveling singly at various distances for three chokes for 1 ounce of 4-shot.

Table 4Percentages of pellets that are traveling singly at various distances for three chokes for 1 ounce of 8-shot.

7.2.Shot string length

Due to small differences in initial velocities and drag forces,the pellets string out along the trajectory forming a cigar-shaped distribution.Even though the practical effect of a longitudinal shot string,in terms of differences in the numbers of pellets hitting a moving or non-moving target,is slight[20],the phenomenon is interesting to briefly study using equation(10).The primary physical explanation for a shot string is that individual pellets exit the undispersed shot column at differing times and velocities during the first few yards of travel[20].In addition,due to variations in pellet size and shape with some pellets deformed in the barrel[13],the pellets experience different drag forces resulting in different flight times.

To investigate the nature of these two effects,the drag coefficient is assumed to be constant in the drag equation(10)such as assumed,for example,in Ref.[1].That is,C（M）=CwhereCis a constant independent ofM.In this case,equation(10)yields the simple solution

and the positionxat timetsatisfies

Suppose that some of the pellets have exited the shot cloud with minimum velocity vaand maximum velocity vbbut the pellets have the same drag coefficientC.Secondly,suppose that other pellets have exited the shot cloud with minimum drag coefficientCaand maximum drag coefficientCbbut the pellets in this case have the same exiting velocity v0.It is assumed for simplicity that vb-vaandCb-Caare small so va≈vb≈v0andCa≈Cb≈C.LetLbe the shot string length,specifically,the difference in the positions between the slowest and fastest pellets.For two pellets with initial velocities vaand vbbut the same drag coefficient,

For two pellets with different drag coefficients but the same exiting velocity,i.e.,the pellets have different sizes or shapes,then,

By equation(30),the shot string length gradually ceases to increase if the exiting velocities differ.However,by equation(31),if the drag coefficients differ due to deformed shapes,then the shot string length continually increases with flight time.As a result,the effect of deformed pellets on shot string length may be of greater interest,especially for soft pellet material,than a spread in velocities after pellets exit a dense shot cloud.s

8.Summary

Several constants are required in order to use formulas(11)through(18)for calculating pellet velocities,flight times,and drop and drift corrections.To find these constants,only the pellet size and density,muzzle velocity,air temperature,air pressure,relative humidity,and crosswind velocity are needed.The tests performed here indicate that the formulas are reasonably accurate for smooth spherical pellets with limited interactions.

In this paper,accurate analytic ballistics formulas are derived for spherical projectiles of shotguns or muzzleloaders using realistic drag coefficients.This is of interest as currently,for example,accurate ballistics formulas for shotguns and muzzleloaders are not known analytically and ballistics curves must be calculated computationally.The formulas may also be useful,for example,in estimating the sensitivity of projectile behavior to changes in parameter values.Furthermore,similar analyses as that given in the present investigation can possibly be performed for other projectile shapes besides spherical,in particular,when an equation similar to(10)holds where the drag coefficient depends solely on Mach number.Finally,the presentation is complete and self-contained.All the formulas are clearly defined and accessible,and can be readily implemented by hunters,military or law enforcement personnel,and shotgun or muzzleloader enthusiasts.

Acknowledgement

The author is grateful to the anonymous referee for helpful comments especially with regard to improving the paper's structure.

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