Conceptual Design Approach for Small-Caliber 

Aeroballistics With Application to 5.56-mm Ammunition 

by Paul Weinacht, James F. Newill, and Paul J. Conroy 

ARL-TR-3620 September 2005 

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Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-3620 September 2005 

Conceptual Design Approach for Small-Caliber 

Aeroballistics With Application to 5.56-mm Ammunition 

Paul Weinacht, James F. Newill, and Paul J. Conroy 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
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Conceptual Design Approach for Small-Caliber Aeroballistics With Application 

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Paul Weinacht, James F. Newill, and Paul J. Conroy 

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13. SUPPLEMENTARY NOTES 

14. ABSTRACT 

A conceptual design approach for small-caliber ammunition is presented. The analysis is based on analytical solutions of the 

point mass trajectory equations obtained using a power law model of the drag coefficient variation with Mach number. The 

method is validated by comparison with numerical trajectory predictions for the 5.56-mm M855. Using existing data, several 

important observations regarding small arms drag characteristics are quantified. The results of the analysis form the basis for a 

conceptual design approach for examining the aeroballistics of small-caliber munitions. Application is then made to 5.56-mm 

ammunition. Results are presented that examine the design boundaries for 5.56-mm ammunition in terms of trajectory match 

relative the 5.56-mm M855. By considering interior ballistics constraints and dependence of muzzle velocity on launch mass, 

the design space is further refined. The results allow the projectile performance to be characterized in terms of impact velocity 

and energy, gravity drop, trajectory mismatch at range, and cross-wind sensitivity. 

15. SUBJECT TERMS

aerodynamics, aeroballistics, small-caliber ammunition 

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Paul Weinacht 

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iii

Contents 

List of Figures iv

List of Tables vii

1. Introduction 1

2. Technical Approach 2

3. Drag Characteristics of Small-Caliber Ammunition 11

4. Trajectory Mismatch 28

5. Combined Exterior and Interior Ballistics Results 39

5.1 Comparison of the Relative Performance of the M855 and Mk262 .............................54 

5.2 Copper and Steel-Cored M855 Variants .......................................................................54 

6. Conclusion 55

7. References 57

List of Symbols, Abbreviations, and Acronyms 59

Distribution List 61
iv

List of Figures 

Figure 1. Fit of drag coefficient for the M855................................................................................3 

Figure 2. Fractional remaining velocity as a function of range, M855...........................................5 

Figure 3. Time of flight vs. range, M855........................................................................................6 

Figure 4. Gravity drop vs. range, M855..........................................................................................6 

Figure 5. Trajectories to hit 300- and 600-m targets, M855...........................................................7 

Figure 6. Cross-wind drift due to 10-m/s cross wind, M855..........................................................7 

Figure 7. Drift due to yaw of repose and aerodynamic jump due to 50-rad/s angular rate at 

muzzle, M855. ...........................................................................................................................8 

Figure 8. Velocity vs. range for baseline M855, M855 with 10% higher muzzle velocity, 

M855 with 10% higher retardation, and M855 with n = 0.25. ................................................10 

Figure 9. Gravity drop vs. range for baseline M855, M855 with 10% higher muzzle velocity, 

M855 with 10% higher retardation, and M855 with n = 0.25. ................................................10 

Figure 10. Cross-wind drift (10-m/s cross wind) vs. range for baseline M855, M855 with 

10% higher muzzle velocity, M855 with 10% higher retardation, and M855 with n = 

0.25...........................................................................................................................................11 

Figure 11. Drag coefficient exponent and form factor for various 0.308-projectiles; bullet 

photographs from Sierra Bullets (8). .......................................................................................18 

Figure 12. Drag coefficient exponent, n, vs. drag form factor with illustration of effect of 

bullet shape; bullet photographs from Sierra Bullets (8).........................................................19 

Figure 13. Scaled impact velocity as a function of scaled range..................................................25 

Figure 14. Scaled impact velocity as a function of scaled range (close-in)..................................25 

Figure 15. Scaled cross-wind drift as a function of scaled range. ................................................26 

Figure 16. Scaled cross-wind drift as a function of scaled range (close-in).................................26 

Figure 17. Scaled gravity drop as a function of scaled range. ......................................................27 

Figure 18. Scaled gravity drop as a function of scaled range (close-up)......................................27 

Figure 19. Gravity drop for M855 and ±1.0 mil mismatch trajectories for 949 m/s muzzle 

velocity.....................................................................................................................................29 

Figure 20. Gravity drop in mils for M855 and ±1.0 mil mismatch trajectories for 949 m/s 

muzzle velocity. .......................................................................................................................30 

Figure 21. Gravity drop for M855 and –1.0 mil mismatch trajectories for different muzzle 

velocities. .................................................................................................................................30 

Figure 22. Acceptable range of muzzle velocity and muzzle retardation to attain ±1.0 mil 

mismatch..................................................................................................................................31
v

Figure 23. Sensitivity of the acceptable range of muzzle velocity and muzzle retardation for 

±1.0 mil mismatch to drag coefficient exponent. ....................................................................32 

Figure 24. Muzzle velocity, muzzle retardation, and trajectory mismatch for M855 and 

M193 and their companion tracer rounds, M856 and M196, respectively..............................33 

Figure 25. Percent increase or decrease in wind drift at 200 m relative to the M855 as a 

function of muzzle velocity for constant values trajectory mismatch at 600 m. .....................34 

Figure 26. Percent increase or decrease in wind drift at 300 m relative to the M855 as a 

function of muzzle velocity for constant values trajectory mismatch at 600 m. .....................34 

Figure 27. Percent increase or decrease in wind drift at 600 m relative to the M855 as a 

function of muzzle velocity for constant values trajectory mismatch at 600 m. .....................35 

Figure 28. Trajectories for M855 and modified ammunition with –1.0 mil mismatch at 

600 m with rezeroing at 200 and 300 m. .................................................................................37 

Figure 29. Gravity drop for M855 and modified ammunition with –1.0 mil mismatch at 

600 m when fired at same gun elevation as M855 or zeroed at 100, 200, and 300 m.............37 

Figure 30. Predicted breech pressure and case mouth pressure (Gage 0.32 in) compared with 

measured case mouth pressure for standard M855 cartridge...................................................40 

Figure 31. Muzzle velocity vs. charge mass for M855 cartridge and propellant..........................41 

Figure 32. Typical projectile shapes for projectiles with form factors of 0.6 and 0.8; bullet 

photographs from Sierra Bullets (8). .......................................................................................42 

Figure 33. Muzzle retardation vs. projectile mass for form factors of 0.6 and 0.8 compared 

with retardation of M855, Mk262, and M193. ........................................................................43 

Figure 34. Impact velocity vs. projectile mass for ranges of 0, 150, 300, 450, and 600 m. .........44 

Figure 35. Impact velocity vs. projectile mass for ranges of 0, 25, 50, 75, and 100 m. ...............44 

Figure 36. Impact energy vs. projectile mass for ranges of 0, 150, 300, 450, and 600 m. ...........46 

Figure 37. Impact energy vs. projectile mass for ranges of 0, 25, 50, 75, and 100 m. .................46 

Figure 38. 3/ 2 mV vs. projectile mass for ranges of 150, 300, 450, and 600 m. ...........................47 

Figure 39. Gravity drop vs. projectile mass for ranges of 150, 300, 450, and 600 m...................47 

Figure 40. Gravity drop relative to M855 for ranges of 150, 300, 450, and 600 m......................48 

Figure 41. Tangent of impact angle vs. projectile mass for ranges of 150, 300, 450, and 

600 m. ......................................................................................................................................49 

Figure 42. Cross-wind drift for 10-m/s cross wind vs. projectile mass for ranges of 150, 300, 

450, and 600 m........................................................................................................................50 

Figure 43. Cross-wind drift relative to M855 vs. projectile mass for ranges of 150, 300, 450, 

and 600 m.................................................................................................................................51 

Figure 44. Impact velocity vs. projectile mass for form factors of 0.6 and 0.8 at ranges of 

150, 300, 450, and 600 m.........................................................................................................51 

Figure 45. Impact energy vs. projectile mass for form factors of 0.6 and 0.8 at ranges of 150, 

300, 450, and 600 m.................................................................................................................52 
vi

Figure 46. Gravity drop of projectiles with form factor of 0.8 relative to M855 vs. projectile 

mass for ranges of 150, 300, 450, and 600 m. .........................................................................53 

Figure 47. Cross-wind drift of projectiles with form factor of 0.8 relative to M855 vs. 

projectile mass for ranges of 150, 300, 450, and 600 m. .........................................................53 
vii

List of Tables 

Table 1. Results from curve-fitting velocity data, 0.224-caliber projectiles. ...............................14 

Table 2. Results from curve-fitting velocity data, 0.243-caliber projectiles. ...............................15 

Table 3. Results from curve-fitting velocity data, 0.308-caliber projectiles. ...............................16 

Table 4. Results from curve-fitting velocity data, 0.311-caliber projectiles. ...............................16 

Table 5. Results from curve-fitting velocity data, 0.323-caliber projectiles. ...............................17 

Table 6. Results from curve-fitting velocity data, 0.338-caliber projectiles. ...............................17 

Table 7. Results from curve-fitting velocity data, 0.375-caliber projectiles. ...............................17 

Table 8. Results from curve-fitting velocity data, 0.458-caliber projectiles. ...............................17 

Table 9. Sensitivity of muzzle velocity on results from trajectory fitting, 0.308 150-gr HPBT 

Match King. .............................................................................................................................20 

Table 10. Sensitivity of muzzle velocity on results from trajectory fitting, 0.224 69-gr HPBT 

Match King. .............................................................................................................................21 

Table 11. Sensitivity of muzzle velocity on results from trajectory fitting, 0.224 52-gr HPBT 

Match King. .............................................................................................................................22 

Table 12. Sensitivity of muzzle velocity on results from trajectory fitting, 0.224 52-gr, G1 

drag curve, constant ballistic coefficient C = 0.225.................................................................23 
viii

INTENTIONALLY LEFT BLANK.
1

1. Introduction 

The current generation of 5.56-mm ammunition for the M16 and M4 rifles has provided the 

U.S. Army with high performance projectiles for several decades. Currently, the U.S. Army is 

investigating possible replacements for the existing family of 5.56-mm ammunition. Some of 

the factors driving the replacement include cost and performance. The reduction of the 

environmental impact associated with the materials currently used in 5.56-mm ammunition, in 

particular the heavy metal lead core, is an additional important consideration and is being 

addressed by the U.S. Army’s Green Ammunition programs. 

As part of the design effort, conceptual design studies are being performed to define the design 

space and highlight the important design variables. A recent study (1) has provided the 

framework for examining the exterior ballistic performance in a conceptual design environment 

where many details of the design have not been fully established. This allows the gross 

characteristics of the round to be examined and the design tradeoffs regarding the aeroballistics 

to be more fully quantified and understood. In the current effort, interior ballistic considerations 

have also been incorporated as well. The analysis also considers relatively simple metrics that 

allow the terminal performance to be assessed in a very general qualitative manner. These types 

of metrics are appropriate for a conceptual design approach, although, clearly, more detailed 

analysis is required to quantify the terminal performance as the design process evolves. 

The basis of the current approach is a simple but accurate method for predicting the trajectories 

for high velocity direct-fire munitions (1). The method allows the trajectories to be characterized 

in terms of three parameters: the muzzle velocity, the muzzle retardation (or velocity fall-off), 

and a parameter defining the shape of the drag curve. The method provides an excellent means 

of assessing the exterior ballistic performance in a conceptual design environment where details 

of the designs have not been completely defined. 

In the following sections, the analytical approach for solving the three-degree-of-freedom 

(3DOF) trajectory equations is briefly described and benchmarked with numerical predictions of 

the trajectory of the 5.56-mm M855 projectile. The drag characteristics of existing small arms 

bullets are then analyzed and several important observations regarding the drag characteristics 

are quantified. Results are then presented for a family of potential 5.56-mm ammunition 

designs. The first set of results examines the performance of 5.56-mm ammunition assuming the 

muzzle velocity is an independent parameter. The boundaries of the design space are defined in 

terms of trajectory match with the existing M855 projectile. The second set of results considers 

the effect of interior ballistic considerations and constraints on the muzzle velocity resulting 

from the dependence of muzzle velocity on projectile mass. Both sets of results provide 

important design guidance for current efforts in 5.56-mm ammunition development. 
2

2. Technical Approach 

The flat fire trajectory of a projectile can be characterized as follows (2): 

 . Drift

Swerve

Epicyclic

Jump

Aerodynamic

Throwoff

Lateral

Point MassTrajectory

The Flat Fire

Flat FireTrajectory

TheComplete

⎥

⎦

⎤ ⎢

⎣

⎡

+⎥

⎦

⎤ ⎢

⎣

⎡

+

⎥

⎦

⎤ ⎢

⎣

⎡

+⎥

⎦

⎤ ⎢

⎣

⎡

+⎥

⎦

⎤ ⎢

⎣

⎡ =⎥

⎦

⎤ ⎢

⎣

⎡

 (1) 

The flat fire point mass trajectory accounts for the most dominant characteristics of the trajectory 

and is most heavily influenced by the mass and drag characteristics of the projectile. It includes 

such effects as gravity drop and cross-wind drift. It is this portion of the trajectory that is the 

focus of the current concept design approach. The aerodynamic jump and lateral throwoff of the 

projectile produce angular deviations of the flight path from the intended line of flight due to 

launch disturbances and mass asymmetries within the projectile, respectively. Bias errors 

associated with these effects are normally removed through the rifle zeroing process. These 

effects can also produce random errors that contribute to the ammunition dispersion. Normally, 

these effects would not be considered in any detail in a conceptual design process as the relative 

magnitude of these effects can only be quantified later in the design cycle. Additionally, the 

relative magnitude of the lateral throwoff is heavily influenced by manufacturing quality 

considerations. The epicyclic swerve represents fluctuating motions of the projectile about the 

trajectory due to the angular motion of the projectile. For a stable projectile, these motions are 

typically small relative to the mean path of the projectile. The drift produces a small horizontal 

deflection of the projectile that is fairly consistent from shot to shot. The deflection is typically 

small compared with cross-wind drift. 

The flat fire point mass trajectory can be obtained by solving the 3DOF equations. This 

approach considers only the linear displacement of the projectile’s center of gravity along the 

flight path and ignores the angular motion of the projectile. Using this approach, the projectile is 

characterized by its muzzle velocity, mass, and the variation of its drag coefficient with Mach 

number. Typically, the integration of the 3DOF equations is performed numerically. However, 

it is also possible to obtain analytical solutions of the 3DOF equations under the assumption of 

direct fire using an assumed form of the drag coefficient variation with Mach number (1). For 

supersonic flight, it can be shown that the drag coefficient variation with Mach number can be 

modeled as a function of Mach number to a power as shown in equation 2. For constant 

atmospheric conditions, the sound speed is constant, and the Mach number and velocity are 

directly proportional. In this case, the drag coefficient varies with Mach number and velocity in 

the same manner. 
3

D n n V

1

M

1 C ∝ ∝ . (2) 

Figure 1 shows the variation of the drag coefficient with Mach number for the 5.56-mm M855. 

The Firing Tables Branch (FTB), U.S. Army Armament Research, Development, and 

Engineering Center (ARDEC) “aeropack” data (3) used to construct firing tables for the round 

are shown along with aerodynamic spark range data (4). The fit of the FTB aeropack drag 

coefficient data obtained using the power law formulation shown in equation 2 is also shown. 

The computed exponent for the M855 is 0.53 for the range between Mach 2.8 (muzzle velocity) 

and Mach 1.1. The power law fit shows excellent agreement over this Mach number range. 

0

0.1

0.2

0.3

0.4

0.5

0.6

012345

Mach Number

Drag Coefficient

FTB Aeropack

Power Law Fit

Range Data

Figure 1. Fit of drag coefficient for the M855. 

Weinacht et al. (1) examined the drag coefficient variation of a wide variety of munitions and 

found that the drag coefficient exponent varied between 0.0 and 1.0 for supersonic flight. 

Additional results specifically aimed at small arms ammunition have been obtained as part of the 

current study. These results are discussed in a subsequent section. 

Using this approach, analytical solution of the governing equations can be obtained resulting in 

closed form equations for the dependent variables characterizing the projectile trajectory. These 

include the velocity V , time of flight t , gravity drop sg-drop , vertical displacement of the 

projectile along the trajectory sy, and vertical deflection due to cross wind sz, as a function of 

range sx, as shown in equations 3–7. 
4

n

1

0

x

0

0 V

s

ds

dV V V 1 n

⎭

⎬

⎫

⎩

⎨

⎧ ⎟

⎠

⎞ ⎜

⎝

⎛ = + . (3) 

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

⎭

⎬

⎫

⎩

⎨

⎧ ⎟

⎠

⎞ ⎜

⎝

⎛ +

⎟

⎠

⎞ ⎜

⎝

⎛ −

=

−

1

V

s

ds

dV 1 n

ds

dV (n 1)

1

t n

1 1

0

x

0

0

. (4) 

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⎟

⎠

⎞ ⎜

⎝

⎛ − − −

⎭

⎬

⎫

⎩

⎨

⎧ ⎟

⎠

⎞ ⎜

⎝

⎛ +

⎟

⎠

⎞ ⎜

⎝

⎛ − −

− =

−

−

0

x

0

n

2(n 1)

0

x

0

2

0

g drop V

s

ds

dV 1 2(n 1) V

s

ds

dV 1 n

ds

dV 2(n 2)(n 1)

g s . (5) 

y x 0 g drop s s tan s = θ + − . (6) 

⎥

⎦

⎤ ⎢

⎣

⎡ = −

0

x

z z V

s

s w t . (7) 

Through the analysis, it can be shown that these trajectory characteristics are functions of only 

three primary variables, the projectile’s muzzle velocity V , muzzle retardation 0 ds 0

dV ⎟

⎠

⎞ ⎜

⎝

⎛ , and the 

exponent defining the shape of the drag curve n . The trajectory is also a function of two 

independent parameters, the gravitational constant g , and the cross-wind velocity wz . The gun 

elevation angle θ0 also appears in equation 6 and can be treated as another independent 

parameter. However, the gun elevation angle θ0 required to hit a target at range can be related 

to the three primary variables: the muzzle velocity, muzzle retardation, and drag coefficient 

exponent. In this regard, the gun elevation angle itself can be treated as a dependent variable. 

The muzzle velocity and muzzle retardation have the strongest influence on the trajectory and the 

exponent defining the shape of the drag curve can be shown to be a higher-order effect whose 

influence is less important that the first two variables, particularly at shorter ranges. 

The muzzle retardation is dependent on the projectile mass and muzzle drag coefficient as shown 

in equation 8. 

V0 0 ref D

0

V S C

2m

1

ds

dV ⎟ = − ρ ⎠

⎞ ⎜

⎝

⎛ . (8) 

Thus, the effect of both projectile mass and muzzle drag coefficient on the trajectory are 

represented by a single parameter, the muzzle retardation. The effect on the trajectory of a 20% 

increase in the drag coefficient is the same as a 20% decrease in the projectile mass. 

The approach was validated in Weinacht et al. (1) for large-caliber tank-fired munition such as 

the M829A1, M830, M865, and M830A1 by comparing the results of the analytical method with 
5

numerical solutions obtained with the 6DOF option of the aerodynamic and trajectory prediction 

code PRODAS (5). Excellent agreement between the two approaches was found. In fact, in 

most cases, the uncertainty or variability in the muzzle velocity and muzzle retardation (in 

particular, the muzzle drag coefficient) produces more error than the approximations made to 

obtain the analytical integrations, although both of these errors are typically small. 

Similar validation for small-caliber munitions was performed as part of the current study. 

Predictions of the trajectory history were made for the 5.56-mm M855 munition using the 

analytical approach and 6DOF predictions from PRODAS. To provide a fair basis for 

comparison, both simulations used the identical muzzle velocity and projectile mass. The drag 

coefficient exponent used in the analytical approach was obtained by fitting the drag coefficient 

variation with Mach number obtained from U.S. Army ARDEC FTB “aeropack” data for the 

M855. The aeropack data is used to compute the firing tables for the round. This data was also 

used to determine the muzzle drag coefficient and, subsequently, the muzzle retardation used in 

the analytical approach. This drag data was also used for the PRODAS simulations. The 6DOF 

simulations also require additional aerodynamic data. Updated aerodynamics based on 

aerodynamic range firings (4) were used for the PRODAS 6DOF simulations, although these 

data have little effect on the point mass trajectory parameters considered here. Figures 2–6 show 

the velocity history, time of flight, gravity drop, vertical displacement for 300- and 600-m range 

trajectories, and cross-wind deflection due to 10-m/s cross wind as a function of range. 

Excellent agreement is seen between the PRODAS simulations and the analytical predictions. 

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250 300 350 400 450 500 550 600

Range - Meters

V/V0

M855-Prodas

M855-Analytical

Figure 2. Fractional remaining velocity as a function of range, M855. 
6

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300 350 400 450 500 550 600

Range - Meters

Time of Flight - Seconds

M855-Prodas

M855-Analytical

Figure 3. Time of flight vs. range, M855. 

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 50 100 150 200 250 300 350 400 450 500 550 600

Range - Meters

Gravity Drop - Meters

M855-Prodas

M855-Analytical

Figure 4. Gravity drop vs. range, M855. 
7

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 50 100 150 200 250 300 350 400 450 500 550 600

Range - Meters

Height - Meters

M855-Prodas

M855-Analytical

M855-Prodas

M855-Analytical

Figure 5. Trajectories to hit 300- and 600-m targets, M855. 

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 50 100 150 200 250 300 350 400 450 500 550 600

Range - Meters

Drift due to Cross Wind - Meters

M855-Prodas

M855-Analytical

Figure 6. Cross-wind drift due to 10-m/s cross wind, M855. 
8

Figure 7 also shows two important contributions to the vertical and horizontal displacement of 

the projectile along the trajectory obtained from the 6DOF predictions. These two contributions 

are the horizontal drift due to yaw of repose and the deflection (vertical or horizontal) due to 

aerodynamic jump. Prediction of these displacements requires additional details of the projectile 

physical properties and aerodynamics beyond those required for 3DOF trajectory predictions 

presented here. Typically, these details are not available in the conceptual design process 

considered here, but are defined as the design evolves. In any event, these effects exist and must 

be considered in the conceptual design even if they can’t be predicted in the conceptual design 

process. 

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0 50 100 150 200 250 300 350 400 450 500 550 600

Range - Meters

Deflection - Meters

Aerodynamic Jump

Yaw of Repose

Figure 7. Drift due to yaw of repose and aerodynamic jump due to 50-rad/s angular rate 

at muzzle, M855. 

There are two notable sources of error or uncertainty in applying the approach of Weinacht et al. 

(1). The first is the approximation made to obtain closed form analytical solution of the point 

mass trajectory equations. This approximation assumes that the component of velocity produced 

by gravity is small and can be ignored when computing the aerodynamic drag of the projectile. 

This approximation is valid for flat fire and has been shown to produce very little error when 

compared with more exact numerical solutions of the governing equations. 

The second source of error is associated with modeling the variation of the drag coefficient with 

Mach number using a power law approach. The analysis of Weinacht et al. (1) shows that the 

drag variation of a wide variety of munitions can be accurately modeled using this approach. 
9

Furthermore, the predicted trajectories obtained using this approach show surprisingly little 

sensitivity to the exponent used in the power law approach. This implies that the shape of the 

drag curve is not a dominant factor in predicting the trajectory, particularly when compared with 

the muzzle velocity and muzzle retardation. This is significant because when the current 

approach is used in a conceptual design approach, the design process can focus on muzzle 

velocity and muzzle retardation as the significant design variables defining the point mass 

trajectories. The second source of uncertainty must be considered whether analytical or 

numerical solutions of the point mass equations are sought. In either case, the variation of the 

drag coefficient with Mach number must be specified. The power law approach represents a 

very useful approach for conceptual design because of its simplicity and accuracy in representing 

the drag variation with Mach number. The geometric details required to obtain a more accurate 

drag variation are unavailable at this point in the design cycle. 

Figures 8–10 show the relative effect of uncertainty in the muzzle velocity, muzzle retardation, 

and the exponent defining the shape of the drag curve on the predicted velocity, gravity drop, and 

cross-wind drift, respectively. Here, the results show differences between the baseline M855, a 

projectile with 10% more muzzle velocity, a projectile with 10% higher retardation, and a 

projectile with a drag coefficient exponent of 0.25 (relative to the baseline drag coefficient of 

0.53). The selected differences are indicative of uncertainties that might be present in a 

conceptual design environment where the physical characteristics of the munition are not 

completely defined. (They should not be interpreted as representative of any variability in 

performance of the M855.) The selected drag coefficient exponents represent a relatively large 

difference given the ability to appropriately estimate this parameter. Analysis presented in the 

following section provides a means of estimating this parameter with greater fidelity than that 

represented by the differences presented here. 

The results show a greater sensitivity to muzzle velocity and muzzle retardation although the 

magnitude of the sensitivity depends somewhat on the dependent variable being considered. 

Gravity drop appears to be more heavily influenced by muzzle velocity than muzzle retardation 

since the gravity drop is strongly affected by the time of flight, which, to first order, is a function 

of muzzle velocity and range. Cross-wind drift shows significant sensitivity to both muzzle 

velocity and muzzle retardation. The greatest sensitivity of the results to the drag coefficient 

exponent relative to the muzzle velocity and muzzle retardation is seen for the velocity history 

shown in figure 8. At ranges up to about 300 m, the difference between predicted velocities for 

the baseline results and the results with a drag coefficient exponent of 0.25 is less than 2%. The 

differences increase to about 10% at 600 m. The sensitivity of the results to drag coefficient 

exponent is less for the cross-wind drift and gravity drop. These results provide further 

demonstration that the muzzle velocity and muzzle retardation are dominant independent 

variables defining the characteristics of the 3DOF trajectory. 
10

0

200

400

600

800

1000

1200

0 50 100 150 200 250 300 350 400 450 500 550 600

Range - Meters

Velocity - (m/s)

M855 Baseline

M855 10% Higher Muzzle Velocity

M855 10% Higher Retardation

M855 n=0.25

Figure 8. Velocity vs. range for baseline M855, M855 with 10% higher muzzle velocity, 

M855 with 10% higher retardation, and M855 with n = 0.25. 

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 50 100 150 200 250 300 350 400 450 500 550 600

Range - Meters

Gravity Drop - Meters

M855 Baseline

M855 10% Higher Muzzle Velocity

M855 10% Higher Retardation

M855 n=0.25

Figure 9. Gravity drop vs. range for baseline M855, M855 with 10% higher muzzle 

velocity, M855 with 10% higher retardation, and M855 with n = 0.25. 
11

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 50 100 150 200 250 300 350 400 450 500 550 600

Range - Meters

Cross Wind Drift - Meters

M855 Baseline

M855 10% Higher Muzzle Velocity

M855 10% Higher Retardation

M855 n=0.25

Figure 10. Cross-wind drift (10-m/s cross wind) vs. range for baseline M855, M855 with 

10% higher muzzle velocity, M855 with 10% higher retardation, and M855 

with n = 0.25. 

3. Drag Characteristics of Small-Caliber Ammunition 

The analysis of Weinacht et al. (1) examined a wide range of munition types, geometric shapes, 

and calibers and found that the exponent defining the variation of the drag coefficient in the 

supersonic regime varied between 0 and 1. For the current study, which is focused on small 

arms munitions, an addition study of the variation of the drag coefficient with Mach number 

(velocity) has been performed using data available from a commercial small arms ammunition 

manufacturer. This information includes ballistic coefficients and trajectory vs. range data. 

The ballistic coefficient C typically provided by small arms ammunition manufacturers is 

defined in equation 9. 

Form

2 i

1

D

m C = . (9) 

The ratio of the mass to reference diameter squared is referred to as the sectional density. The 

sectional density for a given bullet is either quoted by the manufacturer or is easily calculated 

from manufacturer’s data. It depends purely on the physical properties of the bullet and is a 
12

constant for a given bullet type. Generally, the sectional density increases with bullet caliber 

because the bullet mass increases at a rate proportional to the cube of the caliber whereas the 

denominator of the sectional density increases with the square of the caliber. The variation of 

sectional density with bullet caliber reflects only a general trend. Variations of sectional density 

within a fixed bullet caliber are common because the length and material composition (and hence 

mass) vary between different bullet designs. 

The form factor Form i in the ballistic coefficient represents the ratio of the drag coefficient of the 

actual round to the drag coefficient corresponding to a reference drag curve. For most 

commercial arms manufacturers, this reference drag curve is the G1 drag curve. The form factor 

for a low-drag streamlined bullet is generally about 0.6, indicating that the drag coefficient of the 

bullet is 60% of the reference drag coefficient. For bullets with very large nose bluntness, the 

form factor can be larger than 1.0. In contrast to the sectional density, there is likely little 

variation of the form factor between bullet calibers for the same shape bullet. The ratio 

represented by the form factor generally varies with Mach number. Typically, the form factor 

quoted by most small arms ammunition manufacturers represents the average value of this ratio 

over a range of velocities or Mach numbers. In some cases, the velocity regime may be broken 

into several intervals with a different ballistic coefficient and form factor assigned to each 

velocity interval. 

Many manufacturers quote a single ballistic coefficient (and hence form factor) for each bullet. 

Presumably, this single value represents the ballistic coefficient or form factor over a 

representative velocity range for the bullet. If a single form factor is used, the drag curve of the 

actual projectile is simply increased or decreased by a multiplicative constant over the reference 

drag curve. Thus, many commercial small arms ammunition manufacturers essentially assume 

that the shape of the drag curve for their bullet is identical to the shape of the reference G1 drag 

curve. In the context of the power law drag variation used in the current approach, these 

manufacturers essentially assume that the shape of the drag curve is represented by the drag 

coefficient exponent for the G1 drag curve, n = 0.36 for flight velocities between Mach 1.5 and 

3.0. As concluded in Weinacht et al. (1), the exponent defining the shape of the drag curve has a 

relatively small effect on the accuracy of the trajectory prediction compared to the muzzle 

velocity and muzzle retardation, especially at short ranges. In a sense, the analysis of reference 

Weinacht et al. (1) provides some measure of justification of the approach of quoting a single 

ballistic coefficient and representing the shape of the drag curve of all small arms bullets with the 

G1 drag function. 

At least one small arms ammunition manufacturer, Sierra Bullets, quotes several ballistic 

coefficients for each round that are valid over specified velocity regimes. Essentially, the 

manufacturer is attempting to account for differences in the shape of the actual drag curve 

relative to the reference G1 drag curve. This implies that if the drag curve is modeled using a 

power law representation, the drag coefficient exponent may be different from the exponent 

associated with the G1 drag curve. If the quality of the drag data used to derive the form factors 
13

and ballistic coefficients is good, the resulting trajectory predictions are likely to be more 

accurate using multiple ballistic coefficients than the predictions made using a single ballistic 

coefficient. 

An important observation regarding the drag coefficient exponent can be made by examining the 

ballistic coefficient data. For a given bullet, if the ballistic coefficient decreases with decreasing 

velocity in the supersonic regime, the drag coefficient exponent will be greater than the value of 

the exponent for the reference drag curve (n = 0.36). This is because the form factor of the bullet 

is increasing with decreasing velocity (or Mach number) and the drag coefficient is increasing 

faster with decreasing Mach number than the reference drag coefficient. Conversely, if the 

ballistic coefficient increases with decreasing velocity, the drag coefficient exponent will be less 

than the value of exponent for the reference drag curve (0.36). In this case, the form factor of the 

bullet is decreasing with decreasing velocity (or Mach number). This may indicate that the drag 

coefficient is either increasing more slowly with Mach number than the reference drag 

coefficient or perhaps even decreasing with Mach number. 

The database of ballistic coefficients and trajectory data provided by Sierra Bullets provides a 

wealth of data from which very useful information can be derived on the drag properties of small 

arms bullets. An extensive set of ballistic coefficients and trajectory data is provided by the 

manufacturer for a wide variety of projectile shapes and calibers (6, 7). This trajectory data is 

derived from the manufacturer’s ballistic coefficient data and includes velocity, gravity drop, and 

cross-wind drift vs. range. 

In the current study, the velocity vs. range data were curve fit to the functional form shown in 

equation 3 using a nonlinear least-squares approach. Since the muzzle velocity of the bullet is 

specified in the data, only the muzzle retardation and the exponent defining the shape of the drag 

curve were treated as variables in the fitting process. For each bullet type, trajectory data were 

available for a range of muzzle velocities. However, in the fitting process, only a single 

trajectory was typically fit. Where available, the trajectory data corresponding to 3100-ft/s 

muzzle velocities were selected. The 3100-ft/s muzzle velocity is close to the muzzle velocity of 

the M855 which is the focus of the current study. Sensitivity of the results to muzzle velocity 

was examined and is discussed below. In some cases, trajectories were not available at muzzle 

velocities as high as 3100 ft/s and, in these cases, the trajectory for the closest available muzzle 

velocity was selected. Any velocity data below Mach 1.1 (1241 ft/s) was ignored to avoid the 

transonic/subsonic velocity regime where drag coefficient changes rapidly with Mach number. 

For each trajectory, six to nine range locations were fit for each projectile. 

Following the fitting process, the manufacturer’s trajectory data was compared with the 

prediction made with the analytical method using both the muzzle retardation and drag 

coefficient exponent obtained from the corresponding fit and the specified muzzle velocity. 

Comparisons were made for the velocity, gravity drop, and cross-wind drift vs. range and root 

mean square (RMS) errors were computed for the individual trajectories. The average of the 
14

RMS errors from the individual trajectories for the impact velocity, gravity drop, and cross-wind 

drift were 0.85 ft/s, 0.04 in, and 0.03 in, respectively. 

Tables 1–8 display the data obtained from the fitting process for eight different bullet calibers. 

Shown in these tables are the exponent and muzzle retardation obtained from the fitting process 

along with the muzzle velocity. Since the fitting process was accomplished for different muzzle 

velocities for some rounds, the muzzle retardation was normalized to a reference muzzle velocity 

V of 3100 ft/s. This corresponds to the muzzle ve Ref locity of the majority of the trajectory 

selected for the fitting process. It is also close to the muzzle velocity of the M855. The 

normalized muzzle retardation provides a means of comparing the muzzle retardation between 

the rounds on a consistent basis. Muzzle retardation was adjusted for muzzle velocity as shown 

in equation 10 using the power law formulation for the drag coefficient. 

1 n

0

Ref

VRef 0 V

V

ds

dV

ds

dV −

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ⎟

⎠

⎞ ⎜

⎝

⎛ ⎟ = ⎠⎞ ⎜

⎝

⎛ . (10) 

Table 1. Results from curve-fitting velocity data, 0.224-caliber projectiles. 

Results Normalized to VRef

Projectile 

CD

Exponent

(n) 

ds 0

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m) 

V0

(m/s) 

SD

(lb/in2

) 

VRef ds

dV ⎟

⎠

⎞ ⎜

⎝

⎛ 

([m/s]/m) 

VRef

CD iForm

40-gr Hornet 0.099 –2.9038 945.1 0.114 –2.9038 0.521 1.049 

40-gr HP 0.267 –2.1609 945.1 0.114 –2.1609 0.388 0.781 

45-gr Hornet 0.093 –2.2819 823.2 0.128 –2.5865 0.522 1.049 

45-gr SMP 0.266 –2.0273 945.1 0.128 –2.0273 0.409 0.822 

45-gr SPT 0.498 –1.6415 945.1 0.128 –1.6415 0.331 0.666 

50-gr SMP 0.389 –1.7355 945.1 0.142 –1.7355 0.388 0.781 

50-gr SPT 0.429 –1.5034 945.1 0.142 –1.5034 0.336 0.677 

50-gr Blitz 0.429 –1.5034 945.1 0.142 –1.5034 0.336 0.677 

55-gr SPT 0.404 –1.3860 945.1 0.157 –1.3860 0.343 0.690 

55-gr Blitz 0.404 –1.3860 945.1 0.157 –1.3860 0.343 0.690 

55-gr SMP 0.252 –1.6219 945.1 0.157 –1.6219 0.401 0.807 

55-gr FMJBT 0.523 –1.3147 945.1 0.157 –1.3147 0.325 0.654 

55-gr SBT 0.461 –1.3325 945.1 0.157 –1.3325 0.330 0.663 

55-gr HPBT 0.321 –1.7866 945.1 0.157 –1.7866 0.442 0.889 

60-gr HP 0.372 –1.3693 945.1 0.171 –1.3693 0.369 0.742 

63-gr SMP 0.317 –1.4336 945.1 0.179 –1.4336 0.404 0.813 

52-gr HPBT Match King 0.410 –1.5131 945.1 0.148 –1.5131 0.353 0.710 

53-gr HP Match King 0.425 –1.5004 945.1 0.151 –1.5004 0.357 0.718 

69-gr HPBT Match King 0.284 –1.1054 945.1 0.196 –1.1054 0.341 0.687 

80-gr HPBT Match King 0.514 –0.7754 945.1 0.228 –0.7754 0.278 0.560 
15

Table 2. Results from curve-fitting velocity data, 0.243-caliber projectiles. 

Results Normalized to VRef

Projectile 

CD

Exponent

(n) 

ds 0

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m) 

V0

(m/s) 

SD

(lb/in2

)

VRef ds

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m)

VRef

CD iForm

60-gr HP 0.186 –1.8474 945.1 0.145 –1.8474 0.422 0.850 

75-gr HP 0.242 –1.5379 945.1 0.181 –1.5379 0.440 0.884 

80-gr SPTSSP 0.560 –1.1137 914.6 0.194 –1.1299 0.345 0.693 

80-gr SBT Blitz 0.570 –1.0141 914.6 0.194 –1.0285 0.314 0.631 

85-gr HPBT 0.235 –1.1776 945.1 0.206 –1.1776 0.381 0.768 

85-gr SPT 0.499 –1.0492 945.1 0.206 –1.0492 0.340 0.684 

90-gr FMJBT 0.507 –0.8545 945.1 0.218 –0.8545 0.293 0.590 

100-gr SPT 0.587 –0.8786 945.1 0.242 –0.8786 0.335 0.674 

100-gr SBT 0.594 –0.7615 945.1 0.242 –0.7615 0.290 0.584 

100-gr SMP 0.099 –1.2170 945.1 0.242 –1.2170 0.464 0.933 

70-gr HPBT Match King 0.293 –1.2981 945.1 0.169 –1.2981 0.346 0.697 

107-gr HPBT Match King 0.443 –0.6241 945.1 0.259 –0.6241 0.255 0.512 

The drag coefficient at the reference muzzle velocity was extracted from the normalized muzzle 

retardation using equation 11. 

VRef Ref ref

VRef D ds

dV

V S

2m C ⎟

⎠

⎞ ⎜

⎝

− ⎛ = ρ

. (11) 

The form factor evaluated at the reference muzzle velocity was obtained by taking the ratio of 

the drag coefficient obtained from equation 11 and the drag coefficient from the G1 drag curve 

evaluated at the reference velocity, C 0.53

VRef

DG1 = . 

VRef

DG1

VRef D

Form C

C

i = . (12) 

Comparisons were also made between the ballistic coefficients quoted by the manufacturer and 

the ballistic coefficients obtained from the fitting process. In general, the results agreed very 

well, typically within 3% or less. However, several rounds, typically the flat nose projectiles, 

showed differences in the ballistic coefficient coefficients of as much as 12%. For these rounds, 

the manufacturer’s ballistic coefficient shows large changes (20% or greater) across adjacent 

velocity intervals traversed by the projectile for the trajectory used in the fitting process. In these 

cases, the ballistic coefficient derived from the fitting process was bracketed by the 

manufacturer’s ballistic coefficients in adjacent velocity intervals near the muzzle velocity. 
16

Table 3. Results from curve-fitting velocity data, 0.308-caliber projectiles. 

Results Normalized to VRef

Projectile 

CD

Exponent

(n) 

ds 0

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m) 

V0

(m/s) 

SD

(lb/in2

)

VRef ds

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m)

VRef

CD iForm

110-gr RN 0.125 –2.2966 945.1 0.166 –2.2966 0.599 1.206 

110-gr FMJ 0.125 –2.0914 853.7 0.166 –2.2863 0.597 1.200 

110-gr HP 0.157 –1.8672 945.1 0.166 –1.8672 0.487 0.980 

125 -gr HP –0.359 –2.1834 762.2 0.188 –2.9247 0.867 1.745 

125 -gr SPT 0.485 –1.1747 945.1 0.188 –1.1747 0.348 0.701 

150 -gr FN –0.275 –1.3202 701.2 0.226 –1.9315 0.687 1.383 

150-gr RN 0.024 –1.6469 945.1 0.226 –1.6469 0.586 1.179 

150-gr SPT 0.314 –0.9856 945.1 0.226 –0.9856 0.351 0.706 

150-gr SBT 0.522 –0.8628 945.1 0.226 –0.8628 0.307 0.618 

150-gr FMJBT 0.489 –0.8115 945.1 0.226 –0.8115 0.289 0.581 

165-gr SBT 0.269 –0.8343 945.1 0.248 –0.8343 0.327 0.657 

165-gr HPBT 0.453 –0.9097 945.1 0.248 –0.9097 0.356 0.716 

170-gr FN –0.247 –1.1877 701.2 0.256 –1.7236 0.695 1.399 

180-gr RN –0.246 –1.3663 945.1 0.271 –1.3663 0.583 1.174 

180-gr SPT 0.292 –0.8201 945.1 0.271 –0.8201 0.350 0.705 

180-gr SBT 0.368 –0.6592 945.1 0.271 –0.6592 0.282 0.566 

200-gr SBT 0.409 –0.5911 945.1 0.301 –0.5911 0.280 0.564 

220-gr RN –0.109 –1.0000 853.7 0.331 –1.1194 0.584 1.175 

150-gr HPBT Match King 0.519 –0.8003 945.1 0.226 –0.8003 0.285 0.573 

155-gr HPBT Palma Match 

King 0.445 –0.7330 945.1 0.226 –0.7330 0.261 0.525 

168-gr HPBT Match King 0.529 –0.7108 945.1 0.253 –0.7108 0.283 0.570 

175-gr HPBT Match King 0.493 –0.6518 945.1 0.264 –0.6518 0.271 0.544 

180-gr HPBT Match King 0.357 –0.6818 945.1 0.271 –0.6818 0.291 0.586 

190-gr HPBT Match King 0.423 –0.6175 945.1 0.286 –0.6175 0.278 0.560 

200-gr HPBT Match King 0.418 –0.5828 945.1 0.301 –0.5828 0.277 0.556 

220-gr HPBT Match King 0.419 –0.5238 945.1 0.331 –0.5238 0.273 0.550 

240-gr HPBT Match King 0.426 –0.4630 945.1 0.361 –0.4630 0.264 0.530 

Table 4. Results from curve-fitting velocity data, 0.311-caliber projectiles. 

Results Normalized to VRef

Projectile 

CD

Exponent

(n) 

ds 0

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m) 

V0

(m/s) 

SD

(lb/in2

)

VRef ds

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m)

VRef

CD iForm

125-gr SPT 0.267 –1.1366 853.7 0.185 –1.2246 0.356 0.717 

150-gr SPT 0.451 –0.8991 853.7 0.222 –0.9508 0.332 0.668 

174-gr HPBT Match King 0.478 –0.5930 792.7 0.257 –0.6500 0.263 0.530 

180-gr SPT 0.394 –0.7260 792.7 0.266 –0.8077 0.338 0.681 
17

Table 5. Results from curve-fitting velocity data, 0.323-caliber projectiles. 

Results Normalized to VRef

Projectile 

CD

Exponent

(n) 

ds 0

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m) 

V0

(m/s) 

SD

(lb/in2

)

VRef ds

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m)

VRef

CD iForm

150-gr SPT 0.518 –0.9734 945.1 0.205 –0.9734 0.315 0.634 

175-gr SPT 0.445 –0.8611 945.1 0.240 –0.8611 0.325 0.654 

220-gr SBT 0.404 –0.6325 945.1 0.301 –0.6325 0.300 0.604 

Table 6. Results from curve-fitting velocity data, 0.338-caliber projectiles. 

Results Normalized to VRef

Projectile 

CD

Exponent

(n) 

ds 0

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m) 

V0

(m/s) 

SD

(lb/in2

)

VRef ds

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m)

VRef

CD iForm

215-gr SBT 0.430 –0.6777 945.1 0.269 –0.6777 0.287 0.577 

250-gr SBT 0.446 –0.5811 945.1 0.313 –0.5811 0.286 0.576 

300-gr HPBT Match King 0.426 –0.4288 945.1 0.375 –0.4288 0.253 0.510 

Table 7. Results from curve-fitting velocity data, 0.375-caliber projectiles. 

Results Normalized to VRef

Projectile 

CD

Exponent

(n) 

ds 0

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m) 

V0

(m/s) 

SD

(lb/in2

)

VRef ds

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m)

VRef

CD iForm

200-gr FN 0.041 –1.7963 945.1 0.203 –1.7963 0.575 1.157 

250-gr SBT 0.258 –0.9379 945.1 0.254 –0.9379 0.375 0.755 

300-gr SBT 0.413 –0.6782 914.6 0.305 –0.6913 0.332 0.668 

Table 8. Results from curve-fitting velocity data, 0.458-caliber projectiles. 

Results Normalized to VRef

Projectile 

CD

Exponent

(n) 

ds 0

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m) 

V0

(m/s) 

SD

(lb/in2

)

VRef ds

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m)

VRef

CD iForm

300-gr HP/FN –0.481 –2.1156 731.7 0.204 –3.0906 0.995 2.001 
18

In general, the data show that the exponent defining the shape of the drag curve varies between 

–0.5 and 0.6 for these small arms bullets. When comparing the geometric shape of the bullet to 

both the drag coefficient exponent and form factor, there appears to be some correlation. 

Figure 11 shows a tabular listing of the drag coefficient exponent and the form factor along with 

a picture of each round for the 27 0.308-caliber projectiles considered in the study. The tabular 

listing is shown in order of increasing drag coefficient exponent. The negative drag coefficient 

exponents all corresponded to blunt nosed (flat or rounded) projectiles with the largest negative 

values observed for the flat faced projectiles. The negative drag coefficient exponent indicates a 

decreasing drag coefficient with decreasing Mach number as opposed to the more typical 

increasing trend with decreasing Mach number. A decreasing trend of the drag coefficient with 

Mach number in the supersonic regime has been previously documented for flat nose projectiles 

(9, 10). The round nose (spherically blunted) projectiles exhibit a drag coefficient exponent 

close to 0 which is similar to the drag coefficient exponent for a spherical ball projectile 

previously discussed in Weinacht et al. (1). As the projectile becomes more streamlined, the 

drag coefficient exponent increases. The low drag boattailed projectiles with long noses have 

drag coefficient exponents approaching 0.5. Figure 11 also appears to show a correlation 

between drag coefficient exponent and form factor. Although figure 11 shows data for a single 

projectile caliber, similar observations were made for the other projectile calibers examined in 

the study. 

0.16 0.98

0.12 1.21

0.12 1.20

0.02 1.18

-0.11 1.18

-0.25 1.17

-0.25 1.40

-0.27 1.38

-0.36 1.74

I n form

0.16 0.98

0.12 1.21

0.12 1.20

0.02 1.18

-0.11 1.18

-0.25 1.17

-0.25 1.40

-0.27 1.38

-0.36 1.74

I n form

0.42 0.56

0.42 0.55

0.42 0.56

0.41 0.56

0.37 0.57

0.36 0.59

0.31 0.71

0.29 0.70

0.27 0.66

I n form

0.42 0.56

0.42 0.55

0.42 0.56

0.41 0.56

0.37 0.57

0.36 0.59

0.31 0.71

0.29 0.70

0.27 0.66

I n form

0.53 0.57

0.52 0.62

0.52 0.57

0.49 0.54

0.49 0.58

0.49 0.70

0.45 0.71

0.45 0.52

0.43 0.53

I n form

0.53 0.57

0.52 0.62

0.52 0.57

0.49 0.54

0.49 0.58

0.49 0.70

0.45 0.71

0.45 0.52

0.43 0.53

I n form

Figure 11. Drag coefficient exponent and form factor for various 0.308-projectiles; bullet photographs from 

Sierra Bullets (8). 
19

Figure 12 shows the correlation between the drag coefficient exponent and the form factor for 

each of the eight calibers of munitions examined in the current study. The results appear to be 

independent of bullet caliber. A curve fit of the data was obtained using a second-order 

polynomial. The resulting polynomial is shown in equation 13. The maximum error in the drag 

coefficient exponent n from the fit is 0.25 with a standard deviation of 0.085. 

 9841 n 0.0848i 0.932i 0. Form

2 = Form − + . (13) 

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

Drag Form Factor, iForm

Drag coeff. exponent, n

0.224

0.243

0.308

0.311

0.323

0.338

0.375

0.458

Trend Line

Figure 12. Drag coefficient exponent, n, vs. drag form factor with illustration of effect of bullet 

shape; bullet photographs from Sierra Bullets (8). 

The significance of the correlation is that the drag coefficient exponent can be estimated from the 

ballistic coefficient since the form factor is easily determined from equation 9 using the sectional 

density of the projectile. This is particularly important when a manufacturer only quotes a single 

value of the ballistic coefficient and uses the standard G1 drag function to predict trajectory 

data—a common practice within the commercial small arms ammunition industry. The results 

presented here indicate the possibility of improving the trajectory predictions using the power 

law formulation with the exponent estimated from the correlation in equation 13. This is 

particularly important when the exponent estimated using the correlation varies significantly 

from the 0.36 exponent corresponding to the G1 drag function. It is also important to understand 

that the correlation in equation 13 is best suited for small arms axisymmetric bullets, since it was 
20

derived using data for these types of bullets. It may be completely inappropriate for other classes 

of munitions such as kinetic energy projectiles. 

The sensitivity of the results obtained in the fitting process to muzzle velocity was examined by 

fitting several trajectories corresponding to different muzzle velocities for several bullet types. 

Some of the results of this study are shown in tables 9–12. The tables show the drag coefficient 

exponent and muzzle retardation obtained from the fitting process for each muzzle velocity. The 

results also show the muzzle retardation, drag coefficient, and form factor normalized to the 

reference muzzle velocity of 3100 ft/s. Table 9 shows a typical result. The drag coefficient 

exponent shows only minor variation over the range of muzzle velocities available for this round. 

The muzzle retardation from the fit shows a decreasing trend with decreasing muzzle velocity. 

However, when each of the muzzle retardations is normalized to the reference velocity using 

equation 10, the only small variations in the normalized retardation are seen as the muzzle 

velocity is varied. Both the drag coefficient and form factor when normalized to the reference 

velocity also show correspondingly little variation with muzzle velocity. 

Table 9. Sensitivity of muzzle velocity on results from trajectory fitting, 0.308 

150-gr HPBT Match King. 

Results Normalized to VRef

V0

(m/s)

CD

Exponent 

(n)

ds 0

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m) 

SD

(lb/in2

)

VRe f ds

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m)

VRef

CD iForm

1067 0.532 –0.8512 0.226 –0.8042 0.286 0.576 

1037 0.529 –0.8378 0.226 –0.8021 0.285 0.574 

1006 0.519 –0.8259 0.226 –0.8014 0.285 0.574 

976 0.534 –0.8108 0.226 –0.7989 0.284 0.572 

945 0.519 –0.8003 0.226 –0.8003 0.285 0.573 

915 0.532 –0.7875 0.226 –0.7997 0.285 0.573 

884 0.539 –0.7763 0.226 –0.8006 0.285 0.573 

854 0.534 –0.7675 0.226 –0.8048 0.286 0.576 

823 0.562 –0.7502 0.226 –0.7970 0.284 0.571 

793 0.581 –0.7327 0.226 –0.7887 0.281 0.565 

762 0.561 –0.7222 0.226 –0.7937 0.282 0.568 

732 0.524 –0.7122 0.226 –0.8046 0.286 0.576 

For some bullets, trajectory data was available for a very wide range of muzzle velocities. 

Tables 10 and 11 show two such sets of data. In these tables, the drag coefficient exponent and 

the normalized retardation are fairly consistent down to muzzle velocities of 732–762 m/s. 

Below these velocities, the drag coefficient exponent decreases to a value near zero indicating 

that the drag coefficient is relatively constant with Mach number. This behavior can be traced to 

the G1 drag function which is used in conjunction with the form factor contained in the ballistic 
21

coefficient to determine the variation in drag coefficient with Mach number for the individual 

rounds. For both of these rounds, a constant ballistic coefficient is used below 671 m/s. For the 

low muzzle velocities, this constant ballistic coefficient is used for the bulk of the trajectory. 

The drag coefficient exponent for these trajectories at the lower muzzle velocities represents the 

shape of the G1 drag curve at these velocities rather than the shape of the drag curve as modified 

through the variation in the ballistic coefficients with velocity. 

Table 10. Sensitivity of muzzle velocity on results from trajectory fitting, 

0.224 69-gr HPBT Match King. 

Results Normalized to VRef

V0

(m/s)

CD

Exponent 

(n)

ds 0

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m) 

SD

(lb/in2

)

VRe f ds

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m)

VRef

CD iForm

1128 0.271 –1.2611 0.196 –1.1085 0.343 0.690 

1098 0.279 –1.2329 0.196 –1.1070 0.343 0.689 

1067 0.282 –1.2069 0.196 –1.1061 0.342 0.689 

1037 0.283 –1.1814 0.196 –1.1057 0.342 0.688 

1006 0.289 –1.1543 0.196 –1.1041 0.342 0.687 

976 0.290 –1.1292 0.196 –1.1040 0.342 0.687 

945 0.284 –1.1054 0.196 –1.1054 0.342 0.688 

915 0.281 –1.0803 0.196 –1.1061 0.342 0.689 

884 0.271 –1.0568 0.196 –1.1094 0.343 0.691 

854 0.253 –1.0342 0.196 –1.1159 0.345 0.695 

823 0.221 –1.0154 0.196 –1.1308 0.350 0.704 

793 0.189 –0.9942 0.196 –1.1467 0.355 0.714 

762 0.208 –0.9634 0.196 –1.1424 0.354 0.711 

732 0.176 –0.9394 0.196 –1.1598 0.359 0.722 

701 0.226 –0.9012 0.196 –1.1356 0.351 0.707 

671 0.210 –0.8701 0.196 –1.1409 0.353 0.710 

640 0.108 –0.8542 0.196 –1.2090 0.374 0.753 

To demonstrate this point, the trajectories were constructed using the G1 drag curve with a 

constant ballistic coefficient of 0.225 for a 52-gr 0.224-caliber projectile. Essentially, this 

represents a 52-gr 0.224-caliber HPBT Match King projectile with a constant ballistic coefficient 

equal to the value used for the actual round at velocities above 914 m/s. The trajectories were 

then curve fit in a manner identical to the approach used to generate the results in tables 9–11. 

The results of the fits are shown in table 12. The drag coefficient exponent and normal muzzle 

retardation show relatively little variation with muzzle velocity until about 732–762 m/s. The 

values of the drag coefficient exponent are close to the value obtained by directly fitting the drag 

curve (n = 0.36). At lower velocities, the drag coefficient exponent approaches zero. For the 

results in table 12, the drag coefficient exponent represents the shape of the G1 drag curve over 

the velocity regime of the trajectory since the ballistic coefficient is constant. 
22

Table 11. Sensitivity of muzzle velocity on results from trajectory fitting, 

0.224 52-gr HPBT Match King. 

Results Normalized to VRef

V0

(m/s)

CD

Exponent 

(n)

ds 0

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m) 

SD

(lb/in2

)

VRe f ds

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m)

VRef

CD iForm

1250 0.379 –1.8095 0.148 –1.5211 0.355 0.714 

1220 0.401 –1.7681 0.148 –1.5178 0.354 0.712 

1189 0.421 –1.7297 0.148 –1.5144 0.353 0.711 

1159 0.440 –1.6920 0.148 –1.5097 0.352 0.708 

1128 0.454 –1.6578 0.148 –1.5051 0.351 0.706 

1098 0.462 –1.6262 0.148 –1.5006 0.350 0.704 

1067 0.468 –1.5961 0.148 –1.4962 0.349 0.702 

1037 0.464 –1.5713 0.148 –1.4953 0.349 0.702 

1006 0.457 –1.5471 0.148 –1.4954 0.349 0.702 

976 0.438 –1.5283 0.148 –1.5013 0.350 0.704 

945 0.410 –1.5131 0.148 –1.5131 0.353 0.710 

915 0.374 –1.5005 0.148 –1.5316 0.357 0.719 

884 0.443 –1.4520 0.148 –1.5070 0.351 0.707 

854 0.415 –1.4321 0.148 –1.5199 0.354 0.713 

823 0.374 –1.4157 0.148 –1.5436 0.360 0.724 

793 0.321 –1.4004 0.148 –1.5782 0.368 0.741 

762 0.406 –1.3521 0.148 –1.5365 0.358 0.721 

732 0.343 –1.3367 0.148 –1.5813 0.369 0.742 

701 0.255 –1.3258 0.148 –1.6561 0.386 0.777 

671 0.140 –1.3167 0.148 –1.7684 0.412 0.830 

640 0.148 –1.2718 0.148 –1.7722 0.413 0.832 

610 0.024 –1.2465 0.148 –1.9116 0.446 0.897 

579 0.005 –1.1979 0.148 –1.9498 0.455 0.915 

549 –0.011 –1.1445 0.148 –1.9831 0.463 0.931 

It is difficult to determine from the existing Sierra Bullet data whether the drag coefficient at 

lower velocities is well represented by the G1 drag curve. Drag results for the M855 obtained 

from an aeroballistic range (shown previously in figure 1) show a much stronger variation with 

Mach number than does the G1 drag curve. Not withstanding, the results presented here show 

that the drag coefficient exponent is relatively constant over a range of velocities and projectile 

geometries that are of interest for current study and validate the approach of characterizing the 

shape of the drag curve using a power law approach. 

The approach taken here is to fit the velocity vs. range data to obtain the muzzle retardation and 

drag coefficient exponent. It may also be possible to perform fits of the drag curve directly using 

the form factors obtained from the ballistic coefficients and the reference drag curve. This 

approach yields similar results to the results presented here. However, the results show some 
23

Table 12. Sensitivity of muzzle velocity on results from trajectory fitting, 

0.224 52-gr, G1 drag curve, constant ballistic coefficient C = 0.225. 

Results Normalized to VRef

V0

(m/s)

CD

Exponent 

(n)

ds 0

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m) 

SD

(lb/in2

)

VRe f ds

dV ⎟

⎠

⎞ ⎜

⎝

⎛

([m/s]/m)

VRef

CD iForm

1250 0.289 –1.8276 0.148 –1.4980 0.349 0.703 

1220 0.307 –1.7852 0.148 –1.4962 0.349 0.702 

1189 0.325 –1.7441 0.148 –1.4936 0.348 0.701 

1159 0.341 –1.7044 0.148 –1.4903 0.348 0.699 

1128 0.355 –1.6662 0.148 –1.4865 0.347 0.698 

1098 0.367 –1.6297 0.148 –1.4824 0.346 0.696 

1067 0.375 –1.5949 0.148 –1.4785 0.345 0.694 

1037 0.380 –1.5622 0.148 –1.4753 0.344 0.692 

1006 0.379 –1.5318 0.148 –1.4735 0.344 0.691 

976 0.373 –1.5036 0.148 –1.4740 0.344 0.692 

945 0.361 –1.4775 0.148 –1.4775 0.345 0.693 

915 0.341 –1.4535 0.148 –1.4852 0.346 0.697 

884 0.312 –1.4317 0.148 –1.4990 0.350 0.703 

854 0.367 –1.3870 0.148 –1.4793 0.345 0.694 

823 0.337 –1.3646 0.148 –1.4955 0.349 0.702 

793 0.296 –1.3439 0.148 –1.5211 0.355 0.714 

762 0.241 –1.3248 0.148 –1.5598 0.364 0.732 

732 0.318 –1.2748 0.148 –1.5180 0.354 0.712 

701 0.260 –1.2520 0.148 –1.5614 0.364 0.733 

671 0.183 –1.2299 0.148 –1.6276 0.380 0.764 

640 0.179 –1.1910 0.148 –1.6398 0.382 0.769 

610 0.072 –1.1648 0.148 –1.7494 0.408 0.821 

579 0.056 –1.1199 0.148 –1.7775 0.415 0.834 

549 –0.102 –1.0868 0.148 –1.9787 0.461 0.929 

slight sensitivity to the velocity or Mach number range used in fitting the drag curve. By fitting 

the velocity data over the values of range quoted by the manufacturer, a more representative 

velocity range of the bullet is considered. 

As previously discussed, the analytical predictions of the velocity, gravity drop, and wind drift 

using the muzzle retardation and drag coefficient exponent extracted from the fitting process 

were in excellent agreement with Sierra Bullets’s trajectory data. It is also possible to plot all of 

the manufacturer’s data on a family of universal curves using the nondimensionalization that 

results from the analytical method of Weinacht et al. (1). For instance, the analytical method 

shows that the ratio of the impact velocity to muzzle velocity is a function of the nondimensional 

range coordinate s as shown in equations 14 and 15. 
24

1 ns n 0

V

s

ds

dV 1 n

V

V n

1

n

1

0

x

0 0

≠

⎭

⎬

⎫

+

⎩

⎨

⎧ =

⎭

⎬

⎫

⎩

⎨

⎧ ⎟

⎠

⎞ ⎜

⎝

⎛ = + . (14) 

0

x

0 V

s

ds

dV

s ⎟

⎠

⎞ ⎜

⎝

⎛ = . (15) 

Similarly, the gravity drop and the wind drift can be expressed in nondimensional terms as 

functions of the nondimensional range coordinate as shown in equations 16 and 17. The 

formulae for the special cases n = 0,1,2 shown in Weinacht et al. (1) also produce the same 

nondimensionalization as equations 14, 16, and 17, but the functional form is slightly different. 

1 ns 1 2(n 1)s n 0,1,2

2(n 2)(n 1)

1

g

ds

dV

s n

2(n 1)

2

0

g drop

≠

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

− − −

⎭

⎬

⎫

⎩

⎨

⎧

+ − −

− =

⎟

⎠

⎞ ⎜

⎝

⎛ −

−

. (16) 

1 ns 1 n 0,n 1

(n 1)

1

w

ds

dV

s n

1 1

z

0

z

≠ ≠

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

⎭

⎬

⎫

⎩

⎨

⎧

+ − =

⎟

⎠

⎞ ⎜

⎝

⎛ −

. (17) 

When the velocity, gravity drop, and wind drift are nondimensionalized, as previously shown, 

and plotted vs. the nondimensional range, the result is a family of curves in terms of a single 

parameter, the drag coefficient exponent n. Using this approach, the velocity ratio, gravity drop 

and wind-drift data available from the manufacturer for eight different calibers and a variety of 

different projectile masses and shapes can be collapsed on individual plots for each of these three 

variables. Figures 13 and 14 show the velocity ratio as a function of nondimensional range. 

Figure 13 presents the complete range of data, while figure 14 presents a close-up view at shorter 

ranges. Similar plots of the results for gravity drop and wind drift are shown in figures 15–18. 

This data provides additional validation of the approach. Most of the data is contained between 

the n = 0 and n = 0.5 curves from the analytical theory. This is consistent with the results from 

the fitting process. The analytical curves also offer some estimate of the error that results from 

estimating the drag coefficient exponent or from directly assuming that the shape of the drag 

curve is represented by the G1 drag function (n = 0.36 from the fitting). At close ranges, the 

results are relatively insensitive to the drag coefficient exponent. The error increases slightly at 

longer ranges. 
25

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Scaled Range

V/V0

0.224

0.243

0.308

0.311

0.323

0.338

0.375

0.458

n=1

n=0.5

n=0

Figure 13. Scaled impact velocity as a function of scaled range. 

0.7

0.75

0.8

0.85

0.9

0.95

1

0 0.05 0.1 0.15 0.2 0.25 0.3

Scaled Range

V/V0

0.224

0.243

0.308

0.311

0.323

0.338

0.375

0.458

n=1

n=0.5

n=0

Figure 14. Scaled impact velocity as a function of scaled range (close-in). 
26

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Scaled Range

Scaled Wind Drift

0.224

0.243

0.308

0.311

0.323

0.338

0.375

0.458

n=1

n=0.5

n=0

Figure 15. Scaled cross-wind drift as a function of scaled range. 

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.05 0.1 0.15 0.2 0.25 0.3

Scaled Range

Scaled Wind Drift

0.224

0.243

0.308

0.311

0.323

0.338

0.375

0.458

n=1

n=0.5

n=0

Figure 16. Scaled cross-wind drift as a function of scaled range (close-in). 
27

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 0.2 0.4 0.6 0.8 1

Scaled Range

Scaled Gravity Drop

0.224

0.243

0.308

0.311

0.323

0.338

0.375

0.458

n=1

n=0.5

n=0

Figure 17. Scaled gravity drop as a function of scaled range. 

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0 0.1 0.2 0.3

Scaled Range

Scaled Gravity Drop

0.224

0.243

0.308

0.311

0.323

0.338

0.375

0.458

n=1

n=0.5

n=0

Figure 18. Scaled gravity drop as a function of scaled range (close-up). 
28

4. Trajectory Mismatch 

One approach for defining the requirements for new ammunition or modified ammunition based 

on a previous design is to specify that the trajectories must be similar to existing ammunition. 

This allows the new ammunition to be used by the Soldier with little apparent difference in its 

aeroballistic performance. It may also allow compatibility between the new ammunition and the 

existing tracer round. 

For small-caliber ammunition, this requirement is often cited as ±1.0 gunner’s mil mismatch in 

impact location for ranges up to 600 m. It is implied that the gun elevation used to meet this 

requirement is the same for both the modified and baseline ammunition. From a practical 

standpoint, this essentially assumes that the existing sights and sight adjustments for the baseline 

ammunition are used for the modified ammunition, or alternatively, both baseline and modified 

ammunition use the same firing tables. The second part of this requirement can be shown to be 

equivalent to requiring that the gravity drop of baseline and modified ammunition to be within 

±1.0 gunner’s mil mismatch in impact location for range up to 600 m. Equation 6 shows the 

vertical displacement in the trajectory as a function of range. This expression is valid for both 

baseline and modified ammunition. By assuming that the gun elevation Өo is equivalent for 

both, the difference in the vertical impact location can be seen to be produced by the differences 

in the gravity drop alone. 

As previously shown in equation 5, the gravity drop, like the other trajectory characteristics, is a 

function of the muzzle velocity, muzzle retardation and, to a lesser extent, a function of the 

exponent defining the shape of the drag curve. Assuming the shape of the drag curve is the same 

for a similar class of ammunition, the gravity drop is then a function of only the muzzle velocity 

and muzzle retardation. For the results presented here, the exponent defining the shape of the 

drag curve is assumed to be 0.53, the same as the M855. 

Equation 5 provides some very important insights into the trajectory mismatch problem. First, 

for each muzzle velocity there exists a value of the muzzle retardation for the modified 

ammunition that produces the identical impact at 600 m as the M855. For each muzzle velocity 

there are also corresponding values of muzzle retardation that produce impacts 1.0 mils above 

and below the baseline M855 impact at 600 m. The value of the muzzle retardation required to 

obtain a specified value of mismatch at range can be obtained by solving equation 5 for the 

muzzle retardation. Unfortunately, because of the complicated nature of equation 5, closed form 

solutions for the muzzle retardation cannot be obtained and the muzzle retardation must be 

determined in an iterative manner. 

Figure 19 shows the gravity drop as a function of range for the baseline M855 and projectiles 

with a ±1.0 mil mismatch at 600 m for a constant muzzle velocity equal to the baseline M855 
29

muzzle velocity. It can be seen that all three trajectories show little deviation from each other at 

shorter ranges. This deviation increases with range with the maximum deviation occurring at 

600 m. For the ±1.0 mil mismatch trajectories, the muzzle retardation decreases by 27% for 

impacts above the baseline impact and increases by 20% for impacts below the baseline impact. 

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 50 100 150 200 250 300 350 400 450 500 550 600

Range - Meters

Gravity Drop - Meters

M855 Baseline

-1 mil Mismatch

+1 mil Mismatch

Figure 19. Gravity drop for M855 and ±1.0 mil mismatch trajectories for 949 m/s muzzle 

velocity. 

Figure 20 shows the gravity drop in mils for the baseline M855 and projectiles with a ±1.0 mil 

mismatch at 600 m for a constant muzzle velocity. Considering the differences between the 

baseline and mismatch trajectories, the greatest mismatch occurs at maximum range (in this case, 

600 m). Thus, a projectile that meets the ±1 mil mismatch requirement at 600 m also meets this 

requirement at all intermediate ranges. 

Additional trajectories that produce ±1.0 mil mismatches at 600 m can be obtained by varying 

the muzzle velocity. As the muzzle velocity increases, the corresponding values of the muzzle 

retardation increase as well. Figure 21 shows the trajectories for –1.0 mil mismatch for three 

different values of muzzle velocity compared with the baseline M855 trajectory. Although the 

three –1.0 mil mismatch trajectories are relatively similar, as the muzzle velocity increases, the 

trajectory becomes flatter at shorter ranges, but shows much larger gravity drop at longer ranges. 

The range of values of muzzle velocity and muzzle retardation that produce a fixed vertical 

impact mismatch at 600 m compared with the baseline M855 ammunition can be determined 
30

-8

-7

-6

-5

-4

-3

-2

-1

0

0 50 100 150 200 250 300 350 400 450 500 550 600

Range - Meters

Gravity Drop - Mils

M855 Baseline

-1 mil Mismatch

+1 mil Mismatch

Figure 20. Gravity drop in mils for M855 and ±1.0 mil mismatch trajectories for 949 m/s 

muzzle velocity. 

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 50 100 150 200 250 300 350 400 450 500 550 600

Range - Meters

Gravity Drop - Meters

M855 Baseline

850m/s Muzzle Velocity

949m/s Muzzle Velocity

1050m/s Muzzle Velocity

Figure 21. Gravity drop for M855 and –1.0 mil mismatch trajectories for different muzzle 

velocities. 
31

from equation 5 and are represented as a contour plot shown in figure 22. Here, the lines of 

constant vertical impact mismatch are plotted as functions of muzzle retardation and muzzle 

velocity. Any combination of muzzle velocity and muzzle retardation contained between the 

±1.0 mil mismatch lines will produce a vertical impact at 600 m that meets the ±1.0 mil 

requirement. Also shown are the values of muzzle velocity and muzzle retardation that produce 

the same impact location at 600 m as the baseline M855. 

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

850 875 900 925 950 975 1000 1025 1050

Muzzle Velocity - m/s

Retardation - (m/s)/m

-1 mil Mismatch at 600m

+1 mil Mismatch at 600m

0 mil Mismatch at 600m

M855

-1 mil Mismatch

+1 mil Mismatch

Figure 22. Acceptable range of muzzle velocity and muzzle retardation to attain ±1.0 mil 

mismatch. 

The sensitivity of these bounds to the value of the drag coefficient exponent is shown in figure 

23. Here, the bounds on muzzle velocity and muzzle retardation to meet the ±1.0 mil 

requirement are shown for the baseline drag coefficient exponent of 0.53 and a drag coefficient 

exponent of 0.25. These two drag coefficient exponents represents a rather wide variation for 

this parameter, especially for the streamlined low drag shapes typical of tactical small arms 

bullets which are more typically characterized by values of the drag coefficient exponent closer 

to the baseline value of 0.53. For the –1.0 mil mismatch results, the difference in the predicted 

retardation for a given muzzle velocity is less than 8% between the drag coefficient exponents of 

0.25 and 0.53. Somewhat larger differences are observed for the +1.0 mil mismatch trajectory 

results at lower muzzle velocities. However, these low values of muzzle retardation necessary to 

achieve a +1.0 mil vertical impact above the M855 may be very difficult to obtain in practice. 
32

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

850 875 900 925 950 975 1000 1025 1050

Muzzle Velocity - m/s

Retardation - (m/s)/m

M855

-1 mil Mismatch, n=0.53

+1 mil Mismatch, n=0.53

-1 mil Mismatch, n=0.25

+1 mil Mismatch, n=0.25

Figure 23. Sensitivity of the acceptable range of muzzle velocity and muzzle retardation for 

±1.0 mil mismatch to drag coefficient exponent. 

The trajectory mismatch requirement can be examined in light of current ammunition. Muzzle 

velocities, muzzle retardations, and vertical impact locations were obtained from the firing tables 

(11) for the M855, M856, M193, and M196 fired from the M16A2 and M4 rifles. The M856 and 

M196 are tracer rounds for the M855 and M193, respectively. Difference in the impact locations 

and muzzle retardations for identical rounds fired from the M16A2 and M4 rifles are produced 

by the differences in muzzle velocity. Figure 24 displays the mismatch for each of the four 

projectiles and two rifle combinations as a function of muzzle velocity and muzzle retardation on 

mismatch contours similar to those shown in figure 22. It can be seen that the companion tracer 

rounds are well matched to the tactical round for each of the round/rifle combinations with little 

difference in the relative impact locations noted. This occurs despite a rather large variation in 

muzzle velocity produced by firing the same round from a different rifle. 

In addition to trajectory mismatch, the aeroballistician must also consider the wind sensitivity of 

the projectile. In general, firing a projectile with a higher muzzle velocity and lower (less 

negative) muzzle retardation will produce less wind drift. However, the previous results 

demonstrate that for a constant value of trajectory mismatch, as the muzzle velocity increases, 

the muzzle retardation increases (becomes more negative). Equation 7 can be applied to 

determine the relative wind drift as a function of muzzle velocity for constant values of trajectory 

mismatch. 
33

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

850 875 900 925 950 975 1000 1025 1050

Muzzle Velocity - m/s

Retardation - (m/s)/m

M855/M16A2

M856/M16A2

M855/M4

M856/M4

M193/M16A2

M196/M16A2

M193/M4

M196/M4

-1 Mil Mismatch at 600m

+1 Mil Mismatch at 600m

-0.5 mil Mismatch at 600m

+0.5 mil Mismatch at 600m

0 mil Mismatch at 600m

Figure 24. Muzzle velocity, muzzle retardation, and trajectory mismatch for M855 and M193 and their 

companion tracer rounds, M856 and M196, respectively. 

Figures 25–27 show the predicted wind drift at 200, 300, and 600 m, respectively, relative to the 

M855 for constant values of the trajectory mismatch at 600 m. The results shown in figures 

25–27 demonstrate that, for constant trajectory mismatch, the wind drift increases with 

increasing muzzle velocity. These trends are consistent across each of the ranges examined 

although the relative increase is larger at longer ranges. Although decreased wind sensitivity 

should be expected with increasing muzzle velocity, for constant mismatch, the muzzle 

retardation also increases (becomes more negative) with increasing muzzle velocity. The 

combined effects of increasing muzzle velocity and muzzle retardation result in increased wind 

sensitivity. The results also show that the target mismatch varies from –1.0 to +1.0 mils, the 

wind drift decreases. This trend is expected because, for a constant muzzle velocity, the muzzle 

retardation decreases (becomes less negative) as the mismatch varies from –1.0 to +1.0 mils, 

resulting in less wind sensitivity. 

Results are also shown in figures 25–27 for the M855, M856, M193, and M196 fired from the 

M16A2 and M4 rifles. In each case, the M855 and M856 fired from the M16A2 produces the 

least wind sensitivity. A slight increase in wind sensitivity in the M855 and M856 is noted when 

fired from the M4 rifle as a result of the lower muzzle velocity. The M193 and M196 produce 

slightly more wind drift than the M855 and M856 when fired from the same rifle. 
34

-80

-60

-40

-20

0

20

40

60

80

850 875 900 925 950 975 1000 1025 1050

Muzzle Velocity - m/s

Additional Wind Drift - % 

M855/M16A2

M856/M16A2

M855/M4

M856/M4

M193/M16A2

M196/M16A2

M193/M4

M196/M4

-1 Mil Mismatch at 600m

+1 Mil Mismatch at 600m

-0.5 mil Mismatch at 600m

+0.5 mil Mismatch at 600m

0 mil Mismatch at 600m

Figure 25. Percent increase or decrease in wind drift at 200 m relative to the M855 as a function of muzzle 

velocity for constant values trajectory mismatch at 600 m. 

-80

-60

-40

-20

0

20

40

60

80

850 875 900 925 950 975 1000 1025 1050

Muzzle Velocity - m/s

Additional Wind Drift - %

M855/M16A2

M856/M16A2

M855/M4

M856/M4

M193/M16A2

M196/M16A2

M193/M4

M196/M4

-1 Mil Mismatch at 600m

+1 Mil Mismatch at 600m

-0.5 mil Mismatch at 600m

+0.5 mil Mismatch at 600m

0 mil Mismatch at 600m

Figure 26. Percent increase or decrease in wind drift at 300 m relative to the M855 as a function of muzzle 

velocity for constant values trajectory mismatch at 600 m. 
35

-80

-60

-40

-20

0

20

40

60

80

850 875 900 925 950 975 1000 1025 1050

Muzzle Velocity - m/s

Additional Wind Drift - %

M855/M16A2

M856/M16A2

M855/M4

M856/M4

M193/M16A2

M196/M16A2

M193/M4

M196/M4

-1 Mil Mismatch at 600m

+1 Mil Mismatch at 600m

-0.5 mil Mismatch at 600m

+0.5 mil Mismatch at 600m

0 mil Mismatch at 600m

Figure 27. Percent increase or decrease in wind drift at 600 m relative to the M855 as a function of muzzle 

velocity for constant values trajectory mismatch at 600 m. 

Because small arms rifles are zeroed at some specified range, it would be natural to question the 

necessity of the second part of the mismatch requirement that assumes the gun elevation is fixed 

for baseline and modified ammunition. The Rifle Marksmanship Field Manual (FM) (12) for the 

M16/M4 series rifle discusses various approaches for zeroing the rifle. Zeroing is typically 

performed at 25-m range. With proper marksmanship fundamentals, 25-m zeroing will allow the 

Soldier to hit all targets out to 300 m using the existing ammunition. Known distance (KD) 

zeroing at 300 m can also be performed to confirm or refine the 25-m zero. As stated in the FM, 

the zero for the 300-m target is more valid than the zero obtained on the 25-m range when the 

wind is properly compensated for. It would seem that the zeroing process at a known distance 

could eliminate some of the mismatch between the baseline and modified ammunition or allow 

greater flexibility in the selection of the muzzle velocity and muzzle retardation for the modified 

ammunition while still meeting the ±1.0 mil mismatch at 600 m. In other words, allowing 

zeroing at a known distance might be less restrictive than assuming that the gun elevation is 

fixed for the baseline and modified ammunition. 

An analysis was performed allowing zeroing of the weapon at various intermediate ranges 

between 25 and 300 m so that both the baseline and modified ammunition had the same vertical 

impact location at the known distance downrange used for zeroing while allowing a 

±1.0 mil mismatch for the modified ammunition at 600 m. For the baseline M855, the gun 
36

elevation angle required to produce a vertical impact along a horizontal line-of-sight at the 

known distance range was computed using equations 5 and 6. Using this gun elevation angle, 

the vertical impact location was also computed at 600-m range. For the modified ammunition, 

the muzzle retardation required to produce a trajectory that satisfied the following two 

constraints was obtained for fixed values of the muzzle velocity. This trajectory had the identical 

vertical impact location as the M855 at the KD range used for zeroing and a ±1.0 mil mismatch 

at 600 m relative to the M855 impact. Similar to the trajectory mismatch results previously 

presented, an iterative procedure was used to determine the retardation of the modified 

ammunition required to meet these constraints. At each iteration step, the gravity drop and the 

gun elevation angle for the modified ammunition were computed using equations 5 and 6 to 

produce the identical vertical impact location at the KD range used for zeroing as for the baseline 

M855. Based on this gun elevation angle, the vertical impact locations were determined at 

600-m range using equations 5 and 6. The retardation of the modified ammunition was adjusted 

until the proper vertical impact locations were obtained at both the KD range used for zeroing 

and at 600 m. 

Figure 28 shows sample trajectories with KD zeroing for the M855 and the modified ammunition 

at a muzzle velocity of 949 m/s. For the sample trajectories shown in figure 28, the M855 and 

the modified ammunition have been rezeroed at either 200 or 300 m to produce the same vertical 

impact at the respective KD ranges. For each set of trajectories rezeroed at the same KD range, 

the impact locations at 600 m for the M855 and the modified ammunition differ by –1.0 mils. 

Figure 29 compares the gravity drop of the modified ammunition for rezeroing at 100, 200, and 

300 m, with the gravity drop obtained when assuming both the M855 and the modified 

ammunition are fired with the same gun elevation angle. Also shown is the gravity drop for the 

baseline M855. There is very little difference in the gravity drop when rezeroing is allowed at 

ranges less than 300 m, compared to the results obtained by assuming the M855 and the 

modified ammunition are fired with the same gun elevation angle. These differences in gravity 

drop are due to the differences in the predicted muzzle retardation obtained using the procedure 

previously outlined. These results indicate that rezeroing at a known distance offers little 

difference in the modified ammunition performance compared with using the same gun elevation 

for both the modified and baseline ammunition. 

Comparison of the values of the muzzle retardation obtained using rezeroing at 300 m, with the 

muzzle retardation values obtained assuming a fixed gun elevation angle, show less than a 2% 

difference for a muzzle velocity of 949 m/s. Similar results were obtained for muzzle velocities 

between 949 and 1050 m/s. Velocities below 949 m/s showed somewhat more sensitivity to 

rezeroing with differences in the predicted muzzle retardation of about 10% obtained for a 

muzzle velocity of 850 m/s. The differences in the predicted muzzle retardation decreased as the 

KD for rezeroing was decreased. The results indicate that the performance envelope of 
37

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 50 100 150 200 250 300 350 400 450 500 550 600

Range - Meters

Height - Meters

M855 Re-zero at 300m

-1mil Mismatch - Re-zero at 300m

M855 Re-zero at 200m

-1mill Mismatch - Re-zero at 200m

Figure 28. Trajectories for M855 and modified ammunition with –1.0 mil mismatch 

at 600 m with rezeroing at 200 and 300 m. 

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 50 100 150 200 250 300 350 400 450 500 550 600

Range - Meters

Height - Meters

M855 Baseline

Re-zero at 100m

Re-zero at 200m

Re-zero at 300m

Same Gun Elevation as M855

Figure 29. Gravity drop for M855 and modified ammunition with –1.0 mil mismatch 

at 600 m when fired at same gun elevation as M855 or zeroed at 100, 200, 

and 300 m. 
38

allowable retardations (such as that shown in figure 22) for trajectory mismatches of ±1.0 mils is 

not significantly changed by allowing rezeroing at a KD. The results also show that the 

differences in the gun elevations between the M855 and modified ammunition are generally 

quite small when rezeroing is employed. For example, for the baseline muzzle velocity of 

949 m/s, the largest differences in gun elevation occurs for the 300-m rezeroing case, but is less 

than 0.2 mils. 

The results obtained for zeroing the rifle at 25 m were nearly identical to the results obtained by 

assuming the baseline M855 ammunition and the modified ammunition were fired at the same 

gun elevation angle. The differences in the gun elevation angle between the baseline M855 and 

the modified ammunition required to zero the rifle at 25 m were extremely small. Essentially, 

zeroing at 25 m is equivalent to assuming a constant gun elevation angle for the baseline and 

modified ammunition. The predicted muzzle retardation for the modified ammunition obtained 

by allowing zeroing at 25 m was within 1% of the results obtained assuming the modified 

ammunition had the same gun elevation as the baseline M855 ammunition for muzzle velocities 

between 850 and 1050 m/s. 

Clearly, the additional effort of considering rezeroing at ranges less than 300 m does not result in 

significant difference in the performance of the modified ammunition compared with the results 

obtained assuming both the M855 and the modified ammunition are fired at the same gun 

elevation angles. Thus, the second part of the ±1.0 mil mismatch requirement which assumes the 

same gun elevation for both the baseline ammunition and the modified ammunition simplifies 

the analysis process without adding any significant constraint to the modified ammunition 

design. Rezeroing at ranges beyond 300 m for the purpose of removing differences in the 

gravity drop between the baseline and modified ammunition appears to have more of an effect, 

although there is some question whether rezeroing at these ranges could be accomplished in 

practice. 

It is important to note that the previous comments regarding rezeroing only apply to the 

requirements for trajectory mismatch. Zeroing the rifle remains an important approach for 

correcting other sources of inaccuracy produced by launch disturbances which are not considered 

here. It appears then that the requirement for 1.0 mil mismatch in impact location for ranges up 

to 600 m with the additional constraint of fixed gun elevation provides a meaningful but simple 

requirement for trajectory mismatch. It is relatively easy to implement both in practice and in 

engineering analysis and provides nearly the same performance envelope for the modified 

ammunition as requirements which involve rezeroing. 
39

5. Combined Exterior and Interior Ballistics Results 

The previous results show the possible bounds on muzzle velocity and muzzle retardation to 

meet the ±1 mil trajectory mismatch requirement. These results show that the best performance 

(in terms of reduced wind sensitivity and gravity drop and increased impact velocity and energy) 

is obtained by firing bullets with low retardation and high muzzle velocity. These results treat 

the muzzle velocity as an independent variable, but, in fact, the muzzle velocity itself is 

constrained by the interior ballistic constraints and considerations. In particular, for the M16 

rifle, there are limitations on chamber pressure and charge mass. For example, the M855 attains 

nearly the maximum allowable chamber pressure during the interior ballistic cycle. If heavier 

variants of the M855 are fired with the existing charge, the maximum allowable chamber 

pressure would be exceeded resulting in possible catastrophic failure of the gun tube. The result 

is that lower charge weights must be utilized for heavier bullets so that the maximum allowable 

chamber pressure is not exceeded. This typically produces lower muzzle velocity as the 

projectile mass is increased. Additionally, for the M855, the volume available for propellant is 

nearly completely occupied by the current propellant load and it is difficult to add additional 

propellant to increase the current muzzle velocity of the round. This is particularly a problem for 

lighter bullets that can be fired with increased charge masses without exceeding the maximum 

chamber pressure. For these lighter variants of the M855, the maximum charge volume becomes 

the constraint limiting muzzle velocity rather than the maximum allowable chamber pressure. 

Efforts are underway to develop new propellants that will provide increased performance within 

the 5.56 constraints. 

To demonstrate the effect that these constraints have on muzzle velocity and, subsequently, on 

the exterior ballistic performance, interior ballistics computations using the IBHVG2 interior 

ballistics code (13) were performed to obtain predictions of muzzle velocity for different 

projectile masses using the existing WC844 propellant. Two separate series of computations 

were run. The first series of computations constrained the chamber pressure to the maximum 

allowable chamber pressure while allowing the charge mass to vary. Predictions of the resulting 

muzzle velocity were obtained for each projectile mass. A second series of computations 

constrained the charge mass to the existing M855 charge mass while allowing the chamber 

pressure to vary. Again, predictions of the resulting muzzle velocity were obtained for each 

projectile mass. For both series of computations, the change in chamber volume due to projectile 

intrusion resulting from the varying projectile mass was taken into account by assuming that the 

change in projectile mass relative to the M855 was produced by a right circular cylinder with the 

density of lead. Validation of the model for the baseline M855 was obtained by comparing the 

predicted case mouth pressure with experimental measurements as shown in figure 30. The 

predictions are in excellent agreement with the experimental measurements. Also shown is the 

predicted breech pressure which represents one of the constraints on the M855 performance. 
40

-10000

0

10000

20000

30000

40000

50000

60000

0 250 500 750 1000 1250 1500 1750 2000

Pressure (psi)

Predicted Case Mouth Pressure

Case Mouth Pressure - Shot 1

Case Mouth Pressure - Shot 5

Case Mouth Pressure - Shot 10

Case Mouth Pressure - Shot 15

Predicted Breech Pressure

Time (μs)

Figure 30. Predicted breech pressure and case mouth pressure (Gage 0.32 in) compared with 

measured case mouth pressure for standard M855 cartridge. 

Figure 31 shows the predicted muzzle velocities as a function of projectile mass for both series 

of computations. As a benchmark for the results, muzzle velocity measurements for the 56-gr 

M193, the 62-gr M855, and the 77-gr Mk262 are plotted on the same graph. Additional 

experimental results are presented from a recent firing of 5.56-mm prototypes. All of the rounds 

from this set of firings were lighter or weighted the same as the existing M855 and were fired 

using the M855 charge mass. In general, the experimental results confirm the predictions for 

muzzle velocity. 

Obviously, an acceptable charge is one that meets both constraints, so only muzzle velocities that 

are below both curves are acceptable. For the heavier projectiles, the muzzle velocity decreases 

with increasing projectile mass. For lighter projectiles, the muzzle velocity is relatively constant 

showing only a slight increase with decreasing projectile mass. It is interesting to note that the 

62-gr projectile is close to the intersection of both performance constraints, meaning that the 

62-gr projectile has nearly the maximum charge mass and attains a maximum chamber pressure 

that is close to the maximum allowable. 

The interior ballistic performance yields the muzzle velocity as a function of projectile mass. 

The data shown in figure 31 was curve fit using a second-order polynomial to obtain a 

representation of the muzzle velocity vs. charge mass. These fits are shown in equations 18 and 

19 where the velocity is in m/s and the projectile mass is in gr. 
41

0

200

400

600

800

1000

1200

1400

30 40 50 60 70 80 90 100

Projectile Mass (Grains)

Muzzle Velocity - (m/s)

Fixed Chamber Pressure

Fixed Charge Weight

ARDEC Measurements, M855, Mk262

Recent Prototype Firings

Recent ARL Measurements, M855, Mk262, M193

Figure 31. Muzzle velocity vs. charge mass for M855 cartridge and propellant. 

V (m/s) 0.0191m 0.6793m 1009.5 m 61.3gr. 2

0 = − + + < (18) 

 . V (m/s) 0.0026m 6.1077m 1344.2 m 61.3gr 2

0 = − + > (19) 

These results can be used to predict exterior ballistic performance as a function of projectile 

mass. As previously discussed, the velocity history, gravity drop, and cross-wind drift can be 

obtained if the muzzle velocity, muzzle retardation, and exponent defining the shape of the drag 

curve are known. With the muzzle velocity known for each projectile mass, the muzzle 

retardation can be predicted using equation 8. For the prediction shown here, the standard 

atmospheric density of 1.22 kg/m3

 was used. The reference area is easily obtained as shown in 

equation 20 using a reference diameter of 5.56 mm. 

2

ref D

4

S π= . (20) 

The drag coefficient was computed as shown in equation 21. 

VRef

Form DG1

n

0

Ref

V0 D i C

V

V C ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = . (21) 
42

The last two terms of equation 21 represent the drag coefficient as the product of the form factor 

i and the drag coefficient of the standard G1 dr Form ag curve at a reference velocity of 3100 m/s. 

The G1 drag function is a standard drag function used within the commercial small arms 

community. These last two terms represent the drag coefficient of a given bullet geometry at the 

reference velocity of 3100 m/s. The form factor represents the ratio of the actual drag coefficient 

of a projectile to reference drag coefficient. (This is the same form factor that appears in the 

ballistic coefficient quoted for many commercial bullets.) In the current results, the form factor 

was treated as a parameter with results obtained for form factors of 0.6 and 0.8. As previously 

seen, the form factor of 0.6 represents the form factor for a low drag slender boattailed projectile 

with a long ogive. The form factor of 0.8 represents the form factor of a less streamlined 

projectile. A sampling of projectile shapes having form factors of 0.6 and 0.8 is shown in figure 

32. The first term of equation 21 represents the variation of the drag coefficient with velocity (or 

Mach number) for a given bullet geometry. As previously shown, the exponent defining the 

shape of the drag curve can be estimated using the form factor using equation 13. 

Figure 32. Typical projectile shapes for projectiles with form factors of 0.6 and 0.8; bullet photographs from 

Sierra Bullets (8). 

Using the predicted muzzle velocity (equations 18 or 19), the muzzle retardation was obtained 

for each projectile mass using equation 8. Figure 33 shows the muzzle retardation as a function 

of projectile mass for form factors of 0.6 and 0.8 using the predicted dependence of the muzzle 

velocity on projectile mass. As seen in equation 8, the muzzle retardation varies explicitly with 

the inverse of the projectile mass. Because of the dependence of muzzle velocity on projectile 

mass, the muzzle retardation also has an additional implicit dependence on projectile mass 

through the muzzle velocity (equations 18 or 19) and drag coefficient (equation 21). The net 

effect is a decreasing trend of the muzzle retardation with projectile mass. The results for the 

form factor of 0.8 represent an increase in retardation of ~33% over the results for the form 

factor of 0.6. (Generally, the increase in muzzle retardation reflects the proportional increase in 

the factor factor, although this is a relatively minor effect due to small changes in the drag 

coefficient exponent due to form factor.) The results for the M193, M855, and Mk262 are 

reasonably well represented by the results for the form factor of 0.6. 

With the muzzle velocity, muzzle retardation, and exponent characterizing the shape of the drag 

curve defined for each projectile mass, the velocity history, gravity drop, and cross-wind drift 

were predicted. Baseline results are presented for a form factor of 0.6 representing the expected 

Bullets with Iform =0.6 Bullets with Iform =0.8 
43

-3

-2.5

-2

-1.5

-1

-0.5

0

20 30 40 50 60 70 80 90 100 110

Projectile Wt. (grains)

Muzzle Retardation (m/s)/m

Form Factor = 0.6

Form Factor = 0.8

M855

Mk262

M193

Figure 33. Muzzle retardation vs. projectile mass for form factors of 0.6 and 0.8 compared with 

retardation of M855, Mk262, and M193. 

performance for a well-designed, low-drag, streamlined (typically boattailed) projectile. 

Additional results are presented for higher drag shape with a form factor of 0.8 to quantify the 

relative sensitivity of the results to form factor. Results for the dependent variables of interest 

are presented as a function of projectile mass. Each projectile mass is associated with a 

particular muzzle velocity by virtue of the results shown in figure 31. 

Figure 34 shows the impact velocity as a function of projectile mass for ranges of 0, 150, 300, 

450, and 600 m. The results at 0-m range are essentially the muzzle velocity results shown in 

figure 31. Corresponding results for short ranges are shown in figure 35. For short ranges, the 

lighter bullets have higher impact velocity due to the higher muzzle velocity, but these lighter 

bullets decelerate at a higher rate (due to the lighter mass) and the maximum impact velocity at a 

given range occurs for a progressively heavier bullet as the range increases. The impact velocity 

at 450–600 m shows relatively little variation with projectile mass for masses greater than 60 gr. 

The sonic velocity for standard atmospheric conditions is also shown in figure 34. Some care 

must be taken in interpreting results that have impact velocities below the sonic velocity. The 

results presented here do not consider the significant decrease in drag that occurs in the transonic 

to subsonic velocity regime. The current results should overestimate the decrease in impact 

velocity below the sonic velocity. (It is possible to adapt the current approach to more accurately 

predict the flight behavior in this regime. However, this is beyond the scope of the current 

effort.) These comments also applied to the corresponding results for the impact energy, gravity 
44

0

200

400

600

800

1000

1200

20 40 60 80 100 120

Projectile Wt. (grains)

Velocity (m/s)

150 m Range 300 m Range

450 m Range 600 m Range

Muzzle Velocity M855

Mk262 M193

Sonic Velocity

Figure 34. Impact velocity vs. projectile mass for ranges of 0, 150, 300, 450, and 600 m. 

600

650

700

750

800

850

900

950

1000

1050

20 30 40 50 60 70 80 90 100 110

Projectile Wt. (grains)

Velocity (m/s)

25 m Range

50 m Range

75 m Range

100 m Range

Muzzle Velocity

Figure 35. Impact velocity vs. projectile mass for ranges of 0, 25, 50, 75, and 100 m. 
45

drop, and wind drift, which are derived from the same approach. The desirability of designs 

which result in impact velocities below the sonic velocity must also be carefully considered. 

Significant changes in projectile stability occur in the transonic velocity regime and these must 

be investigated as part of the design process to ensure projectile stability for those designs which 

operate in this flight regime. These issues can be avoided by considering only designs that fly at 

supersonic velocities within their operational ranges. 

Results are also shown in figure 34 for the 56-gr M193, the 62-gr M855, and 77-gr Mk262 

projectiles. Muzzle velocities used in the analysis for these rounds were experimentally 

determined from multiple sources using existing data (14) as well as recent in-house 

measurements. The experimentally determined muzzle velocities are slightly lower (2%, 3%, 

and 5% lower for M193, M855, and Mk262, respectively) than the predictions shown in figure 

31. Actual values of form factor (0.65, 0.57, and 0.63, for the M193, M855, and Mk262, 

respectively) and drag coefficient exponent (0.40, 0.53, and 0.43, for the M193, M855, and 

Mk262, respectively) were used rather than the baseline form factor of 0.6 and the corresponding 

drag coefficient exponent of 0.46 computed from equation 13. The form factor and drag 

coefficient exponent for the Mk262 were derived from the 77-gr HPBT Match King ballistic 

coefficients published by Sierra Bullets (7) using an approach similar that that described in 

section 3. The form factor and drag coefficient for the M193 and the M855 was derived from the 

FTB aeropack data using the approach described in section 2. In general, the results for the 

M193, M855, and Mk262 are consistent with the results presented as a function of projectile 

mass, especially when the slight differences in muzzle velocity is taken into account. 

Figure 36 shows the impact energy as a function of projectile mass for ranges of 0, 150, 300, 

450, and 600 m. The impact energy at short range is shown in figure 37. The results show a 

relatively strong decrease in impact energy with decreasing projectile mass below a projectile 

mass of 62-gr due to the relatively constant muzzle velocity for these lighter bullets. The impact 

energy at the muzzle is relatively constant for the heavier bullets, but the heavier bullets have 

higher impact energies at longer ranges due to the lower decelerations associated with the heavy 

projectile mass. Also shown are the results for the M193, M855, and Mk262 projectiles. These 

results follow the general trends of the analysis when the slight differences in muzzle velocity 

are accounted for. The M855 and Mk262 show only minor differences in impact energy relative 

to each other. 

A traditional qualitative measure of lethality has been the product of mass and velocity to the 3/2 

power (15). Results for this metric are shown in figure 38 as a function of projectile mass for 

various ranges. Compared to the impact energy, this metric is slightly less dependent on velocity 

and shows increased values of the metric as the projectile mass increases. 

Figure 39 shows the gravity drop as a function of projectile mass for ranges of 150, 300, 450, 

and 600 m. At shorter ranges, the heavier bullets show the most gravity drop due to their lower 

muzzle velocity and longer time of flight. However, as the range increases, the 62-gr 
46

0

500

1000

1500

2000

2500

20 30 40 50 60 70 80 90 100 110

Projectile Wt. (grains)

Energy (J)

150 m Range

300 m Range

450 m Range

600 m Range

Muzzle Velocity

M855

Mk262

M193

Figure 36. Impact energy vs. projectile mass for ranges of 0, 150, 300, 450, and 600 m. 

750

1000

1250

1500

1750

2000

20 30 40 50 60 70 80 90 100 110

Projectile Wt. (grains)

Energy (J)

25 m Range

50 m Range

75 m Range

100 m Range

Muzzle Velocity

Figure 37. Impact energy vs. projectile mass for ranges of 0, 25, 50, 75, and 100 m. 
47

0

10

20

30

40

50

60

70

80

20 30 40 50 60 70 80 90 100 110

Projectile Wt. (grains)

mV3/2 (kg-(m/s)3/2)

150 m Range

300 m Range

450 m Range

600 m Range

Muzzle Velocity

M855

Mk262

M193

Figure 38. 3/ 2 mV vs. projectile mass for ranges of 150, 300, 450, and 600 m. 

-7

-6

-5

-4

-3

-2

-1

0

20 30 40 50 60 70 80 90 100 110

Projectile Wt. (grains)

Gravity Drop (meters)

150 m Range

300 m Range

450 m Range

600 m Range

M855

Mk262

M193

Figure 39. Gravity drop vs. projectile mass for ranges of 150, 300, 450, and 600 m. 
48

bullet shows the lowest gravity drop due to the combination of higher muzzle velocity relative to 

the heavier bullets and lower retardation compared to the light bullets. Also shown are results 

for the M193, M855, and Mk262. After accounting for differences in muzzle velocity, these 

results compare relatively well with the predictions for the baseline form factor of 0.6. The 

Mk262 shows slightly higher gravity drop relative to the M855 as a result of its lower muzzle 

velocity. On the other hand, the M193 shows a slightly higher gravity drop due to its increased 

retardation (and relatively similar muzzle velocity) relative to the M855. 

Figure 40 shows the gravity drop in mils relative to the M855 as a function of projectile mass. 

Projectile masses between 46 and 95 gr all meet the ±1.0 mil mismatch requirement relative to 

the M855. For projectile masses below 46 gr, the mismatch increases rapidly as the projectile 

mass decreases. The candidate designs which have projectile masses close to 62-gr impact 

above the M855 impacts due to the slight higher muzzle velocity of the candidate designs. 

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

20 30 40 50 60 70 80 90 100 110

Projectile Wt. (grains)

Gravity Drop (mils)

150 m Range

300 m Range

450 m Range

600 m Range

1 Mil Spec

Figure 40. Gravity drop relative to M855 for ranges of 150, 300, 450, and 600 m. 

The effect of gravity drop can be accounted for by proper aiming of the weapon. However, this 

requires the range to target to be correctly determined. Errors or uncertainty in determining 

range in turn will produce deviations from the desired vertical impact location. Ranging errors 

have been shown to be a significant source of impact errors at longer ranges (16). The deviation 

from the vertical impact location can be shown to be the product of the ranging error Rε and the 

tangent of the angle of impact θ I as shown in equation 22. 

y R I ∆s = ε tanθ . (22) 
49

The ranging error may be dependent on range. One approach is to estimate the ranging error as a 

percentage of the range to target (16). However, in the current context, the ranging error should 

be fixed for a given range and only the tangent of the impact angle should vary as the projectile 

mass changes. Figure 41 shows the variation in the impact angle with projectile mass as a 

function of range. A strong variation in the tangent of the impact angle is seen at longer ranges 

for projectile masses below 60 gr. There is much less variation with projectile mass above 60 gr. 

If the ranging error can be estimated as a fraction of the range to target, the effect on the vertical 

miss distance as a function of range is increased significantly over the results in figure 41 which 

just consider the tangent of the impact angle. For ranges less than 150 m, the heavier projectiles 

show increased impact angle relative to the lighter projectiles. However, these trajectories are 

relatively flat and the trajectory is likely considered within point-blank range so that no 

adjustment for gravity drop is required. 

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

20 30 40 50 60 70 80 90 100 110

Projectile Wt. (grains)

Tan(Impact Angle)

150 m Range

300 m Range

450 m Range

600 m Range

M855

Mk262

M193

Figure 41. Tangent of impact angle vs. projectile mass for ranges of 150, 300, 450, and 600 m. 

Figure 42 shows the cross-wind drift for a 10-m/s cross wind as a function of projectile mass for 

ranges of 150, 300, 450, and 600 m. Although the 10-m/s cross wind is relatively high, the 

cross-wind drift varies linearly with the cross-wind velocity, allowing the cross-wind drift for 

other wind velocities to be easily obtained from the presented results. The cross-wind drift 

increases significantly with range. The heavier bullets show less wind sensitivity than the lighter 

bullets particularly at longer ranges. This is because of the stronger influence of muzzle 

retardation compared with muzzle velocity (see equation 17). Results are also shown for the 

M193, M855, and Mk262. The M855 and Mk262 show nearly the same degree of wind 
50

0

1

2

3

4

5

6

7

8

9

20 30 40 50 60 70 80 90 100 110

Projectile Wt. (grains)

Cross Wind Drift (meters)

150 m Range

300 m Range

450 m Range

600 m Range

M855

Mk262

M193

Figure 42. Cross-wind drift for 10-m/s cross wind vs. projectile mass for ranges of 150, 300, 

450, and 600 m. 

sensitivity. The M193 shows increase wind sensitivity compared to the M855 as a result of its 

increased retardation. 

Figure 43 shows the percentage of cross-wind drift relative to the M855 as a function of 

projectile mass for ranges of 150, 300, 450, and 600 m. Bullets heavier than the M855 are less 

sensitive to cross winds. Correspondingly, bullets lighter than the M855 show increased 

sensitivity to cross winds. On a percentage basis, the difference in cross-wind drift relative to the 

M855 increases with range for a constant projectile mass. 

Results were also obtained using a form factor 0.8 to compare with the baseline results obtained 

using a form factor of 0.6. Figure 44 compares the sensitivity of the impact velocity to form 

factor at various ranges as a function of projectile mass. The results show that the impact 

velocity decreases as the form factor increases from 0.6 to 0.8. This is primarily due to the 

increase in muzzle retardation that occurs as the form factor increases. (The results also show a 

slight dependence on the drag coefficient exponent produced by differences in the form factor.) 

The impact energy, shown in figure 45, is similarly affected by the change in form factor, 

although the differences are nearly twice that shown for the velocity because the impact energy 

varies with the square of the impact velocity. 
51

0

50

100

150

200

250

20 30 40 50 60 70 80 90 100 110

Projectile Wt. (grains)

Cross Wind Drift - % of M855 

150 m Range

300 m Range

450 m Range

600 m Range

Figure 43. Cross-wind drift relative to M855 vs. projectile mass for ranges of 150, 300, 450, and 

600 m. 

0

200

400

600

800

1000

1200

20 30 40 50 60 70 80 90 100 110

Projectile Wt. (grains)

Velocity (m/s)

Form Factor=0.6

Form Factor=0.8

Muzzle Velocity

Muzzle

150m

300m

450m

600m

Figure 44. Impact velocity vs. projectile mass for form factors of 0.6 and 0.8 at ranges of 150, 

300, 450, and 600 m. 
52

0

500

1000

1500

2000

2500

20 30 40 50 60 70 80 90 100 110

Projectile Wt. (grains)

Energy (J)

Form Factor=0.6

Form Factor=0.8

Muzzle Velocity Muzzle

150m

450m

300m

600m

Figure 45. Impact energy vs. projectile mass for form factors of 0.6 and 0.8 at ranges of 150, 300, 

450, and 600 m. 

The gravity drop of projectiles with a form factor of 0.8 relative to the gravity drop of the M855 

is shown in figure 46. The vertical impact location is nearly 1.0 mil or greater below the vertical 

impact location of M855 at 600 m for all projectile masses when the form factor is increased to 

0.8. Given the predicted variation in the muzzle velocity with projectile mass, the form factor of 

0.8 represents an upper limit for meeting ±1.0 mil mismatch requirement for projectile masses 

between 62 and 73 gr. A reduction in the form factor below 0.8 is required for projectile outside 

this range of projectile masses to meet the ±1.0 mil mismatch requirement. 

Figure 47 shows the cross-wind drift of the projectile with a form factor of 0.8 as a percentage of 

the cross-wind drift of the M855. The cross-wind drift is greater than the M855 across the range 

of projectile masses considered here. For the 62-gr projectile, the increase in the form factor 

from 0.6 to 0.8 produces 35%–40% more cross-wind drift for ranges of 150–600 m. The relative 

effect decreases slightly as the projectile mass increases. The differences increase more 

substantially as the projectile mass decreases. 

The results indicate some degree of sensitivity to the form factor associated with the drag 

coefficient. The survey of existing designs indicates that form factor of 0.6 can be reasonably 

accomplished, although significant reductions in form factor below 0.6 may be difficult. 

Projectile core configuration and mass distribution may result in designs with have drag form 

factors above 0.6. The current results provide a means of assessing the tradeoffs between 
53

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

20 30 40 50 60 70 80 90 100 110

Projectile Wt. (grains)

Gravity Drop (mils)

150 m Range

300 m Range

450 m Range

600 m Range

1 Mil Spec

Figure 46. Gravity drop of projectiles with form factor of 0.8 relative to M855 vs. projectile 

mass for ranges of 150, 300, 450, and 600 m. 

0

50

100

150

200

250

300

350

400

20 30 40 50 60 70 80 90 100 110

Projectile Wt. (grains)

Cross Wind Drift - % of M855 

150 m Range

300 m Range

450 m Range

600 m Range

Figure 47. Cross-wind drift of projectiles with form factor of 0.8 relative to M855 vs. projectile 

mass for ranges of 150, 300, 450, and 600 m. 
54

performance parameters associated with the point mass trajectory and the overall projectile 

effectiveness of the projectile. 

5.1 Comparison of the Relative Performance of the M855 and Mk262 

Currently within the U.S. Army there is discussion and perhaps some interest in use of the 

Mk262 in lieu of the M855. The Mk262 is based on the Sierra 77-gr match ammunition. The 

results shown here shed some light on the relative performance of the two rounds. In terms of 

impact velocity, impact energy, gravity drop, and cross-wind sensitivity, the Mk262 shows only 

marginal performance improvements in some cases (impact energy at long range, cross-wind 

drift) and slightly lower performance in other cases (gravity drop and impact velocity). These 

performance parameters represent an important subset of the complete set of performance 

parameters. Important considerations such as impact dispersion and lethality are not addressed 

here. Impact dispersion is influenced strongly by manufacturing considerations and there are 

significant differences in the manufacturing approaches and requirements for the high volume 

production of the M855 and the match grade Sierra 77-gr. Differences in the impact dispersion 

of the two rounds may exist. However, these differences must be considered in light of the other 

sources of delivery errors including cross wind and ranging errors that are directly related to the 

cross-wind sensitivity and gravity drop parameters examined here. 

5.2 Copper and Steel-Cored M855 Variants 

Environmental considerations have resulted in suggestions for removing the lead core of M855 

and replacing it with alternative materials. One suggestion is to replace the lead core of the 

M855 with a steel core resulting in a projectile that weighs 52 gr. Another suggestion is to 

replace the lead core and steel penetrator with copper producing a projectile that weighs 57 gr. 

The current design approach can be used to predict the performance of potential designs. 

The results show that over the first 150 m of flight, both the steel and copper variants have 

slightly impact higher velocity than the M855 because of the higher launch velocity. However, 

due to their lighter weight and correspondingly higher retardation, their impact velocity is lower 

at longer ranges and decreases to 83% and 92% of the impact velocity of the M855 at 600-m 

range. Because of the lighter weight of the both variants, the impact energy is less than the 

M855 across the trajectory. Both the copper- and steel-cored M855 variants have trajectories 

early in flight that are slightly above (less than 0.1 mil) the M855 due to the higher launch 

velocity, but the vertical impact point at 600 m is 0.1 mil and 0.5 mil, respectively, below the 

vertical impact point of the M855. The copper and steel cored M855 variants have 8%–12% and 

18%–28% more cross-wind sensitivity, respectively, than the M855 over 150–600-m ranges, 

with the largest difference observed at the longest ranges. 
55

6. Conclusion 

A conceptual design approach for small-caliber munitions has been developed and applied to 

5.56-mm ammunition. The method is based on an analytical approach for predicting the point 

mass trajectories for direct-fire munitions. The method utilizes a power law variation to 

characterize the variation of the drag coefficient with Mach number. High quality aerodynamic 

range data is used to demonstrate the validity of the power law approach for small-caliber 

munitions including the 5.56-mm M855. The method allows closed form solutions to be 

obtained for several important design quantities including the impact velocity and energy, 

gravity drop, and cross-wind drift. From the analytical method, it can be shown that these 

performance parameters are functions of three parameters: the muzzle velocity, muzzle 

retardation, and the exponent defining the power law drag variation. The analytical trajectory 

predictions has been validated by comparison with numerical predictions for the 5.56-mm M855 

and excellent agreement is found. 

An extensive database of trajectory data developed by a commercial small arms bullet 

manufacturer has been analyzed as part of the current study. From the database, a correlation 

that expresses the drag coefficient exponent as a function of the drag coefficient form factor has 

been developed for small arms bullets. The range of the form factor for low drag streamlined 

(typically boattailed) projectiles which represent likely candidates for tactical small arms bullets 

is also obtained from the database. 

The method is then applied to investigate candidate designs for 5.56-mm ammunition. An initial 

set of results examines the performance of candidate designs in terms of muzzle velocity and 

muzzle retardation for a fixed value of the drag coefficient exponent. The results show the 

performance of candidate designs relative to the 5.56-mm M855 for a variety of performance 

metrics including trajectory mismatch and cross-wind sensitivity. In general, the results show 

that the best performance is obtained from projectiles which have high muzzle velocity and low 

muzzle retardation. The relative insensitivity of the results to the drag coefficient exponent is 

also demonstrated. 

Recognizing that muzzle velocity is not truly an independent variable, but is dependent on 

launch mass and interior ballistic constraints, an additional set of results is obtained using 

interior ballistic computations to quantify the relationship between muzzle velocity and projectile 

mass. By considering designs with a drag coefficient form factor of 0.6, the performance of 

candidate designs can be assessed as a function of projectile mass. Performance parameters 

examined include impact velocity and energy, gravity drop, and cross-wind drift. For several 

performance parameters, the optimal performance is obtained for projectile masses close to that 

of the current 62-gr M855. Possible performance improvements are possible for some 

performance metrics with projectiles heavier than 62 gr. However, the performance gains 
56

increase relatively slowly with increasing projectile mass, primarily because of the significant 

decrease in muzzle velocity with increasing projectile mass. Projectile designs lighter than 62 gr 

show improved performance for some metrics such as impact velocity at short range. However, 

the light weight and higher retardation associated with these rounds results generally results in 

poorer performance at ranges beyond 100–150 m. 

The results documented here provide important design guidance for current efforts in 5.56-mm 

ammunition development. The results address primarily interior and exterior ballistic 

considerations, although some qualitative measures of terminal performance are predicted. 

Further analysis is required to ensure that the projectile designs are aerodynamically stable and 

have the required lethality. These considerations are being addressed in the next phase of the 

design effort. 
57

7. References 

1. Weinacht, P.; Cooper, G. R.; Newill, J. F. Analytical Prediction of Trajectories for HighVelocity Direct-Fire Munitions, ARL TR-3567; U.S. Army Research Laboratory: Aberdeen 

Proving Ground, MD, August 2005. 

2. McCoy, R. L. Modern Exterior Ballistics; Schiffer Military History: Atglen, PA, 1999. 

3. Tilghman, B. A. Firing Tables Branch, U.S. Army Armament Research, Development, and 

Engineering Center, Aberdeen Proving Ground, MD. Private communication, 8 December 

2003. 

4. McCoy, R. L. Aerodynamic and Flight Dynamic Characteristics of the New Family of 

5.56-mm NATO Ammunition; BRL-MR-3476; U.S. Army Ballistics Research Laboratory: 

Aberdeen Proving Ground, MD, October 1985. 

5. Arrow Tech Associates. Prodas Users Manual; Arrow Tech Associates: Burlington, VT, 

2002. 

6. Fourth Edition Rifle Exterior Ballistics Tables. http://www.exteriorballistics.com 

/ebexplained /4th-rifle_tables/index.cfm (accessed 28 February 2005). 

7. Sierra Rifle Bullets Ballistic Coefficient Listing. http://www.exteriorballistics.com 

/ebexplained/riflecoefficients.cfm (accessed 28 February 2005). 

8. Bullets. http://www.sierrabullets.com/index.cfm?section=bullets&page=bullets 

&bulletType=0 (accessed 28 February 2005). 

9. Hoerner, S. F. Fluid-Dynamic Drag; Hoerner Fluid Dynamics: Brick Town, NJ, 1965. 

10. Charters, A. C.; Stein, H. The Drag of Projectiles with Truncated Cone Headshapes; report 

no. 624; U.S. Army Ballistics Research Laboratories: Aberdeen Proving Ground, MD, 

March 1952. 

11. Firing Tables for Weapon, 5.56-mm, M16A1, Weapon, 5.56-mm, M16A2, Weapon, 5.56-mm, 

M249, Weapon, 5.56-mm, M4, Firing Cartridge, Ball, M855, Cartridge, Tracer, M856, 

Firing Cartridge, Ball, M193, Cartridge, Tracer, M196; U.S. Army ARDEC Firing Tables; 

Aberdeen Proving Ground, MD, July 1996. 

12. Department of the Army. Rifle Marksmanship M16A1, M16A2/2, M16A4, and M4 Carbine; 

Field Manual No. 3-22.9 (FM 23-9); Headquarters, Department of the Army: Washington, 

DC, 24 April 2003. 
58

13. Anderson, R. D.; Fickie, K. D. IBHVG2 (Interior Ballistics of High Velocity Guns, Version 

2)—A User’s Guide; BRL-TR-2829; U.S. Army Ballistic Research Laboratory: Aberdeen 

Proving Ground, MD, July 1987. 

14. Peterson, B. U.S. Army Research Laboratory, Aberdeen Proving Ground, MD. Private 

communication, 11 January 2005. 

15. Kokinakis, W.; Sperrazza, J. Criteria for Incapacitating Soldiers with Fragments and 

Flechettes; BRL-1269; U.S. Army Ballistics Research Laboratory: Aberdeen Proving 

Ground, MD, January 1965. 

16. Von Wahlde, R.; Metz, D. Sniper Weapon Fire Control Error Budget Analysis; ARL-TR2065; U.S. Army Research Laboratory: Aberdeen Proving Ground, MD, August 1999. 
59

List of Symbols, Abbreviations, and Acronyms 

C Ballistic coefficient 

CD Drag coefficient 

V0

CD Drag coefficient evaluated at the muzzle velocity 

VRef

CD Drag coefficient evaluated at the reference velocity 

VRef

DG1 C Drag coefficient from G1 drag function evaluated at the reference velocity 

D Reference diameter 

g Gravitational acceleration 

i Drag coefficient form factor Form

m Projectile mass 

M Mach number 

n Exponent defining shape of the drag vs. Mach number curve 

s Nondimensional range coordinate 

SD Sectional density, 2

m SD=

D

g drop s − Gravity drop 

x y s Horizontal and vertical ,s displacement along trajectory 

y ∆s Deviation from intended vertical impact location due to ranging error 

z s Out-of-plane displacement along trajectory (normal to x-y plane) 

S Reference area, ref 4

D S

2

ref

π

=

t Time of flight 

V Total velocity 

V Muzzle velocity 0
60

V Reference velocity for comparis Ref on of form factor and retardation 

0 ds

dV ⎟

⎠

⎞ ⎜

⎝

⎛ Muzzle retardation 

VRef ds

dV ⎟

⎠

⎞ ⎜

⎝

⎛ Retardation evaluated at reference velocity 

wz Cross-wind velocity 

Greek Symbols

Rε Ranging error 

θ 0 Initial gun elevation angle 

θ I Trajectory impact angle 

ρ Atmospheric density 
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