A Direct-Fire Trajectory Model for Supersonic, Transonic,

and Subsonic Projectile Flight

by Paul Weinacht

ARL-TR-6998 July 2014

Approved for public release; distribution is unlimited.
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Army Research Laboratory

Aberdeen Proving Ground, MD 21005-5066

ARL-TR-6998 July 2014

A Direct-Fire Trajectory Model for Supersonic, Transonic,

and Subsonic Projectile Flight

Paul Weinacht

Weapons and Materials Research Directorate, ARL

Approved for public release; distribution is unlimited.
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A Direct-Fire Trajectory Model for Supersonic, Transonic, and Subsonic Projectile 

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14. ABSTRACT

This report presents a simple but accurate method of determining the trajectory of projectiles that traverse a flight regime that 

includes supersonic, transonic, and subsonic flight. Closed-form analytical solutions for the important trajectory parameters 

such as the time of flight, velocity, gravity drop, and wind drift are presented. The method makes use of individual power-law 

descriptions of the drag variation with Mach number within the supersonic, transonic, and subsonic regimes. The method 

demonstrates that the free-flight trajectory can be characterized with as few as six parameters: the muzzle velocity, muzzle 

retardation, a power-law exponent that describes the drag variation in supersonic flight, the transition Mach numbers between 

supersonic and transonic flight, the transition Mach number between transonic and subsonic flight, and the retardation at the 

subsonic transition Mach number. The accuracy and simplicity of the method make it very useful for preliminary design or 

performance assessment studies where rapid prediction of projectile trajectories is desired. Sample results are presented for a 

9-mm pistol bullet that traverses supersonic, transonic, and subsonic flight to demonstrate the viability of the method.

15. SUBJECT TERMS

trajectory prediction, flight mechanics, aerodynamics, ballistic calculator, fire control algorithm

16. SECURITY CLASSIFICATION OF: 

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iii

Contents

List of Figures iv

List of Tables iv

1. Introduction 1

2. Technical Approach 2

2.1 Power-Law Drag Model..................................................................................................3

2.2 Analytical Solution Using a Hybrid Supersonic/Transonic/Subsonic Power-Law 

Drag Model...............................................................................................................................7

2.2.1 Trajectory Model for Supersonic Portion of Flight.............................................8

2.2.2 Transition Between Supersonic and Transonic Flight ........................................9

2.2.3 Trajectory Model for Transonic Portion of Flight.............................................10

2.2.4 Transition Between Transonic and Subsonic Flight..........................................12

2.2.5 Trajectory Model for Subsonic Portion of Flight..............................................13

2.2.6 Extensions of the Method..................................................................................14

3. Results 14

4. Conclusions 23

5. References 24

Nomenclature 25

Distribution List 28
iv

List of Figures

Figure 1. Supersonic power-law drag coefficient model compared with Aeropack and range 

data for the M855.......................................................................................................................4

Figure 2. Hybrid supersonic/subsonic power-law drag coefficient model compared with 

Aeropack and range data for the M855......................................................................................5

Figure 3. Hybrid supersonic/transonic/subsonic power-law drag coefficient model compared 

with Aeropack and range data for the M855. ............................................................................7

Figure 4. Drag coefficient as a function of Mach number: 9-mm pistol bullet. ............................15

Figure 5. Velocity as a function of range: 9-mm pistol bullet.......................................................17

Figure 6. Time of flight vs. range: 9-mm pistol bullet...................................................................18

Figure 7. Gravity drop vs. range: 9-mm pistol bullet. ...................................................................18

Figure 8. Wind drift vs. range: 9-mm pistol bullet. .......................................................................19

Figure 9. Gravity drop velocity vs. range: 9-mm pistol bullet.......................................................19

Figure 10. Modeling of drag coefficient as a function of Mach number with different 

power-law models....................................................................................................................20

Figure 11. Sensitivity of velocity vs. range to selection of transonic power-law model: 9-mm 

pistol bullet...............................................................................................................................21

Figure 12. Sensitivity of time of flight vs. range to selection of transonic power-law model: 

9-mm pistol bullet....................................................................................................................21

Figure 13. Sensitivity of gravity drop vs. range to selection of transonic power-law model: 

9-mm pistol bullet....................................................................................................................22

Figure 14. Sensitivity of wind drift vs. range to selection of transonic power-law model: 

9-mm pistol bullet....................................................................................................................22

List of Tables

Table 1. Parameters for 9-mm ball projectile used in hybrid supersonic/transonic/subsonic 

model........................................................................................................................................15

Table 2. Parameters for 9-mm ball projectile used in hybrid supersonic/subsonic model. ...........16
1

1. Introduction

Several recent studies have demonstrated there is a need for characterizing the free-flight 

performance of direct-fire systems including combat rifles, pistols and sniper weapons. For many 

of these systems, combat rifles for example, the trajectory is characterized by supersonic flight 

over the useful operational range of the weapon system. Recent studies (1, 2) have provided a 

framework for examining the free-flight trajectory within the constraints of supersonic flight. 

The basis of this 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 exterior ballistic performance in a conceptual design environment where details of the 

designs have not been completely defined or in assessment studies where the simple and rapid 

predictions of the trajectory are desired. Although developed for supersonic flight, the method 

also allows the prediction of trajectories in subsonic flight where the drag coefficient is constant 

with Mach number.

While the method of reference 1 has proven very useful, there are some applications where the 

trajectory involves supersonic flight during early phases of the trajectory and subsonic flight later 

in the trajectory when the projectile slows due to drag. Examples of these include pistol and long 

range rifle systems. To address these applications, the method presented in reference 1 was 

extended so that it could be applied to trajectories with mixed supersonic and subsonic flight (3).

A key assumption of the method is that there is an abrupt transition from supersonic to subsonic 

flight which essentially ignores the effect of the variations in drag in the transonic regime. For 

many applications, such as long range rifle systems, the duration of the flight in the transonic 

regime is small compared with the overall length of the flight and the effects on the trajectory are 

negligible. However, in some cases, such as pistol applications, the projectile may spend a 

significant portion of the flight in the transonic regime. Although results in reference 3 show a 

modest effect of an abrupt transition between supersonic and subsonic flight on pistol bullet 

trajectories, it appears possible to further extend the approach presented in references 1–3 to

allow effects of the drag variation in the transonic regime to be included. This is the subject of 

the current technical report.

In the following sections, the analytical approach for solving the 3 degree-of-freedom (3DOF) 

trajectory equations is briefly described and benchmarked with numerical predictions of the 

trajectory of a ball projectile fired from a 9-mm pistol. The 9-mm ball projectile example 

illustrates a case where the flight includes supersonic, transonic, and subsonic flight over the 

trajectory. Comparisons are made with results obtained from the current method, the methods 

presented in references 1–3, and 4DOF numerical prediction. The results demonstrate the 

performance and accuracy of the method.
2

2. Technical Approach

The flat-fire trajectory of a projectile can be characterized as follows (4):











































































Drift

Swerve

Epicyclic

Jump

Aerodynamic

Throwoff

Lateral

Point MassTrajectory

TheFlat 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 crosswind drift. It is this portion of the trajectory that is 

the focus of the current conceptual 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. It is 

possible to utilize the current method to predict the mean trajectory and statistical methods to 

address issues associated with ammunition dispersion to determine the pattern of impacts at 

range. 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 (or alternatively, spin drift) produces a 

small horizontal deflection of the projectile due to interaction of the projectile’s gyroscopic 

behavior with the trajectory’s curvature due to gravity drop. The deflection is fairly consistent 

from shot to shot and is typically small compared with crosswind drift.

The flat-fire point-mass trajectory (shown in equations 2–9) can be obtained by solving the 

point-mass or 3DOF equations, which are obtained from Newton’s second law. The flat-fire 

point-mass equations assume that the transverse aerodynamic forces such as the lift and Magnus 

forces are small and that the Coriolis acceleration due to the earth’s rotation can be neglected. If 

the total yaw of the projectile is small, the transverse aerodynamic forces can be assumed to be 

small. Using this approach, the projectile is characterized by its muzzle velocity, mass, and the 

variation of its drag coefficient with Mach number. 

V

V

V S C

2

1

dt

dV

m

x

ref D

x 2

  

(2)

mg

V

V

V S C

2

1

dt

dV

m

y

ref D

y 2

    (3)
3

x

x V

dt

ds



(4)

y

y

V

dt

ds



(5)

The initial conditions are 

x 0 0 V (t  0)  V cos

(6)

y 0 0 V (t  0)  V sin

(7)

s (t 0) 0 x 

(8)

s (t 0) 0 y 

(9)

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). 

2.1 Power-Law Drag Model

The power-law formulation assumes that the drag coefficient variation with Mach number can be 

modeled as a function of Mach number to a power as shown in equation 10. 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.

D n n V

1

M

1

C  

(10)

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 (5) used to construct firing tables for the round 

are shown along with aerodynamic spark range data (6). Clearly, the power-law description is 

not capable of describing the drag variation with Mach number over the complete Mach number 

regime. However, the drag variation can be modeled with the power-law description in specific 

regimes. The fit of the FTB aeropack drag coefficient data obtained using the power-law 

formulation in the supersonic regime is shown in equation 10. 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. 

Reference 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. This was 

further confirmed in reference 2 specifically for small-arms ammunition and a correlation 
4

developed for small-arms ammunition relating the drag coefficient exponent to the aeroballistic 

form factor. In general, as small-arms projectiles become more blunt and less streamlined, the 

drag coefficient exponent decreased.

Figure 1. Supersonic power-law drag coefficient model compared with Aeropack and 

range data for the M855.

For subsonic flight (nominally 

M  0.8

), the projectile typically exhibits a constant drag 

coefficient with Mach number as shown in figure 1. This type of variation of the drag coefficient 

can also be modeled with the functional relationship shown in equation 10 where n = 0. Thus, the 

drag of a projectile flying in either purely supersonic flight or subsonic flight can be described

using a power-law formulation within each regime as described in reference 1. 

Reference 3 proposed a drag description that allowed supersonic to subsonic flight to be 

considered by using a piece-wise description of the supersonic and subsonic drag variation in the 

supersonic and subsonic regimes with the transonic drag rise represented by an abrupt transition 

between the supersonic and subsonic regimes as show in figure 2. This allowed the development 

of closed-form expressions to compute the trajectory from supersonic through subsonic flight. 

For many projectiles (for example, projectiles fired from rifles) the projectile traverses the 

transonic regime over a small distance compared with the length of its overall flight path. For 

these situations, ignoring the transonic drag variation represents a good assumption, and the 

results of the model compare favorably with more complete 3DOF predictions that include the 

effect of the transonic regime.
5

Figure 2. Hybrid supersonic/subsonic power-law drag coefficient model compared with 

Aeropack and range data for the M855.

There are, however, situations where the transonic regime represents a significant portion of the 

overall flight. A common example is the flight of a pistol bullet. Here it is desirable to have a 

model that describes the drag behavior in the transonic regime. It appears that the power-law 

approach can be adopted to treat the subsonic, transonic, and supersonic regimes with separate 

power-law drag models. When the transonic drag behavior is treated with a power-law model, it

appears to provide a reasonable approximation to the transonic drag rise, although this probably 

represents more of a curve-fitting technique than a model with a strong physical basis. This 

power-law representation should allow the development of closed-form solutions similar to those 

developed in references 1–3.

The transonic power-law model is shown in equation 11. For constant atmospheric conditions, 

the sound speed is constant, and the Mach number and velocity are directly proportional. If the 

upper bound of the velocity of the subsonic regime (and the lower bound of the velocity of the 

transonic regime) is represented by 

Vsub

and the upper bound of the velocity of the transonic 

regime is represented by 

Vtran

and the drag coefficients at the lower and upper bounds of the 

transonic regime are

Dtransonic lower C 

and

Dtransonic upper

C 

, then the drag coefficient in the transonic 

regime can be expressed as 

tran

transonic upper

tran

transonic upper

n

tran

D

n

tran

D D V

V

C

M

M C C 











 













 

(11)
6

































 





tran

sub

D

D

tran M

M ln

C

C

n ln

transonic upper

transonic lower

(12)

There is no requirement for the drag curve to be continuous with the Mach number from the 

subsonic to the supersonic regimes. In other words, 

Dtransonic lower C 

does not have to be equal to 

the subsonic drag, and 

Dtransonic upper

C 

does not have to be equal to the supersonic drag at the 

lower end of the supersonic regime.

If there is a discontinuity in the drag curve at both the upper and lower bounds of the transonic 

regime, there are eight parameters required to define the entire drag curve:

• A reference velocity in the supersonic regime (typically taken as the muzzle velocity) and a 

reference drag coefficient at the reference velocity 

• The drag coefficient exponent in the supersonic regime 

• The Mach number (or velocity) at the boundary between the transonic regime and the 

supersonic regime

• Mtran

• The Mach number (or velocity) at the boundary between the subsonic and transonic regime

• Msub

• The two drag coefficients at bounds of the transonic regime

• The subsonic drag coefficient

If the drag curve is assumed to be continuous, the number of parameters required to describe the 

drag curve is reduced to six. Here, the two drag coefficients at the bounds of the transonic regime 

can be eliminated as parameters, since the drag coefficient at the lower bound of the transonic 

regime is equal to the subsonic drag coefficient and the drag coefficient at the upper bound of the 

transonic regime can be computed from the supersonic power-law model. From a practical 

standpoint, the assumption that the drag curve is continuous may be the preferred approach in 

most cases and is consistent with implementations made in most other trajectory models. Also, 

incorporating the transonic drag rise adds one additional parameter, for a total of six parameters

(if the drag curve is assumed continuous), to define the drag curve compared to the five 

parameters required for the hybrid supersonic/subsonic model of reference 3. 

Figure 3 shows an example of the hybrid supersonic/transonic/subsonic power-law drag model 

for the M855 projectile compared with FTB, ARDEC “aeropack” data (5) and aerodynamic 

spark range range data. The computed power-law exponent for the M855 is 0.53 for the 
7

supersonic regime and a constant drag coefficient (n = 0) is used in the subsonic regime as 

discussed previously. When equation 11 is used and a continuous drag curve is assumed,

0.9 Msub

, and 

1.05 Mtran

, a drag coefficient exponent of 

3.63 tran n  

was computed for the 

transonic regime. The power-law model of drag curve represents the ARDEC aeropack fairly 

well. 

Figure 3. Hybrid supersonic/transonic/subsonic power-law drag coefficient model 

compared with Aeropack and range data for the M855.

2.2 Analytical Solution Using a Hybrid Supersonic/Transonic/Subsonic Power-Law Drag 

Model

For mixed supersonic/transonic/subsonic flight, the initial supersonic portion of the flight can be 

modeled using the analytical approach shown in reference 1. The analytical solutions are 

obtained using two main approximations. First, the gravity drop contributes little to total velocity 

during the flight. This assumption implies that the vertical displacement along the trajectory is 

small (direct-fire) and allows the 3DOF equations to be decoupled. 

V V V V

2

y

2



x  

(13)

Second, the vertical and lateral displacements are small in relation to the downrange 

displacement and the total displacement along the trajectory (or slant-range) is nearly equal to 

the downrange displacement. This second assumption allows the downrange displacement to be 

treated as one of the primary independent variables. 
8

x

2

z

2

y

2

x

s  s  s  s  s

(14)

2.2.1 Trajectory Model for Supersonic Portion of Flight

Reference 1 shows that by using the power-law approach, an analytical solution for the 

supersonic portion of flight 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 

g drop s 

, vertical displacement of the projectile along the trajectory 

y

s , and vertical 

deflection due to crosswind 

z

s

as a function of range 

x

s . Equations 15–22 present closed-formed 

solutions for these dependent variables for a variable drag coefficient exponent (

n  0,1,2

) as 

presented in reference 1. (Special case solutions for 

n  0, 1 or 2

are also shown in reference 1. 

Inclusion of these solutions in the present analysis is straightforward.) The displacement due to 

wind drift in equation 19 is a generalized formula based on the integration of the 3DOF

equations shown in McCoy (7).

x 0 V  Vcos

(15)

y 0 Vg drop V  Vsin  

(16)

n

1

0

x

0

0 V

s

ds

dV V V 1 n

























 

(17)

y x 0 g drop s s tan s     (18)













 

0

x

z z V

s

s w t

(19)



































































1

V

s

ds

dV 1 n

ds

dV (n 1)

1

t

n

1

1

0

x

0

0

(20)

































  







































 









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

(21)
9

n 0,1,2

V

s

ds

dV 1 n

V

s

ds

dV 1 n

ds

dV (n 2)

g

V

n

1

0

x

0

n

n 1

0

x

0

0

g drop





































 



























































(22)

In the supersonic regime, these trajectory characteristics are functions of only three primary 

variables, the projectile’s muzzle velocity 

V0

, muzzle retardation 

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 crosswind velocity

wz

. The gun elevation angle 

0

also appears in equations 15, 16, and 18 and can be treated as another independent parameter. 

However, the gun elevation angle 

0

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 23.

V0

0 ref D

0

V S C

2m

1

ds

dV    











(23)

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

represented by a single parameter, the muzzle retardation.

2.2.2 Transition Between Supersonic and Transonic Flight

For the composite supersonic-transonic-subsonic model, equations 15–22 are valid until the 

projectile reaches the transition Mach number between supersonic and transonic flight. The 

transition velocity is determined from the Mach number as shown in equation 24.

tran Mtran V  a

(24)

When equations 17 and 20–22 are used, the range, time of flight, gravity drop, and gravity drop 

velocity where the transition occurs between the supersonic and transonic flight can be 

determined.
10

















 



























 1

V

V

n

ds

dV

V

s

n

0

tran

0

0

xtran

(25)



































































1

V

s

ds

dV 1 n

ds

dV (n 1)

1

t

n

1

1

0

xtran

0

0

tran

(26)

































  







































 









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

tran tran

tran

(27)



































 



























































n

1

0

x

0

n

n 1

0

x

0

0

g drop

V

s )

ds

dV 1 n

V

s

ds

dV 1 n

ds

dV (n 2)

g

V

tran

tran

tran

(28)

2.2.3 Trajectory Model for Transonic Portion of Flight

When equations 25–28 are used as initial conditions, the governing 3DOF equations can be 

integrated to determine the trajectory in the transonic portion of flight. The solution has a similar 

form to the solution for the supersonic portion of flight as shown in equations 29–32. As 

discussed previously, a different power-law exponent is used to describe the drag behavior in the 

transonic regime.

x 0 V  Vcos

(29)

y 0 Vg drop V  Vsin   (30)

y x 0 g drop s s tan s     (31)













 

0

x

z z V

s

s w t

(32)

where
11 tran n

1

tran

tran x x

tran

tran tran V

s s

ds

dV V V 1 n











 













 

(33)

tran

n tran

1

1

tran

x x tran

tran

tran

tran

tran

1 t

V

(s s )

ds

dV 1 n

ds

dV (n 1)

1

t 

































 

































(34)

x x tran g droptran

tran

g droptran

tran

x x tran

tran

tran

n tran

1) 2(n tran

tran

x x tran

tran

2 tran

tran

tran tran

g drop

(s s ) s

V

V

V

(s s )

ds

dV 1 2(n 1)

V

(s s )

ds

dV 1 n

ds

dV 2(n 2)(n 1)

g

s









  





 













  





















 



























 





(35)

n 0,1,2

V

(s s )

ds

dV V 1 n

V

(s s )

ds

dV 1 n

V

(s s )

ds

dV 1 n

ds

dV (n 2)

g

V

n tran

1

tran

x x tran

tran

g droptran tran

tran n

1

tran

x x tran

tran

tran

n tran

n tran 1

tran

x x tran

tran

tran

tran

tran

g drop













 













 





















 













 





















 







































(36)

xsub x tran g droptran

tran

g droptran

tran

xsub x tran

tran

tran

n tran

1) 2(n tran

tran

xsub x tran

tran

2 tran

tran

tran tran

g dropsub

(s s ) s

V

V

V

(s s )

ds

dV 1 2(n 1)

V

(s s

ds

dV 1 n

ds

dV 2(n 2)(n 1)

g

s









  





 













  





















 



























 





(37)

Here the retardation at the transition point is determined from equation 38.

tran ref Dtransonic upper

tran

V S C

2m

1

ds

dV



   











(38)
12

Equation 38 allows for a discontinuity in the drag curve at the transition between supersonic and 

transonic flight. However, if the power-law representation of the drag in the supersonic regime 

provides a good representation of the drag coefficient at transition between supersonic and 

transonic, the drag coefficient at the transition Mach number can be evaluated using the 

supersonic power-law model, and the retardation at the transition Mach number can be obtained

using the muzzle velocity, muzzle retardation, the supersonic power-law drag coefficient 

exponent, and the transition Mach number. This effectively eliminates the drag coefficient at the 

transition Mach number as an input parameter for the model.

0

1 n

0

tran

0

1 n

0

tran

tran ds

dV

M

M

ds

dV

V

V

ds

dV 



























 



























 











 

(39)

2.2.4 Transition Between Transonic and Subsonic Flight

Once the projectile enters subsonic flight, equations 29–37 are no longer valid. The transonic to 

subsonic transition velocity is determined from the Mach number as shown in equation 40.

sub Msub V a 

(40)

When equations 33–37 are used, the range, time of flight, gravity drop, and gravity drop velocity 

where the transition occurs between the transonic and subsonic flight can be determined.

tran x

tran n

tran

sub

tran

tran

tran

sub x 1 s

V

V

V

n

ds

dV

1

s 

















 





























(41)

tran

n tran

1

1

tran

xsub x tran

tran

tran

tran

tran

sub 1 t

V

(s s )

ds

dV 1 n

ds

dV (n 1)

1

t 

































 

































(42)

xsub x tran g droptran

tran

g droptran

tran

xsub x tran

tran

tran

n tran

1) 2(n tran

tran

xsub x tran

tran

2 tran

tran

tran tran

g dropsub

(s s ) s

V

V

V

(s s )

ds

dV 1 2(n 1)

V

(s s )

ds

dV 1 n

ds

dV 2(n 2)(n 1)

g

s









  





 













  





















 



























 





(43)
13

n 0,1,2

V

(s s )

ds

dV V 1 n

V

(s s )

ds

dV 1 n

V

(s s )

ds

dV 1 n

ds

dV (n 2)

g

V

n tran

1

tran

xsub x tran

tran

g droptran tran

tran n

1

tran

xsub x tran

tran

tran

n tran

n tran 1

tran

xsub x tran

tran

tran

tran

tran

sub g drop













 













 





















 













 





















 







































(44)

2.2.5 Trajectory Model for Subsonic Portion of Flight

When equations 41–44 are used as initial conditions, the governing 3DOF equations can be 

integrated to determine the trajectory in the subsonic portion of flight. The solution has a similar 

form to the solution for the supersonic and transonic portions of flight as shown in equations 45–52.

x 0 V  Vcos

(45)

y 0 Vg drop V  Vsin   (46)

y x 0 g drop s s tan s     (47)













 

0

x

z z V

s

s w t

(48)

where











 















sub

x xsub

sub

sub V

(s s )

ds

dV V V exp

(49)























 













 













 

sub

x xsub

sub

sub

sub V

(s s )

ds

dV 1 exp

ds

dV

1

t t

(50)

sub x xsub g drop

sub

sub g drop

sub

x xsub

sub sub

x xsub

sub

2

sub

g drop

(s s ) s

V

V

1

V

(s s )

ds

dV

exp 2

V

(s s )

ds

dV 2

ds

dV 4

g

s







  





























 













 































(51)
14















 









































 













  













 

































sub

sub x x

sub

g dropsub

sub

x xsub

sub sub

x xsub

sub

sub

g drop

V

(s s )

ds

dV V exp

V

(s s )

ds

dV

exp

V

(s s )

ds

dV

exp

ds

dV 2

g

V

(52)

Here the retardation at the transition point is determined from equation 53.

sub ref Dsub

sub

V S C

2m

1

ds

dV    











(53)

Here 

CDsub

is the drag coefficient in the subsonic regime.

2.2.6 Extensions of the Method

The method presented here addresses the transition from supersonic to transonic to subsonic 

flight as a specific application of the method. It should be noted that equations and methodology 

presented here could be easily adapted to address situations where the drag coefficient variation 

requires multiple piece-wise descriptions of the drag curve. For example, the equations in 

sections 2.2.2 and 2.2.3 could be applied directly to the case of supersonic flight where the drag 

coefficient is better described by two separate power-law variations. Such applications may 

result from the natural variation of the drag curve over a range of Mach numbers or perhaps from 

the transition between powered and glide phases of flight.

3. Results

The hybrid supersonic/transonic/subsonic trajectory model is applied here to the flight of a 9-mm 

pistol bullet that is launched at a supersonic velocity and reaches transonic or subsonic velocities 

later in flight. This application was addressed in reference 3 with the hybrid model that has an 

abrupt transition between supersonic and subsonic flight. The current model provides more 

fidelity in modeling the transonic regime. 

Comparisons were also made with trajectory predictions made with the 4DOF model within 

Prodas (8) using an existing drag profile shown in figure 4. Also shown in figure 4 is the drag 

variation used in the hybrid supersonic/transonic/subsonic power-law model. Table 1 shows the 

parameters used in the supersonic/transonic/subsonic model. These parameters were evaluated at 

standard atmospheric conditions. A transonic drag coefficient exponent of –5.4 was computed 

from the input parameters.
15

Figure 4. Drag coefficient as a function of Mach number: 9-mm pistol bullet.

Table 1. Parameters for 9-mm ball projectile used in hybrid 

supersonic/transonic/subsonic model.

Muzzle velocity, 

V0

416 m/s

Muzzle retardation, 

0

dV

ds

     

–1.74 (m/s)/m

Drag coefficient exponent, n 0.27

Transition Mach number, Mtran

1.05

Subsonic Mach number, 

Msub

0.85

Subsonic retardation at 

Mtran

, 

sub

dV

ds

     

0.4043 (m/s)/m

Results are also presented here that were obtained previously using the hybrid 

supersonic/subsonic model (3). The parameters used for this model are shown in table 2.
16

Table 2. Parameters for 9-mm ball projectile used in hybrid 

supersonic/subsonic model (3).

Muzzle velocity, 

V0

416 m/s

Muzzle retardation, 

0

dV

ds

     

–1.74 (m/s)/m

Drag coefficient exponent, n 0.27

Transition (supersonic to subsonic) Mach number 0.925

Subsonic retardation at transition Mach number –0.440 (m/s)/m

Figure 5 shows the predicted velocity as a function of range obtained with the supersonic

/transonic/subsonic model compared with Prodas predictions. Also shown are predictions using 

the supersonic power-law and the supersonic/subsonic power-law models. Each of the models 

shows excellent agreement from launch to the sonic velocity. It should be noted that for the 

current application, the results are relatively insensitive to the drag coefficient exponent (for the 

supersonic drag power-law) because the supersonic portion of flight is relatively short. The 

supersonic/transonic/subsonic model shows a gradual change in slope across the transonic

regime that is nearly the same variation shown in the 4DOF Prodas predictions. The current 

model appears to provide a better representation of the velocity variation in the transonic regime 

than the supersonic/subsonic model, which shows an abrupt change in the velocity fall-off as the 

model transitions from the supersonic regime to the subsonic regime. The supersonic/transonic

/subsonic model and the Prodas predictions show similar velocity variations in the subsonic 

regime where the drag coefficient is constant.
17

Figure 5. Velocity as a function of range: 9-mm pistol bullet.

Figures 6–9 show the predicted time of flight, the gravity drop, the wind drift, and gravity drop 

velocity as a function of range. The hybrid supersonic/transonic/subsonic model shows nearly 

the same variation with range as the Prodas predictions for these quantities. The hybrid 

supersonic/transonic/subsonic model eliminates the small discrepancies between the hybrid 

supersonic/subsonic model and the Prodas prediction that are due to the inability of the 

supersonic/subsonic model to account for the proper drag variation in the transonic regime. 

Compared with the velocity variation with range, these differences are somewhat less significant.
18

Figure 6. Time of flight vs. range: 9-mm pistol bullet.

Figure 7. Gravity drop vs. range: 9-mm pistol bullet.
19

Figure 8. Wind drift vs. range: 9-mm pistol bullet.

Figure 9. Gravity drop velocity vs. range: 9-mm pistol bullet.
20

Provided that the supersonic and subsonic drag is adequately modeled with the power-law 

parameters, the most significant issue with the selection of input parameters for the current 

model concerns the span of the range regime (or alternatively the selection of transonic and 

subsonic Mach numbers 

Mtran

and 

Msub

). Figure 10 shows a drag coefficient model with three 

different power-law variations in the transonic regime. In addition to the baseline model 

(transonic regime between Mach 0.85 and Mach 1.05), two additional variations with narrower 

(Mach 0.925–0.975) and wider (Mach 0.8–1.1) variations are shown. The narrower and wider 

variations selected here represent a range of possible power-law models for the actual drag 

variation which cannot be modeled exactly with a power-law variation.

Figure 10. Modeling of drag coefficient as a function of Mach number with different 

power-law models.

Figures 11–14 show the predicted variation of the velocity, time of flight, gravity drop, and wind 

drift versus range for the three different power-law models. The predicted results show only 

minor differences between the predicted results using the various models and represent the range 

of uncertainty with the current model. Though not shown, results obtained with the span of the 

transonic regime between Mach 0.9 and 1.0 showed very little difference with the results 

obtained with the baseline model.
21

Figure 11. Sensitivity of velocity vs. range to selection of transonic power-law 

model: 9-mm pistol bullet.

Figure 12. Sensitivity of time of flight vs. range to selection of transonic power-law 

model: 9-mm pistol bullet.
22

Figure 13. Sensitivity of gravity drop vs. range to selection of transonic power-law 

model: 9-mm pistol bullet.

Figure 14. Sensitivity of wind drift vs. range to selection of transonic power-law model:

9-mm pistol bullet.
23

Compared with the hybrid supersonic/subsonic model, which has an abrupt transition between 

the supersonic and subsonic regimes, the current model allows more direct modeling of the 

transonic drag variation and appears to require less user judgment (and resulting uncertainty)

compared with the hybrid supersonic/subsonic model, which requires the selection of the single 

transition Mach number. However, the transonic drag behavior is often not completely 

characterized and therefore is a subject of some uncertainty itself. For many projectiles, the 

transonic regime may not be of significant enough interest to warrant complete characterization 

of the aerodynamics. Often, an assumed form for transonic drag rise is used, which introduces its 

own uncertainty in the trajectory prediction. As shown in the results presented here, this 

assumption may only produce minor differences in the predicted trajectory.

The selection of the hybrid supersonic/subsonic or the hybrid supersonic/transonic/subsonic 

model may be driven by the need to better define the variation of the predicted quantities within 

the transonic regime—most likely the velocity variation. As the results have shown, the current 

model provides an improved capability for doing that with only small overhead and additional 

inputs.

4. Conclusions

An approach for predicting the trajectory of projectiles with mixed supersonic, transonic, and 

subsonic flight is presented. The analysis presented shows that projectile flight with mixed 

supersonic, transonic, and subsonic flight regimes can be predicted with as few as six

parameters: the muzzle velocity, the muzzle retardation, a power-law exponent that describes the 

shape of the drag curve in supersonic flight, the transition Mach number between the supersonic 

and transonic regimes, the transition Mach number between the transonic and subsonic portions 

of flight, and the subsonic retardation at the transition Mach number (or alternatively the 

subsonic drag coefficient). Closed-form expressions for the velocity, time of flight, gravity drop,

and wind drift are developed to provide critical information about the trajectory of projectiles in 

direct-fire (low gun elevation angles). The current report extends previous work to incorporate 

modeling improvements when the details of the trajectory in the transonic regime are required. 

The method is simple, requires a minimum of input data, and allows rapid prediction of 

trajectory information across the complete span of supersonic, transonic, and subsonic flight.

The methodology and equations presented here are also easily adapted to address other situations 

where the drag coefficient variation requires multiple piece-wise descriptions of the drag curve. 

For example, the equations presented can be applied with little modification to the case of 

supersonic flight where the drag coefficient is better described by two separate power-law 

variations include applications where there is a natural variation of the drag curve over a range of 

Mach numbers or perhaps from the transition between powered and glide phases of flight.
24

5. 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. Weinacht, P.; Newill, J. F.; Conroy, P. J. Conceptual Design Approach for Small-Caliber 

Aeroballistics With Application to 5.56mm Ammunition; ARL-TR-3620; U.S. Army 

Research Laboratory: Aberdeen Proving Ground, MD, September 2005.

3. Weinacht, P. A. Hybrid Supersonic/Subsonic Trajectory Model for Direct Fire Applications;

ARL-TR-5574; U.S. Army Research Laboratory: Aberdeen Proving Ground, MD, June 

2011.

4. McCoy, R. L. Modern Exterior Ballistics; Schiffer Military History: Atglen, PA, 1999.

5. Tilghman, B. A. M855 Aeropack Data, Firing Tables Branch; U.S. Army Armament RD&E 

Center, Aberdeen Proving Ground, MD, 8 December 2003.

6. McCoy, R. L. Aerodynamic and Flight Dynamic Characteristics of the New Family of 

5.56mm NATO Ammunition; BRL-MR-3476; U.S. Army Ballistics Research Laboratory:

Aberdeen Proving Ground, MD, October 1985.

7. McCoy, R. L. The Effect of Wind on Flat-Fire Trajectories; BRL Report No. 1900; U.S. 

Army Research Laboratory: Aberdeen Proving Ground, MD, August 1976.

8. Arrow Tech Associates. Prodas Users Manual; Burlington, VT, 2002.
25

Nomenclature

a

free-stream speed of sound

CD

drag coefficient

V0

CD

drag coefficient evaluated at the muzzle velocity

Dsub C

drag coefficient in the subsonic regime

Dtransonic lower C 

drag coefficient evaluated at the lower end of the transonic regime

Dtransonic upper

C 

drag coefficient evaluated at the upper end of the transonic regime

D

reference diameter

g

gravitational acceleration

m

projectile mass

M

Mach number

M0

Mach number at muzzle

Msub

Mach number at transition between transonic and subsonic flight

Mtran

Mach number at transition between supersonic and transonic flight

n exponent defining the variation of the drag coefficient with Mach number in the

supersonic regime

tran n

exponent defining the variation of the drag coefficient with Mach number in the

transonic regime

s

total downrange displacement

g drop s  gravity drop

sub g drop s  gravity drop evaluated at the transition between transonic and subsonic flight

tran g drop s  gravity drop evaluated at the transition between supersonic and subsonic flight

x y

s ,s

horizontal and vertical displacement along trajectory
26

xsub s

horizontal (downrange) displacement where transition between transonic and

subsonic sonic flight occurs

xtran

s

horizontal (downrange) displacement where transition between supersonic and

transonic sonic flight occurs

z

s

out-of-plane displacement along trajectory (normal to x-y plane)

Sref

reference area, 

4

D

S

2

ref





t

time of flight

sub t

time where transition between transonic and subsonic flight occurs

tran t

time where transition between supersonic and transonic flight occurs

V total velocity

V0

muzzle velocity

Vgdrop

vertical velocity component due to gravity

sub Vgdrop

vertical velocity component due to gravity evaluated at the transition between 

transonic and subsonic flight

g droptran

V 

vertical velocity component due to gravity evaluated at the transition between 

supersonic and transonic flight

Vsub

velocity at transition between transonic and subsonic flight

Vtran

velocity at transition between supersonic and transonic flight

x Vy V ,

downrange and vertical velocity components, respectively

0

ds

dV 











Muzzle retardation

sub ds

dV 











retardation evaluated at upper end of subsonic regime

tran ds

dV 











retardation evaluated at upper end of transonic regime

wz

crosswind velocity
27

Greek Symbols

 0

initial gun elevation angle



atmospheric density
28

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