FLEVEL

MEMORANDUM REPORT ARBRL-MR-02891

MUZZLE-BLAST-INDUCED TRAJECTORY

PERTURBATION OF NONCONICAL ANDCONICAL BOATTAIL PROJECTILES

CL.Kevin S. Fansler

Edward M. Schmidt

C = January 1979

m VUS ARMY ARMAMENT RESEARCH AND DEVELOPMENT COMMANDBALLISTIC RESEARCH LABORATORY

ABERDEEN PROVING GROUND, MARYLANDiV

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The fiudings in this report are not to be construed asan official Department - the Army position, unlessso designated by other authorized documents.

rim• ja o/' et.vk name op, mcmfaetwur,' ,mo in th'a :vpo,.tdove we ••et.tucm inrdoemrawen of •ny ocwwr'oial prmodoe.

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," . -.TI.A ... ...... ................... ..... ..... . . . .. S. TVOE OF REPORT & PERIOD COVERED

ZZLE- LAST-JNDUCED ,RAJECTORY .CRTURBATION OF ,NONCON CAL AND CONICAL BOATTAIL PROJECTILE$-- / 1 2 .Final /

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I.Kevin S./ Fansler 90Edward M./SchmidtI

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Spin-stabilized Projectie.4 BTransitional BallisticsFree-flight BaliisticsProjectile Accuracy

/.-•> •zzle-blast loadings on F'pin-stabilized projectiles are analyzed ar.d used tocompute rnsultent trajectory deviations. Both conical and nonconical boattailrounds av:' treated. Approximations are made that permit the expression for theforce on the projectile to be integrated; the resulting equation is used todevelop a universal momentum transfer function that can be directly related toprojectile jump. Although nonconical boattail configurations are more sensitiveto muzzle blast than conical designs, the computed trajectory deviation ineither case is small compared to the total measured dispersion of typical systemsq

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TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS .. .. ...... ............... . ........

1. INTRODU'CTION .. ......... . .............. ............. 7

II. ANALYSIS OF GASDYNAMIC LOADS. .. .. .. .................... 8

A. Region 1: Projectile Base Still Within the Bore . .. 8

B. Region 2: Projectile Base Out of the Gun Tube . . . 11i

III. RESULTS AND DISCUSSION .. ....... . ................ ......12

IV. SUMM4ARY AND CONCLUSIONS....... ... . .. .. .. .. .. .. ...

REFERENCES .. ........... . .............. ................ 19

AIPENDIX A. Expression for Transverse Momentum. .. ......21

LIST OF SYMBOLS. .. ......... . ................ .......... 23

IDISTRIBUTION LIST. .. ....... . ................ ........ 25

.3

I! i

LIST OF ILLUSTRATIONS

Figure Page

1. Flowfield sch'!matic ......... .................. ... 16

2. Experimental and extrapolated lift data for sphere-cones ................ ......................... ... 16

3. Momentum function for the conical S49 boattail ... ..... 17

4. Momentum function for the square boattail ........... ... 17

5. Momentum function for the triangular boattail .... ...... 18

6. Cross-section of nonconical boattail in muzzle plane 18

5

3 3OZI*3PA=I

1. INTRODUCTION

On the modern battlefield, weapon systems must shoot projectileswith a high degree of precision to lessen the chances of enemy returnfire. Ideally, the weapon designer should be capable ofquantifying the sources of dispersion in order to meet imposed pre-cision standards. One source of perturbation to the desired trajectoryiF the gasdynamic force experienced in the weapon muzzle blast. Forfin-stabilized projectiles, dispersion caused by the blast environmenthas been investigated 1,2; however, treatment of spin-stabilizedprojectiles has been largely neglected. With the introduction ofnovel boattail designs3 ,there is also interest in comparing thesensitivity to muzzle blast of the nonconical boattail configurationswith the sensitivity of the conical boattail configurations.

The present paper eramines the influence of muzzle-blast loadingsupon the trajectory of spin-stabilized rounds. With both conical andnonconical boattail configurations, the analytical approach is similar tothat used previously for fin-stabilized projectiles: namely, the mostsignificant gasdynamic loadings are assumed to occur within the quasi-steady, supersonic core of the propellant gas jet. Pressure distribu-tions on the projectiles are estimated utilizing steady-state flowtheory and experiments. This procedure is not locally exact; however,it produces a reasonable estimate of the overall muzzle-blast-inducedimpulse. Transverse angular and linear momentums imparted by theblast are calculated for triangular, square and conical boattailsand used to determine the contribution to the dispersion of 155mm,M549 type projectiles.

1. E. M. Schmidt, K. S. Fansler, and D. D. Shear, "TrajectoryPerturbations of Fin-Stabilized Projectiles Due to Muzzle Blast,"Journal of Spacecraft and Rockets, Vol. 14, No. 6, June 1977,pp. 339-344.

2. K. S. Fansler, and E. M. Schmidt, "Trajectorj Perturbations ofAsymnetric Fin-Stabilized Projectiles Caused by Muzzle Blast,"Journal of Spacecraft and Rockets, IVl. 15, No. 1, January-February 1978, pp. 62-64.

3. A. S. Platou, "An Improved Projectile Boattail," BRL MR 2395,US Army Ballistic Research Laboratories, Aberdeen Proving Ground,Maryland, July 1974. AD 785520.

L7

FA=Vs u mJ.

ng Il. ANALYSIS' OF GASDYNAMIC LOADS

Two distinct regions are considered. The first region for whicha model is developed covers the flow following exit of the rotating/obturating band fron the gun tube. When this band passes the muzzle,the gas seal is released, but the boattail is still within the tubeand the forward portion of thM projectile is immersed in the expandingpropellant gases. The existence of asymmetry in this fiow field dueto projectile angle of attack results in transverse loads. For conicalboatcail rounds, these loads alter the transverse momenium; however,due .o the c lindrical 'wheelbase' characteristic of non-conical boat-tail designs , momentum exchange occurs only when the non-planarsurfaces are not contacting the bore. The second region of interestcommences as the projectile base clears the muzzle and terminates asthe base passes through the Mach disc of the propellant gas jet.

In both regions, the analysis seeks an upper-bound estimate ofthe propellant gas loadings. This approach is supported by previousupper-bound estimates 1 ' 2 showing that muzzle-blast- induced dispersionof fin-stabilized projectiles is a negligible component of the totplmeasured dispersion levels. Thus, it was not worthwhile to seek moreaccurate, but lower magnitude, estimates.

A. Region 1: Projectile base still within the bore

A schematic of the flow field is shown in Figure 1. For con-venience, the flow behind the projectile is assumed to be sonic (occursfor V a 700 m/s). At separation of the obturating band, the pro-

pellant gases are released to expand into the atmosphere, generatingthe characteristic muzzle-blast wave. Due to the presence of theprojectile, this free expansion is constrained and if the projectileis at an angle of attack, a1 , relative to the bore-line, the propellant

gases will be deflected. Assuming that the projectile acts somewhatlike a plug nozzle and that all of the flow is deflected through theangle a1 at the sonic line, the resultant transverse lift force cn the

projectile can be simply estimated from momentum considerations:L1 = (p* + p*V. Au4 (I)

ul

where Au is the area of the muzzle that is unplugged (a function of

boattail geometry), p* is the pressure at the muzzle, p* is the gasdensity at the muzzle and V* is the gas velocity at the muzzle.

Equation (1) is an estimate for a stationary projectile. It maybe corrected to account for projectile velocity, Consider a coordinatesystem moving with the projectile. The velocity of the fluid in thissystem is given as Vr. If the fluid near the projectile is deflected

r

8i

to flow parallel to the projectile surface, the transformation betweenthe stationary and moving coordinates is

Vr exp (ia) =V exp (ia) - Vp (2)

where a is the turning angle for the fluid in the gun-tube coordinatesystem, V is the local velocity and V is the projectile velocity.

* pWe obtain:

(I = ( V /V) a1 (3)

The largest value for a corresponds to the Mach number of the fluidapproaching infinity. For Vp = V*, the limiting value of V is 3 V

p pand thus,

a = 2a /3 (4)

Thus to correct the lift force for projectile motion, we replace a 1 by2a /3 in Equation (1). Using the isentropic flow relations, we

obtain

2L= (y+l) p* AuaI (5)

The transverse momentum transferred to the projectile during thetime between obturator separation and base emergence from the muzzle isthe integral of the lift force over the time of passage. Transformingthe variable of integration from time to distai,,e, t = (X/D) (D/Vp),

pallows the momentum integral to be written:

2 (y+l) (DIV p)p*a, IAJ(X/D) (6)

where X is the distance traveled since unplugging started and D is thebore dlameter of the gun tube.

Nonconical Boattails.

The value of P1 depends upon the boattail geometry because of

the appearance of the vent area term, Au, under the integral sign.

This expression is addressed in detail in Appendix A where it isfound for nonconical boattail.- that

P1 = (16n/45) (y+l) (p*D3 /Vp)803 /2a 1 (X/D) 5/

2 (7)

where n is number of plane boattail surfaces and a is the boattailangle. The total amount of momentum transferred to the projectileby the venting flow is obtained by evaluating this expression at

k the maximum travel of the round, i.e., when the projectile base is at

I.

.. .........-...7,.. . .." .

the muzzle exit plane,. For nonconical boattails that terminate in apolygon shape:

PIt = [5 /(90n ) (y+l)[pD 3/(0Vp)]a (8)

It is useful to define a nondimensiona) momentum transfer function,P, similar to that developed in Reference 4,

T P/[(D/V )(Y+l)p*Aeqa 1] (9)

For nonconical boattail projectiles, the surfaces are treated asminiature airfoils. In Reference 4, it was shown that the equivalentlifting surface area for a multiple-finned missile is A n= A/2 for

the lift vector in the angle-of-attack plane. Since the nonconicaldesigns have only one surface to the "airfoil", the following definitionis used:

k ei1 = PA/4. (10)

where A is the area of each plane surface.

This definition permits the momentum transfer functions to bewritten (See Appendix A, Equations A9 and All)

TP = (128/15)(n/r) 3 (OX5D)5/2 (11)

and

P = (47 2 )/(15n 2 ) (12)it

Conical Boattails.

Using the same approach, the momentum transfer function for aconical boattail design is

P1 = (n/3) (y+l) (p*D 3/Vp)O6a1 (X/D)2 (13)

2and in nondimensional form, with Aeq •D /4:

4. K. S. Fans Ler and E. M. Schmidt, "The Influence of MuzzleGasdynamics Upon the Trajectory of Fin-Stabilized Projectiles,"BRL R 1793, U. S. Army Ballistic Research Laboratory, AberdeenProving Ground, MD, June 1975. AD B0O5379L.

I

i ...........

I (4 0/3(X/D)) (14)

B. ýeion 2: Projectile Base of the gun tube

Nonconical Boattails.

There are no data available describing the aerodynamics ofnonconical boattails in reverse flow. Since the three-dimensionalnature of the muzzle flow makes direct computation difficult, the liftforce on the boattail will be estimated using two-dimensional airfoiltheory. This approximation neglects any effects of base bluntnessdue to truncation of the boattail; however, the side force cn a bluffbody is generally smaller than that on a slender body at an equal angleof attack. Therefore, the current approximation should produce anupper bound on the transverse force exerted on these nonconicalsurfaces by muzzle gas loadings. The value of P2(X/D) can be obtained

[2directly from earlier results using the definition of A in

eqEquation (10). The center of force is taken as the area centroid forthe plane surfaces. This center of force value would be correctfor two-dimensional supersonic airfoil theory but may be quite differentfrom the possible actual results. Nevertheless, for this analysis. theapproximation should be sufficient since the percent error betweendiffering possible moment-arm values would be small.

Conical Boattails.

The transverse force on conical boattail projectiles is estimatedusing experimental datas acquired on blunt, flared sphere-conesat a variety of Mach numbers. These data were obtained using a sphere-cone where the ratio of the diameter of the cone at the sphere-cone junction to the base diameter was 0.8. The smallest cone half-angle of this set of data was 100. For the purposes of the presentanalysis, it is desired to compare the muzzle blast sensitivity of a7 nonconical boattail with the sensitivity of a similar half-anglecone. Thus, the data 5 are extrapolated to this angle, Figure 2. Thecenter of force is taken to be halfway along the boattail length.

5. A. D. Foster, "A Compilation of Longitvdinal AerodynamicCharacteristics Includin9 Pressure Information for Sharp andBlunt-nose Cones Having Flat and Modified Bases," SandiaCorporation Report No. SC-R-64-7311, January 196.5.

i'

II1. RESULTS AND DISCUSSION

The M549 projectile is used as a basis for the computations sinceflight data are available on this round with both the standard 7° boat-tail and a triangular boattail. The properties of the projectiles aregiven in the tables below.

Table I: NS49 Parameters Independent of Boattail

M = 43.0 kgP

D = 0.155m

V = 670m/sP

p* = 1.52 x 107 Pa

y = 1.25

CM = 3.30a

CL = 2.95L

Ix = 0.13 kg-ra2

Table 11. MS49 Parameters Dependent on Boattail

Standard Triangular SquareConical NCBT NCBT

A (M2) 0.0189 0.028 0.016eq

@(deg) 7 7 7

AW(m) 0.295 0.293 0.340

0 m) 0.097 0.347 0.195

2I y(kg-m2) 1.93 1.71 1.71

Here A is the distance between the force acting on the planar surfaceand the center of gravity;Z is the length of the boattail.

The values of P through both the in-bore and exterior regionsof interest are shown in Figures 3-5 for the three boattails considered.

12

i. ilL•............................. ........................ ~.....,, .... •,'L~a. .~

The coordinate X is the location of the projectile center of pressurealong the bore axis, with X = 0 corresponding to the muzzle. Sincethe momentum transfer function is cummulative, the asymptotic value ofeach curve is equal to the total momentum P transferred in the blast.

From Equation (14), it is apparent that the Pit for the conical boat-

tail has an explicit dependence on Z/D and 0; however, for the non-conical boattails considered (those taken back to the intersection ofplanar surfaces), this is not the case. Pi and P are independent of

it tall boattail and projectile parameters, i.e.r the curves are 'universally'applicable to any projectile with an n-sided nonconical boattaillaunched with the velocity of the exit-gas speed of sound.

The momentum functions are used to compute the transverse angularand linear velocities imparted to the projectiles in the blast region,using the approach of Reference 4. These velocities are input to theaerodynamic jump relations 6 for conical boattails, resulting in

E/a1 =lI+(CL A)/(CM D)] (y+l)p*A 2D/(MpVp)I t (i5)1 l(Leq '(p~p t

Fcr nonconical boattails, the working equation, using Equations (10)and (A12), is

3 3 2 -2O/al = [71 (y+l)/ 2 4 ] [p*D /(MpVp )] [ l-(CL A)/(CM D)]I[Pt/n 0)] (16)a a

An expression for the first maximnum yaw may also be derived6 : (17)

ICmaxI/a, = [2(y+l) A eqAD -2rpD CM /(21Jy)

For nonconical boattails, the working equation is, using Zquations (A12)and (10),

I~ax/a =I 4 (1 2 2I¢maxI/a I f 7 3(y+l)/12] [AD p* Pt/(lyV n 0)]

• [(Ix't/1) - TpD SCM /(21 )1½ (18)2ay

Here o is the initial spin in radians per caliber.

6. C. H. Murphy, "Free Flight Motion of Symmetric Missiles,"U.S. Army Ballistic Research Laboratories Report Number 1216,Aberdeen heoving Ground, Maryland, July 19G3. AD 442757.

13

fhho projectile parameters from Tab' ;s I and I I with the above

relations are used to determine the va,;.e• in the following table:

Table III. Launch Dynamic Characteristics

Standard Square TriangularConical NCBT NCBT

P 0.21 0.52 0.48t

k (deg/deg) 0.167 0.37 0.57Imax I /"lO/a! (mil/deg) 0.051 0.11 0.18

kýmaxl (degs for lowith a= 0.1°) 0.02 0.04 0.06

Comparing these results forc/aI with those obtained for fin-stabilizedF 1

projectiles1 , we find that the values are similar.

Table III shows that the jump sensitivity of the nonconicalboattail rounds is more than twice that of the standard conical boattail.However, the last row gives the contribution to the first maximum yawof this niuzzle-blast-induced jump. For a reasonable distribution ofthe in-bore yaw level (lo,a 1 0.1°), the resultant contribution to

the first maximum yaw is 0.060 in the worst case. This value is judgednegligible compared with the frequentlg observed first-maximum yawvalues for the M549 of approximately 5 . Since the first-maximum-yawvalue is proportional to the aerodynamic jump, we may infer that muzzleblast contributes only a small amount to the total observed dispersion.

IV. SUMMARY AND CONCLUSIONS

A model is developed to determine the effect of muzzle gasdynamicloadings upon the trajectory of spin-stabilized projectiles for bothconical and nonconical boattail configurations. The analysis consistsof two parts. First, the passage of the boattail out of the gunfollowing separation of the obturator is addressed. Second, the regionexterior to the muzzle but prior to the projectile's passage throughthe Mach disc is modelled. The conical boattail aerodynamics aredetermined from experimental data acquired on sphere cones; two-dimensional airfoil theory is applied to approximate the loadings onnonconical boattails.

Expressions for t'.. loadings :re integrated to determine themomentum transferred to each of the M549 projectile configurationslaunched at a muzzle velocity of 670 m/s from a 155mm gun. The nor,-conical boattail configurations are found to be more than twice as

14

do

sensitive to blast as the conical design; however, in neither case isthe contribution to dispersion due to muzzle gasdynamic loadingssignificant when compared to the total dispersion of the gun system.

r

15

I.!

SONIC LINE

t '~BLAST WAVE

PATH LINES

MUZZLEEXIT PLANE

Figure 1. Flowfield schematic

3.0--EXPERIMENTS- REF. 8

- --- XTRPOLTEDCURVE

.0-

sUj

Uj

.0 102030405

MACH NUMBER

Figure 2. Experimental and extrapolated lift data for sphere-cones

16

0. 25-

I: ~0.2 --

0.15-

I• 8.85f' 0. e

8-1 9 1 2 3 4 5 6

X IDe

Figure 3. Momentum function for the conical 549 boattail

0.6-

0.5

0.4 -

0.3

0.2 + IN-BORE PHASE- EXTERIOR PHASE

0 1 1 ,1. -

-1 0 1 2 3 4 5 6

,( Xo/D

f Figure 4. Momentum function for the square boattallf17 A

........................ •

1 0.5

0.4-

0.3

0.2 + IN-BORE PHASE

,- EXTERIOR PHASE

0[1_

[1 -2 -1 0 1 2 3 4 5I•: Xe /D

Figure 5. Momentum function for the triangular boattail

1,

I

I'

Figure 6. Cross-section of nonconical boattail in muzzle plane

18

i•, r-.

S. ....... . .. . ,, ... .. .. .. , .. ,, . . = ,•. , ,i,• •. =,u• L ;

-. . .

". . .. . .: .- .....

REFERENCES

1. E. M. Schmidt, K. S. Fansler, and D. D. Shear, "TrajectoryPerturbations of Fin-Stabilized Projectiles Due to Muzzle Blast,"Journal of Spacecraft and Rockets, Vol. 14, No. 6, June 1977,pp. 339-344.

2. K. S. Fansler, and E. M. Sch:didt, "Trajectory Perturbations ofAsymmetric Fin-Stabilized Projectiles Caused by Muzzle Blast,"Journal of Spacecraft and Rockets, Vol. 15, No. 1, January-February 1978, pp. 62-64.

3. A. S. Platou, "An Improved Projectile Boattail," BRL MR 2395,[ US Army Ballistic Research Laboratories, Aberdeen Proving Ground,Maryland, July 1974. AD 785520.

4. K. S. Fansler and E. M. Schmidt, "The Influence of MuzzleGasdynamics Upon the Trajectory of Fin-Stabilized Projectiles,"BRL R 1793, U.S. Army Ballistic Research Laboratories, AberdeenProving Ground, MD, June 1975. AD B005379L.

5. A. D. Foster, "A Compilation of Longitudinal AerodynamicCharacteristics Including Pressure Information for Sharp andBlunt-nose Cones Having Flat and Modified Bases," SandiaCorporation Report No. SC-R-64-1311, January 1965.

6. C. H. Murphy, "Free Flight Motion of Symmetric Missiles,"U.S. Army Ballistic Research Laboratories Report Number 1216,Aberdeen Proving Ground, Maryland, July 1963. AD 442757.

19

19

APPENDIX A. EXPRESSION FOR TRANSVERSE MOMENTUM

In order to evaluate the integral in Equation (6), it isdesirable to obtain closed-form expressions for A

U

A. Nonconical Boattail

Figure 6 shows the cross section of a boattail in the muzzleplane. The bore radius is R and the area of the circular segment canbe readily determined to be

SR2RA (a - sin2a/2) (AI)Ul

The expression in brackets may be expanded in series yielding:

Au1 = D2 3a a 2 a- **)(2i~ Au5u '6- (a -T + I-0 a .(A2)

In keeping with the upper bound approach,the series is truncated afterthe first term to give

A =D2a/36 (A3)u1

The distance BC, Figure 6, is

BC - R[l-cos(a)]

T (a2 _ a4/12 + .. )(A4)R2 4

However, BC is also related to the boattail angle and the length, X,of the nonconical surfaces protruding beyond the muzzle plane.

BC =_X tan8 (AS)

Assuming small e and taking the first term in the series ofEquation (A4) gives

a2 = 40 X/D (A6)

which can be substituted into Equation (A3):

A (4D 2 /3) (X/D) 3 2 0 3/2(A7)

For an n-sided boattail,

A nA =(4nD2 /3)(X 3/2 3/2 (A8)u u1I

21

!.Ii..,,.

j -.-- -~ P -MM-PA." Aa&.*

This formulation for A may be substituted into liquation (6)uand integrated to provide

P1 = (y+l)[16 nD3 p*al/45 V 1 O3/2 (X_/)S5 2 (A9)p

If the boattail terminates where the planar surfaces intersect, then

am = it/n. (A10)

This permits the evaluation of the total momentum transferred in-bore:

P = (n5/90 n4 )(y+l)(p*D3/AV ) a1 (All)

This type of boattail has a surface area that can be readily computed.Since the surface is an ellipse formed by intersecting the cylindricalbody with a plane at angle 0 to the axis,

A = A /0 (evaluated at the muzzle station)

= D2 (Tr/n) 3 /(60) (A12)

B. Conical Boattail

The unplugged area 4s

2 2Au = ir(R - r2) (A13)

where r is the diameter of the section of the boattail passing themuzzle plane. Since R - r = X tan a :X 0,

Au =r(2R - X 6) Xe , (A14)

which may be substituted into Equation (6) to give

[l = (7/3) (y+l) (p*D 3/Vp) 0 (X/D) 2 1 (A1S)

22

L. ....

LIST OF SYMBOLS

a =angle defined in Figure 6

A = area of a plane surface of the nonconical boattail

A = -the equivalent area of an airfoil. in two-,eq dimensional flow

A = the area at the muzzle that is unpluggedu

C DR = projectile drag coefficient in muzzle blast

CL slope of -che lift coefficient curve for forwarda flight

CM f slope of the static moment coefficient curve ina forward flight

D~ th., diameter of the bore and projectile

I =axial moment of inertia

I =transverse w-iment of inertiaY

L = the lift force on the projectile

L 1 = lift force on projectile In the first part of itstransit

£ = length of boattail

M p= projectile mass

n = the number of plane surfaces possessed by the boattail

p =pressure

P =the momentum given the projectile .

Pl momentum given the projectile during the first part ,:of the projectile's transit :

S= the value of P nondimensionalized according toEquation (10) :'

Pt the total amount of P imparted by the muzzle blast

t;

t

&23

LIST OF SYMBOLS (CONTINUEI))

t = time variable

V = local velocity of gases

V = velocity of projectilep

V = velocity of gases relative to a coordinate systemr located on the projectile

X =position along the axis of the center of force

for the projectile relative to the muzzle

X = distance traveled since unplugging started

y = axis coordinate in the angle of attack plane that isperpendicular to the x-axis

= maximum angle between projectile and gun-tube axis--angle of attuck

= angle determined by a and fluid flow veloci~y relativeto the ptojectile

= angle of attack upon emergency from muzzle

= angle cf attack upon emergency from muzzle blast

a, = derivativ, of a w•.th respect to distance in calibers

a/ = value of a' upoi emergency from muzzle bl.st0

= rr.tio of specific heats

A = distance from center of force to center of gravity forflight through muzzle-blast region

= comrleY angle of attack

= roll angle of projectile

= boattail angle

e = aerodynamic jump

p local density of gases

24

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