mathematical model for bullet ricochet

7/22/2019 Mathematical Model for Bullet Ricochet

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Journal of Criminal Law and Criminology

Volume 61 | Issue 3 Article 12

1971

Mathematical Model for Bullet RicochetMohan Jauhari

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Recommended CitationMohan Jauhari, Mathematical Model for Bullet Ricochet, 61 J. Crim. L. Criminology & Police Sci. 469 (1970)
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Tim JounNA, 07 CnmnAL LAw CnaflOLOar AxD PoLmE ScrmoECopyright 0 1970 by Northwestern University School of Law

VoL 61, No. 3Psint in U S

MATHEMATICAL MODEL FOR BULLET RICOCHET*MOHAN JAUHARI

Mohan Jauhari, M.Sc. is Assistant Director, Central Forensic Science Laboratory, Governmentof India, Calcutta, India. Mr. Jauhari is an experienced firearms examiner and has made an extensivestudy of the problem of bullet ricochets. Among the papers which he has published on this subjecthis article, Bullet Ricochet from Metal Plates, appeared in this Journal in September 1969. EITOM.

Bullet ricochet is defined as the deflection of abullet from its course, while maintaining its in-tegrity, as a result of impact on a target of suffi-cient solidarity. Jauhari 1, 2, 3) was the first toundertake a systematic experimental study of thephenomenon of bullet ricochet by firing low veloc-ity handgun cartridges on targets of such diversenature as wood, plastics, metal plates, glass, andbrick. These experimental studies, apart from pro-viding useful bullet ricochet data, suggested guide-lines for building a mathematical model. In thepresent paper a mathematical model for bulletricochet has been proposed. The experimental datacollected earlier has been interpreted in the light ofthis model and experiments have been suggestedto determine the values of unknown parametersinvolved in the equations.

M TMEML TIC L MODELLet a bullet strike a plane target with velocityV, and ricochet with a velocity Vn. We assume thatthe velocity vectors Va, Vi and the normal to thetarget at the point of impact lie in one and the sameplane which may be called as the plane of fire. Let

i and r be the angles of incidence and ricochet asmeasured from the surface of the target in theplane of fire see diagram 1).The incident velocity VI of the bullet can be re-solved into two components; one along the targetVrT) and one perpendicular to it Vn) in theplane of fire. The following relations follow:

Vrr V, Cos i 1)V I = VI Sin i. 2)

Similarly the velocity Vn of the bullet after ricochetcan be resolved into components VRT and VNgiven by the relations:

Read on the 10th March, 1969 during WinterSchool on Forensic Science held under the auspices ofNational Institute of Sciences, India, at Nangal,Punjab.

VET Vn Cos r (3)VEN Vu Sin r. 4)

At this stage we introduce two dimensionless pa-rameters a and 1 defined by

ViT Vi Cos rVIT V1 Cos i (5)VRN VR Sinr (6)ViN ViSini

i.e. they are the ratios of the moduli of componentvelocity vectors after and before ricochet.

Condition for ricochet From equations 5) and6) it can be seen that the condition for ricochet is

a _ 0, ft __ 7)Relations between the angle of incidence and rico

chet Dividing (5) by (6), we ge tTan i a 6 Tan r. 8)

Equation (8) gives the dependence of the angle ofricochet on the angle of incidence in terms of pa-rameters a and fl. It can be seen from (8) that

ccording. (9)

DIAGRAx 1


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MOHAN JA UHAR6Thus depending on the values of parameters a andfi the angle of ricochet can be less, equal, or greaterthan the angle of incidence.Velocity and energy after ricochet Using equa-tions (5) and (6) and eliminating the terms in-volving r, we get

VI V, [02 Cos i + 2 Sin2 i]11. (I0)If W is the mass of the bullet and En its kineticenergy after ricochet, we haveER X WV2 = X WVJ2[a Cos i + 2 Sin2 i]= E, [ Cose i + 02 Sin7 i], 11)where E1 = IV s initial kinetic energy of thebullet. If EL is the loss of energy due to ricochet,

EL Ex - ER= E 1 - a2 Cos2 i 2Sin2 i]. 12)

Impulsive force during ricochet The impact of abullet on a target is in the form of an impulse, i.e. alarge force acting for a very short time. The valueof the average force acting on the bullet can be de-rived by calculating the rate of change of mo-mentum undergone by it during ricochet. If FTand FN are the moduli of the components of thisforce acting along and perpendicular to the target,then

T WVIT WVRT 13)t

andWVIN + WVRNFN =(14)

where t is the time for which the impact lasts. Apositive sign has been used before WVRN in 14)because VnN is directed opposite to VI. Substi-tuting the values of Vrr, VRT, VI & V fromequations 1), 5), 2) and 6) respectively, we get

WVi(1 - a) Cos iT 15)tand

FN WV 1-)Sini 16tThe decceleration corresponding to FT and FNwill be

I FT Vi t - a) Cosi 7S 17) t

fN=FN Vz 1+ 0) Sini

If L is the length of the ricochet mark along theline into which the plane of fire and the target planeintersect, we have

vT - VIT 2fT.L.Substituting the values of VuT, Vrr and fT fromequations 5), 1) and 17) respectively, we get

2Lt VI + a) Cosi 19)Substituting the value of t in equations 15), (16),17) and (18) we get,FT WVI, i a Cos i2L 20)

F, WV] + )(1 + a) Sin, Cosi2L 21)fT IT VI( - a2) Coe i 22)

W 2L (andFN =FN _ VI2 1 +,6)(1 + a) Sin i Cos i 23)

WRelation between the length and depth of ricochet

mark If D is the maximum depth of the ricochetmark then Vn i.e. the initial component of bulletvelocity normal to the target will be reduced tozero at this depth by decceleration fN given byequation (23). Hence

0 V N 2fN.D.Substituting the values of Vm & fN from equations2) and 23) respectively, we get

LTaniD -- -)(1 - 24)Equation 24) gives the relationship between thedepth and length of a ricochet mark.

If t1 is the time during which Vm is reduced tozero and the maximum depth D is attained, then0 V- fx ti

Substituting the values of Vn and fN from equa-tions 2) and 23) respectively we get,

2L= (I + 0) 1 + a)Vx Cos i (25)

[Vol. 6f


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M THEM TIC L MODEL FOR BULLET RICOCHET

If L, is the distance along the line of intersection ofthe plane of fire and the target plane from the start-ing end of the ricochet mark to the point wheremaximum depth of ricochet mark is attained, then

L, Vrr t 3 fr ti.Substituting the values of Vrr t, and fT fromequations 1), 25) and 22) respectively, we ge t

Lfix + P + 1] 26I = (2+ ( /) 26)If L2 is the corresponding distance from the termi-nating end of the ricochet mark, then

L 1 + a + 2a/ )L=L-L 1 1 +a) 1+ )I 27)

It can be seen from 26) and 27) that the posi-tion of maximum depth is asymmetrically placedwith respect to the two ends of the ricochet mark.It is also clear that so long a and P are less thanunity

L, > L2,which means that the position of maximum depthwill be nearer to the terminating end of the ricochetmark as compared to the starting end. At the sametime it can be visualized that, due to the symmetryof the problem, the ricochet mark will be sym-metrical about the line into which the plane of fireintersects the target plane.

EXPERmENTAL DETE=NmoATION OF a iThe mathematical treatment given above is

based on the introduction of two dimensionlessparameters a and i defined by equations 5) and6). It is possible to study their dependence onvarious factors experimentally and determine theirexperimental values. In earlier papers 1, 2, 3) theauthor had devised a simple experimental set up tostudy bullet ricochet. It was possible by this set upto measure the angle of ricochet at various anglesof incidence with sufficient degree of accuracy. Byadopting a similar set up the velocity of a bulletafter ricochet can also be measured by a counterchronograph provided with photoelectric screens.Knowing the incident velocity, angle of incidenceand ricochet the parameters a and Pi can be calcu-lated by the formulae 5) and 6). The velocityafter ricochet will have to be determined in closeproximity of the target using minimum possibleseparation of screens. This is so because a bullet isnot only deformed but becomes unstable after

ricochet and is therefore likely to follow an erraticpath and lose velocity at a rapid rate. In closeproximity of the target, i.e. before the bullet hastraversed any considerable length, it will be travel-ling almost in the true direction of ricochet. Theseconsiderations had earlier prompted the author torecord the point of impact of the bullet after rico-chet on a deal board placed very near the targetand calculation of the angle of ricochet was donewith reference to this point. As the velocity meas-urements will have to be done with minimum sep-aration of screens, a counter chronograph capableof measuring short time intervals will be ideallysuited. It has not been possible to undertake ex -periments of this nature due to lack of chrono-graphing facilities in the author's laboratory. It- is,however, proposed to undertake this work in futurewhen the required facilities are available.

Another practical method by which the values ofparameters a and/8 can be calculated is with thehelp of equations 8) and 24). As i, r, D and L aremeasurable quantities, equations 8) and 24) canbe solved simultaneously for a and P in terms ofi, r, L and D. Solving equations 8) and 24) fora and ft we ge t

Tan i 12Tanr 228Tan i Tanr\2 4LTanr

Tanr Tani + Dn

Tanr 12Tani 2 29)

/ Tanr1Tan / + E Tan rOnly the positive values of a and,6 are of iiterest tous as only then the bullet ricochets.

Thus if i, r, L and D are known the values of aand can be calculated. Once the values of a andare known, the values of Vu, En EL, t etc. are allautomatically known if V, is known in advance.The value of V, is generally known from the datagiven by cartridge manufacturers.

A reference to equation 7) shows that it is pos-sible to determine the ratio a 9 if only i and r areknown. As mentioned earlier this data can be ob -tained by the method used by. he author for bulletricochet studies. It is generally not possible to ob-tain the.values of r for a large number of values of i

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MOHAN JA UHARIat suitable intervals because a bullet fails to rico-chet not only by penetrating the target but also bydisintegration. Sometime both the conditions pre-vail simultaneously.

DISCUSSIONThe mathematical model proposed above isbased on the introduction of two dimensionless

parameters a and i which can be determined ex-perimentally either by resorting to velocity meas-urements or by making measurements on the rico-chet mark left by a bullet on a target. The ratioa/fl can, however, be calculated if only i and cor-responding r are known. To show the order of mag-nitudes involved, the ratio a fl has been calculatedwith the help of i, r data obtained earlier in respectof metal plates (2). The calculations are being givenin respect of X6 n thick aluminium platesfired upon by 178 grain .380, ball, MK2, K.F. re-volver jacketted bullet at an incident velocity of600 /Sec.

Target Uin rin tan RemarksegreesI degrees 9 tan-

Aluminium X6 5 5 3.120 10 2.125 Bullet dis-integrates

Aluminium 15 4 3.830 7 4.735 8 5.040 - Bullet dis-integratesIt can be seen from above that a fl does not remainconstant at different angles of incidence. This im-plies that either a or fi or both vary with the angleof incidence. On the basis of physical reasoningboth a and flare expected to vary. Examination ofricochet marks on metal plates showed that thesewere in the form of dents with permanent deforma-tion of the target at the point of impact. In case ofwooden targets the marks were sometime in theform of dents and sometimes showed the evidenceof bullet penetrating the target material to somedepth. In cases where the marks are in the form ofdents without actual penetration of the target ma-terial the component of bullet velocity along thetarget Vrr) is likely to be reduced by forces pre-dominantly frictional in nature. If the forces arefrictional in nature the variation of a with the angleof incidence is obvious; since as this angle changes

the component of bullet velocity along the targetalso changes. At high velocities such as are en-countered in bullet ricochet the frictional force islikely to vary with velocity. It will also be not inde-pendent of the area of contact between the bulletand the target due to the phenomenon of seizingin which there is mutual transfer of material be-tween the two. If a bullet while ricocheting pene-trates the target material to some depth the forcesretarding the component of velocity along the tar-get are again expected to be velocity dependent. Ineither case variation in the value of a at differentangles of incidence is expected. The parametermay be identified as the coefficient of restitution.As the impact of a bullet on a target leads to de-formation of both and as the deformation is veloc-ity dependent, f is expected to vary with the angleof incidence too. One can visualize that for a sta-tionary target there is nothing to increase the twocomponents of bullet velocity along and normal tothe target Vrr and Vx ). In fact both will be de-creased. Thus both a and l are expected to be lessthan unity. Further, for ricochet a >_ 0 and l 0;hence for ricochet

0_a


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M THEM TIC L MODEL FOR BULLET RI O H T

incidence when fired upon by 40 grain .22, LongRifle, K.F. lead bullet the loss of angle was only 20.When the same experiment was repeated undersimilar conditions on brick the loss was as much as8 .This shows that the ratio a fl was much higherin the latter case as compared to the former.It has been mentioned earlier that the values ofa and f can be determined if i, r, L and D areknown. The length of a ricochet mark can be deter-mined quite accurately. Some difficulty may beexperienced in determining the depth if the ricochetmark is too shallow or if the target gives way atthe point of impact thereby creating a hole withoutthe passage of bullet 1, 2, 3). In an actual crimesituation the value of i and r can be approximatedby knowing the position of the shooter and thevictim with respect to the ricochet mark. L and Dcan be determined by performing measurementson the ricochet mark itself. The incident velocitycan be ascertained from the data provided by themanufacturer of the cartridge used in the commis-sion of crime. All this data can be utilized to com-pute bullet velocity and energy after ricochet. Thiscalculation may be of great help in assessing thewounding power of a bullet after ricochet and thushelp in confirming or refuting a particular theorypropounded to reconstruct the shooting incident.An example from the experiments conducted bythe author 2) is being given here to show how thecalculations are to e performed. During bulletricochet experiment it was found that the 230grain, .45 ACP, Rem-Umc bullet ricocheted at anangle of 12 r) when fired on a % thick aluminiumplate at 25* i) incidence with an incident velocityof 800 ft./sec. Vr). Measurements on the ricochetmark showed that its length was 6.5 cms.(L) andmaximum depth 1.2 cms. (D). Using relations 28)and (29), we find that

a = .85, fl= 8Substituting the values of VI, a, fi and i in equa-tions (10), (11), 12), (15), (16) and (19), we ge tVn 624 ft./sec.; En 6397 foot poundals;EL 4117 foot poundals; FT 11355 poundals;Fj 47839 poundals and t - .003 secs. Thus inthis case the impact of the bullet lasted for .003secs. and the bullet lost 4117 foot poundals ofenergy during ricochet. The loss of energy by thebullet coupled with the fact that the bullet becomesunstable and its ballistic shape is deformed dearlyshows that the lethal range of this bullet is likelyto decrease considerably after ricochet. Further,

if we substitute a .85 and l .38 in equation26) we find that

L 4.8 cms.The value of L, as measured on the ricochet markwas found to be 4.6 cms. which is fairly close to thatcalculated on the basis of mathematical model de-veloped in the preceding paragraphs. In this con-nection it may be mentioned that if we substitutethe values of a and ft from equations 28) and (29)in equation (26), we get a relationship between theangle of incidence i) and the angle of ricochet r).This relationship also involves L, LI and D. Thus ifi L, L. and D are known, one can calculate thevalue of r which will indicate approximately thepath of the bullet immediately after ricochet. Thisdetermination can also be of practical interest in thefield of investigation.

It can thus be seen that all important features ofbullet ricochet observed experimentally 1, 2, 3)are explainable within the framework of this simplemathematical model. It is admitted that the modelis rather over simplified in as much as the wholemathematical treatment is based on particle dy-namics. A bullet is not a mere particle but a bodyhaving finite dimensions. It is also spinning at thetime of impact. It i% therefore, necessary to exam-ine how far the quantitative results obtained on thebasis of this model differ from those observed ex -perimentally. It would be interesting to determinethe velocity after ricochet experimentally by achronograph and see how much it differs from thatcalculated by performing measurements on thericochet marks. If the deviation is excessive themodel will have to be suitably modified for prac-tical work. It has been seen earlier that the value ofL as calculated on the basis of this simplifiedmodel in a particular case does not differ much fromthat obtained experimentally. It is felt that all suchcalculations will have to be performed in a largenumber of control firings and the results comparedwith the experimental values. Work on these linesis in progress.

REFERENCES1. JADHaRI M., Ricochet of Cartridge, S.A., Ball,Revolver, .380, MK2, K.F. Bullet, jona. INDAcAD. FoR. Sc., Vol. 5, No. 1, pp. 29-33, 1966.2. JAviuA7 M., Bullet Ricochet from Metal Plates,JouR. CRam. LAW CHmmN. PoLIcE So., Vol. 60,No. 3, pp. 387-394, 1969.3. JADHmAI M., On the Ricochet of .22, Long Rifle,K.F. Bullet JouR. IND. AcAD. For. Sc. Vol. 9,No. 1, pp. 14-18, 1970.

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mathematical model for bullet ricochet

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