ROFESSOR MIKE INVESTIGATES...

THE BALLISTICCOEFFICIENT EXPLAINEDProfessor Mike Wr,ight provides a technical explanation of ballistic coefficientand how it affects our pelletsfo most shooters the idea of a

| ' ba l l i s t i c coe f f i c i en t ' (BC) i sbo th f am i l i a r and ba f f l i ng a t t hesame t ime, Many of us regular lyuse software packages which usethe ba l l is t ic coef f ic ient o f ourchosen pe l le t to ca lcu la tetrajectory, energy retention andother var iab les a t var ious ranges,but how many of us rea l lyunders tand what the BC of a pe l le t

actua l ly is? In essence, thebal l is t ic coef f ic ient o f a pe l le t is ameasure of how wel l i t res is ts a i rdrag and re ta ins ve loc i ty as i ttravels downrange: the larger thecoef f ic ient , the lower the ve loc i tyloss. The bas ic sc ience of how a i rdrag af fec ts pe l le t ve loc i ty is qu i te

straightforward, but i t has beenobscured by the long history ofbu l l e t ba l l i s t i c s , f o r wh i ch t heconcept o f the ba l l is t ic coef f ic ient

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In the 1860 's , a Br i t ishclergyman, the wonderful ly namedFrancis Bashforth, was conductingvelocity loss experiments on art i l lerypro ject i les us ing a ba l l is t icpendulum. Owing to the d i f f icu l tyof measuring down-range velocit iesfor a whole range of bul lets, heconceived the idea'of a 'standard

bul le t ' f rom which the ba l l is t icperformance of al l other bul letscould be ca lcu la ted, by sca l ing upor down, without recourse to furthertests, Consequently, he proposed

that the bal l ist ic coeff icient (or,

more specif ical ly, i ts reciprocal the'drag factor ') be based on astandard cyl indrical project i le whichwas one inch in d iameter andweighed one pound. The BC of th iss tandard bu l le t was def ined as 1 .0 .

This is a very long way from theai rgun pe l le t , which has muchsmal ler d imensions and weighs inat around one 500th or"so of theweight of the standard bul let. l t isnot surprising therefore that theBCs of the pellets we use are very,very much lower than 1.0. Thevalue of BC for most avai lablepel le ts fa l ls w i th in a span of va luesranging f rom about 0 .01 to 0 .04,depending on the ca l ibre , weightand shape of the pel let concerned.In practice, the actual BC of thepel le t in f l ight a lso depends on theri f l ing pattern of the gun's barreland the mechanism of oowerdel ivery; spring or PCP.

In many ways i t is unfortunatethat the bal l ist ic coeff icient idea,which has subsequently beenrefined and redefined by a variety ofresearchers, was ever carried over

in to the a i rgun f ie ld ; i t would bemuch simpler to work with

Figure 2: Velocity Loss due to Air Drag

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straightforward drag factors likeevery other branch of engineeringscience. But the notion of theball istic coefficient is now firmlyembedded in the l iterature andsoftware of airgun ball istics and weneed to understand what it reallymeans. The best way to get a handleon what the BC is all about is totake a look at the way in which airdrag affects a pellet in fl ight.

&ilp ffiraRFigure 1 i l lustrates the fl ight of apellet which has been fired from ahorizontally aligned barrel. Windeffect apart, the pellet is subjectedto two external forces: a constantvertical force due to gravitY and avarying force due to air drag. Thecombination of these forcesoroduces the familiar downward-curving trajectory of the pellet shownin the figure.

ln order to understand how thegravity and drag forces combine toproduce a pafticular trajectory, weneed to know how air dragvariesduring the pellets' f l ight.

The drag force increases withvelocity in a way that depends on the ffi

Figure l: Air Drag

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nature of the air flow over the pelletsurface. For the subsonic regime,with air f low velocit ies substantial lybelow Mach 1 ( i .e, for velocit ies upto about 950 fVsecond) that appliesto virtual ly al l airgun pel lets, the dragforce increases with the square of thevelocity of the pel let. Since thedeceleration of the pel let, or the 'air

drag', is proportional to this force,the fol lowing simple formula appl ies:

Air Drag =Change in Velocity _ Vetocityz

Tine Ballistic Ratngeof Pellet

The constant in this for:mula, the'Bal l ist ic Range of the Pellet ' , is veryclosely related to the bal l ist iccoefficient, BC, as we shall see.

In practical shooting we are, ofcourse, more interested in the wavthat pel let velocity varies withdistance rather than t ime and theair drag formula is better recast inthose terms:

Change in Vetocity _ DistanceVelocity Ballistic Ranqe

0r o/o.Reduction in Velocity =

of Pellet

lA0 x DistanceBallistic Ranoe

of Pellet'

Note that, for a given distance,the percentage reduction in velocitydue to air drag is constant. This isthe essential point to grasp aboutvelocity retention and it is the keyfact that determines the shape ofthe velocity/ distance curve. Toil lustrate this point, let's look at anexample for a very poor pellet.Consider a pellet with a muzzlevelocity of 600 fVsec, which isslowed down by air dragto 492ft/sec at 10 yards range i.e. thepellet has lost 18 % of its velocityin 10 yards. Over the next 10 yards

i t wi l l lose another 18%, arr iving átthe 20 yard mark with 403 ftlsec,At 30 yards, the velocity wi l l be afurther 18% less at 337 ft/sec andso on, in s imi lar fash ion, as thepellet moves down range. Figure 2shows how air drag robs the pel let ofvelocity as it moves down range.

HxpmlrerltimB Veimcüty H-*ssThis pattern of constant percentageloss over a given interval governsmany phenomena in the world aroundus. For example, it is the way thatradioactivity decays and is the basisof the carbon dating method for'ancient artefacts. lt is also the patternfor ring separation in a tree trunk andshows how the value of moneydeclines during a period of constantinf lat ion. l t is cal led an exponentialdecay curve and the formula for italways contains the same number

number 3.1416, which is label led aspi ( ) because it comes up sofrequently. Using'e',,the velocityformula is:

Muzzle VelocityVelocitY at Range'd'= --¡------,

elBail¡stlc?ilge '

This is all very well, but we need toknow what the Ballistic Range of aparticular pellet is before we can doany calculat ions. By putt ing thevelocity formula into logarithmic form,we find that the following formulaemerges for the Ballistic Range:

dBallistic Range =

,-tn(ffi¡,)

Where ' ln ' denotes the naturallogarithm of the velocity ratio

You can f ind the 'ex' and ' ln '

buttons on al l scienti f ic calculators,including the one that I purchasedlast week for a pound from a well-

We can now use this value toestimate the velocity at anotherrange, say 35 yards:

vetocity at 3s rards = frl=

29}ntse,

This velocity value is marked up ,Figure 2, where it can be seen thatl ines up precisely with the 35 yardrange. The Ball istic Range formulaprovides us with a simple method oconverting two measured velocities,taken at two different ranges into ameasurement of the value of theBall istic Range.,But how does thisconstant, the Ball istic Range tie upwith the commonly used'' Ball isticCoeff icient'?

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(2.7I83 to four decimal places)irrespective of the field of application.The equation for the velocity declineof our pel let with distance is:

Velocity at Range'd' = #ffi

Because the number 2.7783'occurs over and over again in al lforms of dynamic analysis, it is agiven the 'shorthand' label 'e ' , where'e' is shorthand for 'exoonential

constant ' . This is just l ike the

known discount chain. Thesefunctions are also bui l t into a varietyof computer spread sheet packages.

Let's take a look at how theseformulae apply to our example. Wewil l f i rst f ind the value of the Ball ist icRange. We know that the muzzlevelocity of 600 fVsec has fallen to403 fVsec at20 yards, so theBall ist ic Range wil l be given by:

Battistic nanle = ffi

= Slyar¿s

Ballistic Coefficient, l th:ink it 'shelpful to tr,ace a l itt le bit of thehistory behind it.

Al though the not ion of a bal l is t icoefficient in its present form, asmeasure of bul let f l ight ef f ic iency,originated with the ReverendBashforth, it was Professor PeterGuthr ie Tai t , working in Cambridgrand Ed inburgh, who pub l ished thefirst accurate trajectory formulae ithe 1860s, inc lud ing the idea o fBal l is t ic Range. Some 20 yearslater, the Krupp Company inGermany, made the first accuratemeasurements on the effect of airdrag on bullet trajectory in the1881. This was done by test f i r inglarge flat-based, blunt-nosedbullets manufactured to a standar<design. Fol lowing the Kruppexper iments, mi l i tary engineers inRussia (Mayevski) and l ta ly (Siaccworked to derive a mathematicalmodel to predict bullet trajectory,apparently unaware that ProfessorTate had already broken the backof the problem (there was noGoogle in those daysl). TheMayevski modell was subsequentlytaken up by Colonel James Ingal lsof the United States Army, who

74m

TECHNICAL AIRGUN

at tempted to s impl i fy Mayevsk i 'sresults for use by non=mathemat ica l army personnel . l twas Ingal ls who ra t iona l ised thebal l is t ic coef f ic ient as a measure o fthe ba l l is t ic e f f ic iency of anypro ject i le , re la t ive to a s tandardbu l l e t and pu t t he BC i n t he f o rmsti l l used today. The results werepubl ished in what are s t i l l knowna s ' l n g a l l s T a b l e s ' .

The idea of sca l ing the ba l l is t ice f f ic iency of a s tandard bu l le t todetermine the performánce ófanother chosen pro ject i le ( i .e . ana i r gun pe l le t ) requ i res that thephysics of the problem be takenfu l ly in to account . I have prev ious ly

wr i t ten a deta i led ar t ic le on th is inAi rgun Spor t , a summary o f whichis as fo l lows:

The air f lowing over the pel let

creates a dynamic air pressure,

which depends on the square of thevelocity of the pel let and the densityof the air. This is depicted l

diagrammatical ly in Figure 3.This pressure multiplied by the

effective frontal area of the pellet gives

the drag force on the pellet, which,when divided by the pellet mass,determines the deceleration air drag;

PettetEffective Area .. Vetocitv,

AIr Drag = prttrt uuu

x atr oensnY z:

The term in square brackets iseffectively the reciprocal of theball ist ic coeff icient, which is usuallygiven,as:

Pettet MassD l r - -

and is always calculated in lb- inchunits.

The factor; 'i 'is the Form Factor ofthe pel let, which determine5 i tseffective frontal area and the amountof drag.

Using BC and compar ing theformulae for a i drag i t should beapparent that the Bal l is t icCoeff icient is simply a scaledvers ion of the Bal l is t ic Range. l f wefactor in the correct units, anappropriate f igure for air densityand the correct base for ' i ' i t worksout that :

Battistic coefficient - Ballistic Range oarto8000

So for our example pel let, the BCis 50/8000 = 0.0063. ( l said that i twas a poor pel let!) Let 's summarisethe key facts so far:

ffi At subsonic speeds, ai dragvarieswith the square of pel let speed.

ffi Velocity reduction with increasingdistance is such that a constantpercentage of velocity is lost foreach equal increment of range.

ffi The velocity versus distance curveis an exponential decay.

ffi The rate of decay depends on theBal l is t ic Range (BR) o f the pe l le t ;the higher the BR, the better thepellet retains its velocity.

ffi The BR can be found from '

measuring the pel let velocity attwo range points.

ffi The Batlistic Coefficient is theBall ist ic Range in yards divided by8000,

ffi mm*unir'*g tfrts ffi mfr $istl*#neff icierutIn his companion art icle in this issue,

Jim Tyler recounts the trials andvicissitudes of, measuring the ballistic

coefficient for a particular pellet. You

might think that it is unnecessary togo to the trouble, because the BC's of

a vast range of pellets have alreadybeen published on the internet and

are available at the click of a button.

You might indeed argue that point, butyou would be wrong. The ballistic

coefficient of a pellet depends on itsprecise weight and dimensionaltolerances and its exact shape after it

has been fired from a barrel. So, with

the best wi l l in the world, thepublished figures can only be regarded

as an approximation within plus or

minus about 15%. To get an accuratefigure, you really need to measure it

for your'orrrrn pel leVrif le combination.The most direct method for

rneasuring the bal l ist ic coeff icient is

to fire a oellet over twochronographs which are separated

by a known distance, 1d' and to use

the formulae: '

Velocity Ratio =Velocity fron Nearest Chrono

Velocity from Furthest Chrono

Ballistic Coefficient =800 x In(Velocity Ratio)

Figure 4 shows a very convenientset of curves which do the maths foryou for three typical chronograPhseparation distances; 15, 20 and

25 yards. They are very easy to use.Figure 5 is a diagram of the processinvolved. Two chronos are set up acarefu I ly measured distance apart.The distance 'd ' is the separat ion inyards between the centres of thesensors of the two instruments. Thenearest chrono (red in the diagram)gives a velocity reading V1. The,further (green) chrono reads V2. Tof ind the veloci ty rat io, Vl is d iv idedby V2. l f the distance 'd ' is 15, ,20or 25 yards, the value of BC canthen be read off the appropriatecurve. For example, suppose the Vlreading was 750 ft lsec and thereading on the second chrono, 20yards away, V2, was 647 ft/sec. Thevelocity ratio is 750/647 = 1.16.The bal l is t ic coeff ic ient (0.017) cannow be read directly off the 20 yardcurve in Figure 4. ,

Alternatively, as Figure 5 shows,we can use the formula to get thesame result:

:Bc=

2o =o.ol78000 x ln l.16

Th i s a r t i c l e , , a l ong w i t h J im :

Ty ler 's ' companion p iece, shouldprov ide a reasonable in t roduct ionto the ba l l is t ic coef f ic ient and ,

exp la in what 's ' beh ind the l i t t lenumber we type in to our ba l l is t , icprogrammes. But , as J im's ar t ic lewarns, there are more than a few

wr ink les to be i roned out yet ,

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Figure 3: Veloclty Lsss due to Air Drag

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