- External ballistics
External ballistics is the part of the science of
ballisticsthat deals with the behaviour of a non-powered projectile in flight. External ballistics is frequently associated with firearms, and deals with the behaviour of the bulletafter it exits the barrel and before it hits the target.
Forces acting on the projectile
When in flight, the main
forces acting on the projectileare gravity, air resistanceand if present wind. Gravity imparts a downward acceleration on the projectile, causing it to drop from the line of sight. Drag or the air resistance decelerates the projectile with a force proportional to the square of the velocity (or cube, or even higher powers of "v", depending on the speed of the projectile). Wind makes the projectile deviate from its trajectory. During flight, gravity, drag and wind have a major impact on the path of the projectile, and must be accounted for when predicting how the projectile will travel.
For medium to longer ranges and flight times, besides gravity, air resistance and wind, several meso variables described in the external factors paragraph have to be taken into account.
For long to very long ranges and flight times, minor effects and forces such as the ones described in the long range factors paragraph become important and have to be taken into account. The practical effects of these variables are generally irrelevant for most firearms users, since normal group scatter at short and medium ranges prevails over the influence these effects exert on firearms projectiles
At extremely long ranges,
artillerymust fire projectiles along trajectories that are not even approximately straight; they are closer to parabolic, although air resistance affects this.
In the case of
ballistic missiles, the altitudes involved have a significant effect as well, with part of the flight taking place in a near-vacuum.
mall arms external ballistics
Drag resistance modelling and measuring
Mathematical models for calculating the effects of air resistance are quite complex and for the simpler mathematical models not very reliable beyond 500 m (500 yd), so the most reliable method of establishing trajectories is still by empirical measurement.
Fixed drag curve models generated for standard-shaped projectiles
Use of ballistics tables or ballistics software based on the Siacci/Mayevski G1 drag model, introduced in 1881, are the most common method used to work with external ballistics. Bullets are described by a
ballistic coefficient, or BC, which combines the air resistance of the bullet shape (the drag coefficient) and its sectional density(a function of mass and bullet diameter).
The deceleration due to drag that a projectile with mass "m", velocity "v", and diameter "d" will experience is proportional to BC, 1/"m", "v²" and "d²". The BC gives the ratio of ballistic efficiency compared to the standard G1 projectile, which is a 1 pound (454 g), 1 inch (25.4 mm) diameter bullet with a flat base, a length of 3 inches (76.2 mm), and a 2 inch (50.8 mm) radius tangential curve for the point.
Sporting bullets, with a
calibre"d" ranging from 0.177 to 0.50 inches (4.50 to 12.7 mm), have BC’s in the range 0.12 to slightly over 1.00, with 1.00 being the most aerodynamic, and 0.12 being the least. Very-low-drag bullets with BC's ≥ 1.10 can be designed and produced on CNC precision lathes out of mono-metal rods, but they often have to be fired from custom made full bore rifles with special barrels [ [http://www.lima-wiederladetechnik.de/Englisch/LM-Class-Bullets.htm LM Class Bullets, very high BC bullets for windy long Ranges] ] . Sectional densityis a very important aspect of a bullet, and is the ratio of frontal surface area (half the bullet diameter squared, times pi) to bullet mass. Since, for a given bullet shape, frontal surface increases as the square of the calibre, and mass increases as the cube of the diameter, then sectional density grows linearly with bore diameter. Since BC combines shape and sectional density, a half scale modelof the G1 projectile will have a BC of 0.5, and a quarter scale model will have a BC of 0.25.
Since different projectile shapes will respond differently to changes in velocity (particularly between
supersonicand subsonicvelocities), a BC provided by a bullet manufacturer will be an average BC that represents the common range of velocities for that bullet. For riflebullets, this will probably be a supersonicvelocity, for pistol bullets it will be probably be subsonic. For projectiles that travel through the supersonic, transonicand subsonicflight regimes BC is not well approximated by a single constant, but is considered to be a function "BC(M)" of the Mach numberM; here M equals the projectile velocity divided by the speed of sound. During the flight of the projectile the M will decrease, and therefore (in most cases) the BC will also decrease.
Most ballistic tables or software takes for granted that one specific drag function correctly describes the drag and hence the flight characteristics of a bullet related to its ballistics coefficient. Those models do not differentiate between flat-based, spitzer, boat-tail, very-low-drag, etc. bullet types. They assume one invariable drag function as indicated by the published BC. These resulting drag curve models are referred to as the Ingalls, G1 (by far the most popular), G2, G5, G6, G7, G8, GI and GL drag curves.
How different speed regimes affect .338 calibre rifle bullets can be seen in the .338 Lapua Magnum product brochure which states Doppler radar established BC data. [ [http://www.lapua.com/fileadmin/user_upload/esitteet/Lapua.338LapuaMagnum.pdf .338 Lapua Magnum product brochure] from Lapua] The reason for publishing data like in this brochure is that the Siacci/Mayevski G1 model can not be tuned for the drag behaviour of a specific projectile. Some ballistic software designers, who based their programs on the Siacci/Mayevski G1 model, give the user the possibility to enter several different BC constants for different speed regimes to calculate ballistic predictions that closer match a bullets flight behaviour at longer ranges compared to calculations that use only one BC constant.
More advanced drag models
Besides the traditional Siacci/Mayevski G1 drag model other more advanced drag models exist. The most prominent alternative ballistic model is probably the model presented in 1980 by Prof. Dr.
Arthur J. Pejsa. [http://home.sprintmail.com/~pejsa/aboutartpejsa.htm Mr. Pejsa] claims on his website that his method was consistently capable of predicting (supersonic) rifle bullet trajectories within 2.54 mm (0.1 in) and bullet velocities within 0.3048 m/s (1 ft/s) out to 914.4 m (1000 yd) when compared to dozens of actual measurements. [ [http://www.pejsa.com/ Pejsa Ballistics] ]
The Pejsa model is an analytic closed-form solution that does not use any tables or fixed drag curves generated for standard-shaped projectiles. The Pejsa method uses the G1-based ballistic coefficient as published, and incorporates this in a Pejsa retardation coefficient function in order to model the retardation behaviour of the specific projectile. Since it effectively uses an analytic function (
drag coefficientmodelled as a function of the Mach number) in order to match the drag behaviour of the specific bullet the Pesja method does not need to rely on any prefixed assumption.
Besides the mathematical retardation coefficient function, Pejsa added an extra slope constant factor that accounts for the more subtle change in retardation rate downrange of different bullet shapes and sizes. It ranges from 0.1 (flat-nose bullets) to 0.9 (
very-low-drag bullets). If this deceleration constant factor is unknown a default value of 0.5 will predict the flight behaviour of most modern spitzer-type rifle bullets quite well. With the help of test firing measurements the slope constant for a particular bullet/rifle system/shooter combination can be determined. These test firings should preferably be executed at 60% and for extreme long range ballistic predictions also at 80% to 90% of the supersonic range of the projectiles of interest, staying away from erratic transonic effects. With this the Pejsa model can easily be tuned for the specific drag behaviour of a specific projectile, making significant better ballistic predictions for ranges beyond 500 m (547 yd) possible.
Some software developers offer commercial software which is based on the Pejsa drag model enhanced and improved with refinements to account for normally minor effects (Coriolis, gyroscopic drift, etc.) that come in to play at long range. The developers of these enhanced Pejsa models designed these programs for ballistic predictions beyond 1000 m (1094 yd). [ [http://www.precisionworkbench.com/index.htm Lex Talus Corporation Pejsa based ballistic software] ] [ [http://www.patagoniaballistics.com/index.html Patagonia Ballistics Pejsa based ballistic software] ]
6 degrees of freedom (6 DOF) model
There are also advanced professional ballistic models like [http://www.prodas.com/ PRODAS] available. These are based on 6 Degrees Of Freedom (6 DOF) calculations. 6 DOF modelling needs such elaborate input, knowledge of the employed projectiles and long calculation time on computers that it is unpractical for non-professional ballisticians and field use where calculations generally have to be done on the fly on PDA's with relatively modest computing power. 6 DOF is generally used by military organizations that study the ballistic behaviour of a limited number of (intended) military issue projectiles. Calculated 6 DOF trends can be incorporated as correction tables in more conventional ballistic software applications.
For the precise establishment of BC's or maybe scientifically better expressed
drag coefficients Doppler radar-measurements are required. Weibel 1000e Doppler radars are used by governments, professional ballisticians, defence forces and a few ammunition manufacturers to obtain real world data of the flight behaviour of projectiles of their interest. Correctly established state of the art Doppler radar measurements can determine the flight behaviour of projectiles as small as airgun pellets in three-dimensional space to within a few millimetres accuracy.
Doppler radar measurement results for a lathe turned monolithic solid .50 BMG
very-low-drag bullet(Lost River J40 .510-773 grain monolithic solid bullet / twist rate 1:15 in) look like this:
This table demonstrates that, even with a very aerodynamic bullet fired at high velocity, the "bullet drop" or change in the point of impact is significant. This change in point of impact has two important implications. Firstly, estimating the distance to the target is critical at longer ranges, because the difference in the point of impact between 400 and convert|500|yd|abbr=on is 25–32 in (depending on zero), in other words if the shooter estimates that the target is 400 yd away when it is in fact 500 yd away the shot will impact 25–32 in (635–813 mm) below where it was aimed, possibly missing the target completely. Secondly, the rifle should be zeroed to a distance appropriate to the typical range of targets, because the shooter might have to aim so far above the target to compensate for a large bullet drop that he may lose sight of the target completely (for instance being outside the field of view of a telescopic site). In the example of the rifle zeroed at convert|200|yd|abbr=on, the shooter would have to aim 49 in or more than 4 ft (1.2 m) above the point of impact for a target at 500 yd.
Freeware small arms external ballistics software
* [http://sourceforge.net/projects/balcomp GNU Exterior Ballistics Computer] (GEBC) - An open source 3DOF ballistics computer for Windows, Linux, and Mac - Supports the G1, G2, G5, G6, G7, and G8 drag models.
* [http://nerdulator.net/ Nerdulator.net Exterior Ballistics Computer] - A web-based small arms ballistics interface based on the open source GNU Exterior Ballistics Library, originally written by Derek Yates - Supports the G1 and G8 drag models.
* [http://accurateshooter.com/ accurateshooter.com Ballistics section] links to / hosts these 4 freeware external ballistics computer programs:
* [http://www.cronander.net/CRONXR1A.zip] 2DOF & 3DOF R.L. McCoy / Gavre exterior ballistics (zip file) - Supports the G1, G2, G5, G6, G7, G8, GS, GL, GI, GB and RA4 drag models
* [http://www.huntingnut.com/files/pointblank/PointBlankCRBSv18a.zip] PointBlank Ballistics (zip file) - Siacci/Mayevski G1 drag model
* [http://www.eskimo.com/~jbm/ballistics/traj_basic/traj_basic.html] JBM's real-time Basic Trajectory interactive online ballistics calculator
* [http://www.eskimo.com/~jbm/ballistics/traj/traj.html] JBM's real-time Trajectory interactive online ballistics calculator
* [http://www.eskimo.com/~jbm/ballistics/mpm/mpm.html] JBM's real-time Modified Point Mass Trajectory interactive online ballistics calculator - This program uses an approximation for the yaw of repose to simplify the 6DOF equations. It provides estimates of spin drift and magnus effects. It also takes aerodynamic jump and Coriolis effects in account.
* [http://accurateshooter.net/Downloads/pejsajacksonballistics.xls] Pejsa Ballistics (MS Excel spreadsheet) - Pejsa model
* [http://www.bestpalmsoftware.com/mathematic/misc/pocketssf_sharpshooterfriend344.htm] Sharpshooter Friend (Palm PDA software) - Pejsa model
Evaluation small arms external ballistics software
* [http://www.precisionworkbench.com/index.htm Lex Talus Corporation] "Precision Shooter's Workbench©" and "Field Firing Solutions©" fully functional 30-day free evaluation programs for PC and PDA - Pejsa model
Internal ballistics- The behaviour of the projectile and propellant before it leaves the barrel.
Terminal ballistics- The behaviour of the projectile upon impact with the target.
Trajectory of a projectile- Basic external ballistics methematic formulas.
General external ballistics
* (Simplified calculation of the motion of a projectile under a drag force proportional to the square of the velocity)
* - basketball ballistics.
Small arms external ballistics
* Speer Reloading Manual Number 11, Omark Industries, 1987 (no
* [http://www.nennstiel-ruprecht.de/bullfly/index.htm How do bullets fly? by Ruprecht Nennstiel, Wiesbaden, Germany]
* [http://www.exteriorballistics.com/ebexplained/index.cfm Exterior Ballistics.com]
* [http://www.frfrogspad.com/extbal.htm A Short Course in External Ballistics]
* [http://bryanlitz.bravehost.com/ Articles on long range shooting by Bryan Litz]
* [http://bryanlitz.bravehost.com/BalProgs.html How External Ballistics Programs Work by Bryan Litz]
* [http://www.cronander.net/ 2DOF and 3DOF Exterior Ballistics in MS Excel by Hans Cronander, Goteburg, Sweden]
* [http://www.pejsa.com/ Website of Pejsa Ballistics]
* [http://www.lima-wiederladetechnik.de/Weite-Schuesse/Weite-Schuesse-4.htm Weite Schüsse - part 4, Basic explanation of the Pejsa model by Lutz Möller de icon]
* [http://www.patagoniaballistics.com/balengine.html Patagonia Ballistics ballistics mathematical software engine]
Artillery external ballistics
* [http://members.tripod.com/~nigelef/fc_ballistics.htm BRITISH ARTILLERY FIRE CONTROL - BALLISTICS & DATA]
* [http://www.nvbmb.nl/downloads/b-gl-306-006fp-001.pdf FIELD ARTILLERY, VOLUME 6, BALLISTICS AND AMMUNITION]
Doppler radar tracking systems
* [http://www.weibel.dk/Default.aspx Weibel Doppler radar company website]
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