Information about External Ballistics

External ballistics is the part of the science of ballistics that 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 bullet after it exits the barrel and before it hits the target. When in flight, the main forces acting on the projectile are gravity and air resistance.

Forces acting on the projectile

Gravity imparts a downward acceleration on the projectile, causing it to drop from the line of sight, and 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). Over long periods of flight, these forces have a major impact on the path of the projectile, and must be accounted for when predicting where the projectile will travel.

Target shooters must be very aware of the external ballistics of their bullets. When shooting at long ranges, bullet drop can be measured in tens of feet within the accurate range of many rifle cartridges, so knowledge of the flight characteristics of the bullet and the distance to the target are essential for accurate long range shooting. At extremely long ranges, artillery must fire projectiles along trajectories that are not even approximately straight; they are closer to parabolic, although air resistance affects this. For the longer ranges and flight times, the Coriolis effect becomes important. 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.

Small 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.

The formula for calculating the ballistic coefficient is as follows:

:

where:

• BC = ballistic coefficient
• sd = sectional density
• i = form factor
• m = mass of the bullet, lb or kg
• d = diameter of the object, in or m

Alternately:

:

where:

• BC = ballistic coefficient
• m = mass of the bullet
• A = cross-sectional area
• Cd = drag coefficient
• d = average density of the bullet
• l = bullet length

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. Sectional density is 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 model of 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 supersonic and subsonic velocities), a BC provided by a bullet manufacturer will be an average BC that represents the common range of velocities for that bullet. For rifle bullets, this will probably be a supersonic velocity, for pistol bullets it will be probably be subsonic. For projectiles that travel through the supersonic, transonic and subsonic flight regimes BC is not well approximated by a single constant, but is considered to be a function BC(M) of the Mach number M; 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.^[1] 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

Pejsa model

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. Arthur J. Pejsa. 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.

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 coefficient modelled 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 can be determined. These test firings should preferrably be executed at 75% to 80% of the supersonic range of the projectiles of interest, staying away from erratic transonic effects. With this the Pejsa model can easily and accurately be tuned for the specific drag behaviour of a specific projectile, making significant better ballistic predictions for ranges beyond 500 m (546.7 yd) possible.

Some software developers offer commercial software which is based on the Pejsa drag model enhanced with refinements to account for normally minor effects (Coriolis, spin 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 (1093.6 yd).

6 degrees of freedom (6 DOF) model

There are also advanced professional ballistic models like 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.

Doppler radar-measurements

For the precise establishment of BC's or maybe scientifically better expressed drag coefficients Doppler radar-measurements are required. The normal shooting or aerodynamics enthusiast however has no access to such expensive professional measurement devices. Weibel 1000e Doppler radars are used by governments, professional ballisticians, defence forces and a few ammunition manufacturers to obtain exact real world data of the flight behaviour of projectiles of their interest.

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:

Range (m)	500	600	700	800	900	1000	1100	1200	1300	1400	1500	1600	1700	1800	1900	2000
Ballistic coefficient	1.040	1.051	1.057	1.063	1.064	1.067	1.068	1.068	1.068	1.066	1.064	1.060	1.056	1.050	1.042	1.032

The initial rise in the BC value is attributed to a projectiles always present yaw and precession out of the bore. The test results were obtained from many shots not just a single shot. The bullet was assigned 1.062 for its BC number by the bullet's manufacturer Lost River Ballistic Technologies.

General trends in ballistic coefficient

In general, for small calibre ammunition, a pointed bullet will have a better ballistic coefficient (BC) than a round nosed bullet, and a round nosed bullet will have a better BC than a flat point bullet (the similar is true for large calibre projectiles). Large radius curves, resulting in a shallower point angle, will produce lower drags, particularly at supersonic velocities. Hollow point bullets behave much like a flat point of the same point diameter. Bullets designed for supersonic use often have a slight taper at the rear, called a boat tail, which further reduces drag. Cannelures, which are recessed rings around the bullet used to crimp the bullet securely into the case, will cause an increase in drag.

The transonic problem

In the transonic region, an important thing that happens to most bullets, is that the centre of pressure (CP) shifts forward as the bullet decelerates. That CP shift affects the stability of the bullet. If the bullet is not well stabilized, it cannot remain pointing exactly forward through the transonic region. However, even if the bullet has sufficient stability (static and dynamic) to be able to fly through the transonic region and stays pointing exactly forward, it is still affected. The erratic and sudden CP shift can cause dispersion, even if the bullets flight becomes well behaved again when it enters the subsonic region. This makes accurately predicting the ballistic behaviour of bullets in the transonic region very hard.

External factors

Wind

Wind has a range of effects, the first being the effect of making the bullet deviate to the side. From a scientific perspective, the "wind pushing on the side of the bullet" is not what causes wind drift. What causes wind drift is drag. Drag makes the bullet turn into the wind, keeping the centre of air pressure on its nose. This causes the nose to be cocked (from your perspective) into the wind, the base is cocked (from your perspective) "downwind." So, (again from your perspective), the drag is pushing the bullet downwind making bullets follow the wind. A somewhat less obvious effect is caused by head or tailwinds. A headwind will slightly increase the relative velocity of the projectile, and increase drag and the corresponding drop. A tailwind will reduce the drag and the bullet drop. In the real world pure head or tailwinds are rare, since wind seldom is constant in force and direction and normally interacts with the terrain it is blowing over. This often makes ultra long range shooting in head or tailwind conditions hard. Wind also causes a Magnus effect, whereby the sideways component of the wind combined with the spin of the bullet creates a force acting either up or down, perpendicular to the sideways vector of the wind.

Ambient air density

Air temperature, pressure, altitude and humidity variations make up the ambient air density. Decreased air density will result in a decrease in drag, and increased air density will result in a rise in drag. Humidity has a counter intuitive impact. Since water vapor has a density of 0.8 grams per litre, while dry air averages about 1.225 grams per litre, higher humidity actually decreases the air density, and therefore decreases the drag.

Vertical angles

The vertical angle (or elevation) of a shot will also affect the trajectory of the shot. Ballistic tables for small calibre projectiles (fired from pistols or rifles) assume that gravity is acting nearly perpendicular to the bullet path. If the angle is up or down, then the perpendicular acceleration will actually be less. The effect of the path wise acceleration component will be negligible, so shooting up or downhill will both result in a similar decrease in bullet drop.

Long range external factors

The coordinate system that is used to specify the location of the point of firing and the location of the target is the system of latitudes and longitudes, which is in fact a rotating coordinate system, since the Earth is rotating. For small arms, this rotation is generally insignificant, but for ballistic projectiles with long flight times, such as extreme long-range rifle projectiles, artillery and intercontinental ballistic missiles, it is a significant factor in calculating the trajectory. During its flight, the projectile moves in a straight line (not counting gravitation and air resistance for now). Since the target is co-rotating with the Earth, it is in fact a moving target, relative to the projectile, so in order to hit it the gun must aim slightly ahead of the target, the gun must be aimed to a point where the bullet and the target will arrive simultaneously.

When the straight path of the projectile is plotted in the rotating coordinate system that is used, then this path appear as curvilinear. The fact that the coordinate system is rotating must be taken into account, and this is achieved by adding terms for a "centrifugal force" and a "Coriolis effect" to the equations of motion. When the appropriate Coriolis term is added to the equation of motion the predicted path with respect to the rotating coordinate system is curvilinear, corresponding to the actual straight line motion of the projectile.

Maximum effective small arms range

The maximum practical range of all small arms and especially high-powered sniper rifles depends mainly on the aerodynamic or ballistic efficiency of the spin stabilised projectiles used. Long-range shooters must also collect relevant information to calculate elevation and windage corrections to be able to achieve first shot strikes. The data to calculate these fire control corrections has a long list of variables including^[2]:

• ballistic coefficient of the bullets used
• height of the sighting components above the rifle bore
• the zero range at which the sighting components and rifle combination were sighted in
• bullet weight
• actual muzzle velocity (powder temperature affects muzzle velocity, primer ignition is also temperature dependent)
• range to target
• inclination angle in case of uphill/downhill firing
• target speed and direction
• wind speed and direction (main cause for horizontal projectile deflection and generally the hardest ballistic variable to measure and judge correctly. Wind effects can also cause vertical deflection.)
• air temperature, pressure, altitude and humidity variations (these make up the ambient air density)
• earth's gravity (changes slightly with latitude and altitude)
• gyroscopic drift (horizontal and vertical plane gyroscopic effect — often know as spin drift - induced by the barrels twist direction and twist rate)
• coriolis effect drift (latitude, direction of fire and hemisphere data dictate this effect)
• lateral throw-off
• aerodynamic jump
• the inherent potential accuracy and adjustment range of the sighting components
• the inherent potential accuracy of the rifle
• the inherent potential accuracy of the ammunition
• the inherent potential accuracy of the computer program and other firing control components used to calculate the trajectory

The ambient air density is at its maximum at Arctic sea level conditions. Cold gunpowder also produces lower pressures and hence lower muzzle velocities than warm powder. This means that the maximum practical range of rifles will be at it shortest at Arctic sea level conditions. Another problem is presented by the fact that when the velocity of a rifle bullet approaches the speed of sound it enters the transonic region. In the transonic region most bullets show significant accuracy decay. Because of this marksmen normally restrict themselves to engaging targets within the supersonic range of the bullet used.

The ability to hit a target at great range has a lot to do with the ability to tackle environmental and meteorological factors and a good understanding of exterior ballistics and the limitations of equipment. Without computer support and highly accurate laser range-finders and meteorological measuring equipment as aids to calculate ballistic solutions, long-range shooting beyond 1000 m (1100 yd) becomes guesswork for even the most expert long-range marksmen.^[3]

Interesting further reading:

Using ballistics data

Here is an example of a ballistic table for a .30 calibre Speer 169 grain (11 g) pointed boat tail match bullet, with a BC of 0.480. It assumes sights 1.5 inches (38 mm) above the bore line, and sights adjusted to result in point of aim and point of impact matching ("zeroed") at 200 yards (183 m):

Range (yd)	0	100	200	300	400	500
Range (m)	0	91	183	274	366	457
Velocity (ft/s)	2700	2512	2331	2158	1992	1834
Velocity (m/s)	823	766	710	658	607	559
Height (in)	-1.5	2.0	0	-8.4	-24.3	-49.0
Height (mm)	-38	51	0	-213	-617	-1245

Here's the height information for the same bullet, zeroed for 300 yards (274 m). Velocities will be identical:

Range (yd)	0	100	200	300	400	500
Range (m)	0	91	183	274	366	457
Height (in)	-1.5	4.8	5.6	0	-13.1	-35.0
Height (mm)	-38	122	142	0	-333	-889

From these tables it can be seen that, even with a high velocity, very aerodynamic bullet, drop is very significant, and picking the right zero for the target distance can be quite important. An experienced shooter firing a high quality rifle can easily keep shots within a 10 inch (254 mm) circle at 500 yards (457 m), so if the range is not correctly estimated then the drop (or rise before the zero distance) of the bullet can cause result in a miss on a target that should be easy to hit.

See also

• 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.

Freeware small arms external ballistics software

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 interactive online ballistics calculator
• http://accurateshooter.net/Downloads/pejsajacksonballistics.xls Pejsa Ballistics (MS Excel spreadsheet) - Pejsa model

Notes and references

• Tan, A., Frick, C.H., and Castillo, O. (1987). "The fly ball trajectory: An older approach revisited". American Journal of Physics 55 (1): 37.  (Simplified calculation of the motion of a projectile under a drag force proportional to the square of the velocity)
• The Perfect Basketball Shot. (PDF). Retrieved on September 26, 2005. - basketball ballistics.
• Speer Reloading Manual Number 11, Omark Industries, 1987 (no ISBN)
• 2DOF and 3DOF Exterior Ballistics in MS Excel by Hans Cronander, Goteburg, Sweden
• Website of Pejsa Ballistics
• How do bullets fly? by Ruprecht Nennstiel, Wiesbaden, Germany
• How External Ballistics Programs Work by Bryan Litz
• Articles on long range shooting by Bryan Litz
• Exterior Ballistics.com
• Weite Schüsse - part 4, Basic explanation of the Pejsa model by Lutz Möller
• A Short Course in External Ballistics
• BRITISH ARTILLERY FIRE CONTROL - BALLISTICS & DATA