External Ballistics

Brief

The article explains basics of the external ballistics

Details

This article is based on of the External Ballistic wikipedia article.

External ballistics or exterior ballistics is the part of ballistics that deals with the behavior of a non-powered projectile in flight.

Forces acting on the projectile

When in flight, the main forces acting on the projectile are gravity, drag, and 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. 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.

The effect of gravity

The effect of gravity on a projectile in flight is often referred to as bullet drop. It is important to understand the effect of gravity when zeroing the sighting components of a gun. To plan for bullet drop and compensate properly, one must understand parabolic shaped trajectories.

Bullet drop

In order for a projectile to impact any distant target, the barrel must be inclined to a positive elevation angle relative to the target. This is due to the fact that the projectile will begin to respond to the effects of gravity the instant it is free from the mechanical constraints of the bore. Thus, a bullet fired with a zero elevation angle can never impact a target higher than or at the same elevation as the center axis of the bore. The imaginary line down the center axis of the bore and out to infinity is called the line of departure and is the line on which the bullet leaves the barrel. As the bullet travels downrange, it arcs below the line of departure as it is being deflected off its initial path by gravity. Bullet drop is defined as the vertical distance of the projectile below the line of departure from the bore. Even when the line of departure is tilted upward or downward, bullet drop is still defined as the distance between the bullet and the line of departure at any point along the trajectory. Bullet drop is therefore of little practical use to shooters because it does not describe the actual trajectory of the bullet and is independent of the direction or distance to a target. It is best described as an intermediate parameter, most useful in ballistic computations where it is needed to calculate the values of other parameters. Apart from this, it is mainly useful when conducting a direct comparison of two different projectiles regarding the shape of their trajectories.

Bullet path

For hitting a distant target an appropriate positive elevation angle is required that is achieved by angling the line of sight from the shooter's eye through the centerline of the sighting system downward toward the line of departure. This can be accomplished by simply adjusting the sights down mechanically, or by securing the entire sighting system to a sloped mounting having a known downward slope, or by a combination of both. This procedure has the effect of elevating the muzzle when the barrel must be subsequently raised to align the sights with the target. A bullet leaving a muzzle at a given elevation angle follows a ballistic trajectory whose characteristics are dependent upon various factors such as muzzle velocity, gravity, and aerodynamic drag. This ballistic trajectory is referred to as the bullet path.

Bullet path is of great use to shooters because it allows them to establish ballistic tables that will predict how much elevation correction must be applied to the sight line for shots at various known distances. Bullet path values are determined by both the sight height, or the distance of the line of sight above the bore centerline, and the range at which the sights are zeroed, which in turn determines the elevation angle. A bullet following a ballistic trajectory has both forward and vertical motion. Forward motion is slowed due to air resistance, and in point mass modeling the vertical motion is dependent on a combination of the elevation angle and gravity. Initially, the bullet is rising with respect to the line of sight or the horizontal sighting plane. The bullet eventually reaches its apex (highest point in the trajectory parabola) where the vertical speed component decays to zero under the effect of gravity, and then begins to descend, eventually impacting the earth. The farther the distance to the intended target, the greater the elevation angle and the higher the apex.

The bullet path crosses the horizontal sighting plane two times. The point closest to the gun occurs while the bullet is climbing through the line of sight and is called the near zero. The second point occurs as the bullet is descending through the line of sight. It is called the far zero and defines the current sight in distance for the gun. Bullet path is described numerically as distances above or below the horizontal sighting plane at various points along the trajectory. This is in contrast to bullet drop which is referenced to the plane containing the line of departure regardless of the elevation angle. Since each of these two parameters uses a different reference datum, significant confusion can result because even though a bullet is tracking well below the line of departure it can still be gaining actual and significant height with respect to the line of sight as well as the surface of the earth in the case of a horizontal or near horizontal shot taken over flat terrain. Simply put, one parameter (bullet drop) compares the relative position of an actual bullet with an imaginary bullet that is not impeded by gravity while the other parameter (bullet path) describes the actual path of a bullet through the Earth's atmosphere, taking into account both gravity and aerodynamic effects.

Example bullet path chart [show]

Drag resistance modeling and measuring

Mathematical models for calculating the effects of drag or air resistance are quite complex and often unreliable beyond about 500 meters, 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 1/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 G1 standard projectile originates from the "C" standard reference projectile defined by the German steel, ammunition and armaments manufacturer Krupp in 1881. The G1 model standard projectile has a BC of 1. The French Gâvre Commission decided to use this projectile as their first reference projectile, giving the G1 name.

Sporting bullets, with a calibre d ranging from 0.177 to 0.50 inches (4.50 to 12.7 mm), have G1 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.

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 wadcutter, flat-based, spitzer, boat-tail, very-low-drag, etc. bullet types or shapes. They assume one invariable drag function as indicated by the published BC.

Several drag curve models optimized for several standard projectile shapes are however available. The resulting fixed drag curve models for several standard projectile shapes or types are referred to as the:

How different speed regimes affect .338 calibre rifle bullets can be seen in the .338 Lapua Magnum product brochure which states Doppler radar established G1 BC data. The reason for publishing data like in this brochure is that the Siacci/Mayevski G1 model can not be tuned for the drag behavior of a specific projectile whose shape significantly deviates from the used reference projectile shape. Some ballistic software designers, who based their programs on the Siacci/Mayevski G1 model, give the user the possibility to enter several different G1 BC constants for different speed regimes to calculate ballistic predictions that closer match a bullets flight behavior at longer ranges compared to calculations that use only one BC constant.

The above example illustrates the central problem fixed drag curve models have. These models will only yield satisfactory accurate predictions as long as the projectile of interest has the same shape as the reference projectile or a shape that closely resembles the reference projectile. Any deviation from the reference projectile shape will result in less accurate predictions. How much a projectile deviates from the applied reference projectile is mathematically expressed by the form factor (i). The form factor can be used to compare the drag experienced by a projectile of interest to the drag experienced by the employed reference projectile at a given velocity (range). The problem that the actual drag curve of a projectile can significantly deviate from the fixed drag curve of any employed reference projectile systematically limits the traditional drag resistance modeling approach. The relative simplicity however makes that it can be explained to and understood by the general shooting public and hence is also popular amongst ballistic software prediction developers and bullet manufacturers that want to market their products.

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