External ballistics

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External ballistics is the part of ballistics that refers to the behavior of a 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.

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 insignificant, but for ballistic projectiles with long flight times, such as 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 appears 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.

Gravity imparts a downward acceleration on the bullet, causing it to drop from the line of sight, and the air resistance decelerates the bullet 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). When shooting at long ranges, projectile drop can be measured in tens of feet within the accurate range of the projectile, so knowledge of the flight characteristics of the projectile 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 longest-range artillery, the Coriolis effect becomes important.

Mathematical models for calculating the effects of air resistance are extremely complex and not very reliable, so the most common method of calculating trajectories is by empirical measurement.

Use of ballistics tables or ballistics software based on the G1 drag model 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 and its sectional density. The acceleration 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 mm) diameter bullet with a flat base, a length of 3 inches (76 mm), and a 2 inch (51 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 BCs in the range 0.12 to 0.56, with 0.56 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 caliber, 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 subsonic and supersonic 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 large calibre projectiles BC is not well approximated by a 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. In particular, a major drop in BC occurs when during the flight the projectile speed decreases from supersonic to subsonic.

In general, for small calibre ammunition, a pointed bullet will have a better 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, that 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.

Here is an example of a ballistic table for a .30 caliber 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.

Other factors that affect external ballistics are wind and air density. Wind has a range of effects, the first being the obvious effect of pushing the bullet to the side, as well as the somewhat less obvious effect of a head or tailwind. 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. 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.

Decreased barometric pressure or high altitudes will result in a decrease in drag, and increased barometric pressure or low altitudes will result in a rise in drag. Humidity also has an impact--the opposite of the one that most people expect. Since water vapor has a density of 0.8 grams per liter, while dry air averages about 1.3 grams per liter, higher humidity actually decreases the air density, and therefore decreases the drag.

The vertical angle (or elevation) of a shot will also effect 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 pathwise acceleration component will be negligible, so shooting up or downhill will both result in a similar decrease in bullet drop.

[edit] See Also

Internal ballistics - The behavior of the projectile and propellant before it leaves the barrel.
Terminal ballistics - The behavior of the projectile upon impact with the target.

[edit] 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.