Bullet drop

From OODA WIKI

Main effects in external ballistics

Projectile/bullet drop and projectile path

File:How Forensic Investigators use the evidence from a shooting to locate bullet trajectory.png
Represents how evidence is found in a shooting
File:M855 drop during 25-meter zeroing trajectory M16A2 M4.jpg
Typical trajectory graph for a M4 carbine and M16A2 rifle using identical M855 cartridges with identical projectiles. Though both trajectories have an identical 25 m near zero, the difference in muzzle velocity of the projectiles gradually causes a significant difference in trajectory and far zero. The 0 inch axis represents the line of sight or horizontal sighting plane.

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

Projectile/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. 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 projectile leaves the barrel. Due to the effects of gravity a projectile can never impact a target higher than the line of departure. When a positively inclined projectile travels downrange, it arcs below the line of departure as it is being deflected off its initial path by gravity. Projectile/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, projectile drop is still defined as the distance between the bullet and the line of departure at any point along the trajectory. Projectile drop does not describe the actual trajectory of the projectile. Knowledge of projectile drop however is useful when conducting a direct comparison of two different projectiles regarding the shape of their trajectories, comparing the effects of variables such as velocity and drag behavior.

Projectile/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 projectile 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. If the projectile is spin stabilized, aerodynamic forces will also predictably arc the trajectory slightly to the right, if the rifling employs "right-hand twist." Some barrels are cut with left-hand twist, and the bullet will arc to the left, as a result. Therefore, to compensate for this path deviation, the sights also have to be adjusted left or right, respectively. A constant wind also predictably affects the bullet path, pushing it slightly left or right, and a little bit more up and down, depending on the wind direction. The magnitude of these deviations are also affected by whether the bullet is on the upward or downward slope of the trajectory, due to a phenomenon called "yaw of repose," where a spinning bullet tends to steadily and predictably align slightly off center from its point mass trajectory. Nevertheless, each of these trajectory perturbations are predictable once the projectile aerodynamic coefficients are established, through a combination of detailed analytical modeling and test range measurements.

Projectile/bullet path analysis is of great use to shooters because it allows them to establish ballistic tables that will predict how much vertical elevation and horizontal deflection corrections must be applied to the sight line for shots at various known distances. The most detailed ballistic tables are developed for long range artillery and are based on six-degree-of-freedom trajectory analysis, which accounts for aerodynamic behavior along the three axial directions—elevation, range, and deflection—and the three rotational directions—pitch, yaw, and spin. For small arms applications, trajectory modeling can often be simplified to calculations involving only four of these degrees-of-freedom, lumping the effects of pitch, yaw and spin into the effect of a yaw-of-repose to account for trajectory deflection. Once detailed range tables are established, shooters can relatively quickly adjust sights based on the range to target, wind, air temperature and humidity, and other geometric considerations, such as terrain elevation differences.

Projectile 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 projectile 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 projectile is rising with respect to the line of sight or the horizontal sighting plane. The projectile 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 projectile 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 projectile is descending through the line of sight. It is called the far zero and defines the current sight in distance for the gun. Projectile path is described numerically as distances above or below the horizontal sighting plane at various points along the trajectory. This is in contrast to projectile 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 projectile 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.

Maximum point-blank range and battle zero

Knowledge of the projectile drop and path has some practical uses to shooters even if it does not describe the actual trajectory of the projectile. For example, if the vertical projectile position over a certain range reach is within the vertical height of the target area the shooter wants to hit, the point of aim does not necessarily need to be adjusted over that range; the projectile is considered to have a sufficiently flat point-blank range trajectory for that particular target.[1] Also known as "battle zero", maximum point-blank range is also of importance to the military. Soldiers are instructed to fire at any target within this range by simply placing their weapon's sights on the center of mass of the enemy target. Any errors in range estimation are tactically irrelevant, as a well-aimed shot will hit the torso of the enemy soldier. The current trend for elevated sights and higher-velocity cartridges in assault rifles is in part due to a desire to extend the maximum point-blank range, which makes the rifle easier to use.[2][3][4]

Drag resistance

Schlieren photo/Shadowgraph of the detached shock or bow shockwave around a bullet in supersonic flight, published by Ernst Mach in 1888.

Mathematical models, such as computational fluid dynamics, are used for calculating the effects of drag or air resistance; they are quite complex and not yet completely reliable, but research is ongoing.[5] The most reliable method, therefore, of establishing the necessary projectile aerodynamic properties to properly describe flight trajectories is by empirical measurement.

Fixed drag curve models generated for standard-shaped projectiles

File:G1 Shape Standard Projectile Measurements in Calibers.png
G1 shape standard projectile. All measurements in calibers/diameters.

Use of ballistics tables or ballistics software based on the Mayevski/Siacci method and G1 drag model, introduced in 1881, are the most common method used to work with external ballistics. Projectiles 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 fictitious projectile with a flat base, a length of 3.28 calibers/diameters, and a 2 calibers/diameters 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.[6] The French Gâvre Commission decided to use this projectile as their first reference projectile, giving the G1 name.[7][8]

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.[9]

Sectional density is a very important aspect of a projectile or bullet, and is for a round projectile like a bullet 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 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:

File:G7 Shape Standard Projectile Measurements in Calibers.png
G7 shape standard projectile. All measurements in calibers/diameters.
  • G1 or Ingalls (flatbase with 2 caliber (blunt) nose ogive - by far the most popular)
  • G2 (Aberdeen J projectile)
  • G5 (short 7.5° boat-tail, 6.19 calibers long tangent ogive)
  • G6 (flatbase, 6 calibers long secant ogive)
  • G7 (long 7.5° boat-tail, 10 calibers tangent ogive, preferred by some manufacturers for very-low-drag bullets[10])
  • G8 (flatbase, 10 calibers long secant ogive)
  • GL (blunt lead nose)

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.[11][12] 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.[13][14] How much a projectile deviates from the applied reference projectile is mathematically expressed by the form factor (i).[15] 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.

More advanced drag models

Pejsa model

Another attempt at building a ballistic calculator is the model presented in 1980 by Dr. Arthur J. Pejsa.[16] Dr. Pejsa claims on his website that his method was consistently capable of predicting (supersonic) rifle bullet trajectories within 2.5 mm (0.1 in) and bullet velocities within 0.3 m/s (1 ft/s) out to 914 m (1,000 yd) in theory.[17] The Pejsa model is a closed-form solution.

The Pejsa model can predict a projectile within a given flight regime (for example the supersonic flight regime) with only two velocity measurements, a distance between said velocity measurements, and a slope or deceleration constant factor.[18] The model allows the drag curve to change slopes (true/calibrate) or curvature at three different points.[19] Down range velocity measurement data can be provided around key inflection points allowing for more accurate calculations of the projectile retardation rate, very similar to a Mach vs CD table. The Pejsa model allows the slope factor to be tuned to account for subtle differences in the retardation rate of different bullet shapes and sizes. It ranges from 0.1 (flat-nose bullets) to 0.9 (very-low-drag bullets). If this slope or deceleration constant factor is unknown a default value of 0.5 is used. 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. A practical downside of the Pejsa model is that accurate projectile specific down range velocity measurements to provide these better predictions can not be easily performed by the vast majority of shooting enthusiasts.

An average retardation coefficient can be calculated for any given slope constant factor if velocity data points are known and distance between said velocity measurements is known. Obviously this is true only within the same flight regime. With velocity actual speed is meant, as velocity is a vector quantity and speed is the magnitude of the velocity vector. Because the power function does not have constant curvature a simple chord average cannot be used. The Pejsa model uses a weighted average retardation coefficient weighted at 0.25 range. The closer velocity is more heavily weighted. The retardation coefficient is measured in feet whereas range is measured in yards hence 0.25 * 3.0 = 0.75, in some places 0.8 rather than 0.75 is used. The 0.8 comes from rounding in order to allow easy entry on hand calculators. Since the Pejsa model does not use a simple chord weighted average, two velocity measurements are used to find the chord average retardation coefficient at midrange between the two velocity measurements points, limiting it to short range accuracy. In order to find the starting retardation coefficient Dr. Pejsa provides two separate equations in his two books. The first involves the power function.[20] The second equation is identical to the one used to find the weighted average at R / 4; add N * (R/2) where R is the range in feet to the chord average retardation coefficient at midrange and where N is the slope constant factor.[21] After the starting retardation coefficient is found the opposite procedure is used in order find the weighted average at R / 4; the starting retardation coefficient minus N * (R/4). In other words, N is used as the slope of the chord line. Dr. Pejsa states that he expanded his drop formula in a power series in order to prove that the weighted average retardation coefficient at R / 4 was a good approximation. For this Dr. Pejsa compared the power series expansion of his drop formula to some other unnamed drop formula's power expansion to reach his conclusions. The fourth term in both power series matched when the retardation coefficient at 0.25 range was used in Pejsa's drop formula. The fourth term was also the first term to use N. The higher terms involving N where insignificant and disappeared at N = 0.36, which according to Dr. Pejsa was a lucky coincidence making for an exceedingly accurate linear approximation, especially for N's around 0.36. If a retardation coefficient function is used exact average values for any N can be obtained because from calculus it is trivial to find the average of any integrable function.[22] Dr. Pejsa states that the retardation coefficient can be modeled by C * VN where C is a fitting coefficient which disappears during the derivation of the drop formula and N the slope constant factor.[23]

The retardation coefficient equals the velocity squared divided by the retardation rate A. Using an average retardation coefficient allows the Pejsa model to be a closed-form expression within a given flight regime.

File:Five301906govtBullets.png
Five bullets used in United States military loadings from left to right: M1903 bullet, M1906 ball, M1 ball, M2 ball used by Dr. Pejsa for the second reference drag curve, and M2 armor-piercing (AP) bullet

In order to allow the use of a G1 ballistic coefficient rather than velocity data Dr. Pejsa provided two reference drag curves. The first reference drag curve is based purely on the Siacci/Mayevski retardation rate function. The second reference drag curve is adjusted to equal the Siacci/Mayevski retardation rate function at a projectile velocity of 2600 fps (792.5 m/s) using a .30-06 Springfield Cartridge, Ball, Caliber .30 M2 152 grains (9.8 g) rifle spitzer bullet with a slope or deceleration constant factor of 0.5 in the supersonic flight regime. In other flight regimes the second Pejsa reference drag curve model uses slope constant factors of 0.0 or -4.0. These deceleration constant factors can be verified by backing out Pejsa's formulas (the drag curve segments fits the form V(2 - N) / C and the retardation coefficient curve segments fits the form V2 / (V(2 - N) / C) = C * VN where C is a fitting coefficient). The empirical test data Pejsa used to determine the exact shape of his chosen reference drag curve and pre-defined mathematical function that returns the retardation coefficient at a given Mach number was provided by the US military for the Cartridge, Ball, Caliber .30 M2 bullet. The calculation of the retardation coefficient function also involves air density, which Pejsa did not mention explicitly. The Siacci/Mayevski G1 model uses the following deceleration parametrization (60 °F, 30 inHg and 67% humidity, air density ρ = 1.2209 kg/m3).[24] Dr. Pejsa suggests using the second drag curve because the Siacci/Mayevski G1 drag curve does not provide a good fit for modern spitzer bullets.[25] To obtain relevant retardation coefficients for optimal long range modeling Dr. Pejsa suggested using accurate projectile specific down range velocity measurement data for a particular projectile to empirically derive the average retardation coefficient rather than using a reference drag curve derived average retardation coefficient. Further he suggested using ammunition with reduced propellant loads to empirically test actual projectile flight behavior at lower velocities. When working with reduced propellant loads utmost care must be taken to avoid dangerous or catastrophic conditions (detonations) with can occur when firing experimental loads in firearms.[19]

Manges model

Although not as well known as the Pejsa model, an additional alternative ballistic model was presented in 1989 by Colonel Duff Manges (U S Army Retired) at the American Defense Preparedness (ADPA) 11th International Ballistic Symposium held at the Brussels Congress Center, Brussels, Belgium, May 9–11, 1989. A paper titled "Closed Form Trajectory Solutions for Direct Fire Weapons Systems" appears in the proceedings, Volume 1, Propulsion Dynamics, Launch Dynamics, Flight Dynamics, pages 665–674. Originally conceived to model projectile drag for 120 mm tank gun ammunition, the novel drag coefficient formula has been applied subsequently to ballistic trajectories of center-fired rifle ammunition with results comparable to those claimed for the Pejsa model.

The Manges model uses a first principles theoretical approach that eschews "G" curves and "ballistic coefficients" based on the standard G1 and other similarity curves. The theoretical description has three main parts. The first is to develop and solve a formulation of the two dimensional differential equations of motion governing flat trajectories of point mass projectiles by defining mathematically a set of quadratures that permit closed form solutions for the trajectory differential equations of motion. A sequence of successive approximation drag coefficient functions is generated that converge rapidly to actual observed drag data. The vacuum trajectory, simplified aerodynamic, d'Antonio, and Euler drag law models are special cases. The Manges drag law thereby provides a unifying influence with respect to earlier models used to obtain two dimensional closed form solutions to the point-mass equations of motion. The third purpose of this paper is to describe a least squares fitting procedure for obtaining the new drag functions from observed experimental data. The author claims that results show excellent agreement with six degree of freedom numerical calculations for modern tank ammunition and available published firing tables for center-fired rifle ammunition having a wide variety of shapes and sizes.

A Microsoft Excel application has been authored that uses least squares fits of wind tunnel acquired tabular drag coefficients. Alternatively, manufacturer supplied ballistic trajectory data, or Doppler acquired velocity data can be fitted as well to calibrate the model. The Excel application then employs custom macroinstructions to calculate the trajectory variables of interest. A modified 4th order Runge-Kutta integration algorithm is used. Like Pejsa, Colonel Manges claims center-fired rifle accuracies to the nearest one tenth of an inch for bullet position, and nearest foot per second for the projectile velocity.

The Proceedings of the 11th International Ballistic Symposium are available through the National Defense Industrial Association (NDIA) at the website http://www.ndia.org/Resources/Pages/Publication_Catalog.aspx Archived 2012-01-26 at the Wayback Machine.

Six degrees of freedom model

There are also advanced professional ballistic models like PRODAS available. These are based on six degrees of freedom (6 DoF) calculations. 6 DoF modeling accounts for x, y, and z position in space along with the projectiles pitch, yaw, and roll rates. 6 DoF modeling needs such elaborate data input, knowledge of the employed projectiles and expensive data collection and verification methods that it is impractical for non-professional ballisticians,[26] but not impossible for the curious, computer literate, and mathematically inclined. Semi-empirical aeroprediction models have been developed that reduced extensive test range data on a wide variety of projectile shapes, normalizing dimensional input geometries to calibers; accounting for nose length and radius, body length, and boattail size, and allowing the full set of 6-dof aerodynamic coefficients to be estimated. Early research on spin-stabilized aeroprediction software resulted in the SPINNER computer program.[27] The FINNER aeroprediction code calculates 6-dof inputs for fin stabilized projectiles.[28] Solids modeling software that determines the projectile parameters of mass, center of gravity, axial and transverse moments of inertia necessary for stability analysis are also readily available, and simple to computer program.[29] Finally, algorithms for 6-dof numerical integration suitable to a 4th order Runge-Kutta are readily available.[30] All that is required for the amateur ballistician to investigate the finer analytical details of projectile trajectories, along with bullet nutation and precession behavior, is computer programming determination. Nevertheless, for the small arms enthusiast, aside from academic curiosity, one will discover that being able to predict trajectories to 6-dof accuracy is probably not of practical significance compared to more simplified point mass trajectories based on published bullet ballistic coefficients. 6 DoF is generally used by the aerospace and defense industry and military organizations that study the ballistic behavior 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.

Though 6 DoF modeling and software applications are used by professional well equipped organizations for decades, the computing power restrictions of mobile computing devices like (ruggedized) personal digital assistants, tablet computers or smartphones impaired field use as calculations generally have to be done on the fly. In 2016 the Scandinavian ammunition manufacturer Nammo Lapua Oy released a 6 DoF calculation model based ballistic free software named Lapua Ballistics. The software is distributed as a mobile app only and available for Android and iOS devices.[31] The employed 6 DoF model is however limited to Lapua bullets as a 6 DoF solver needs bullet specific drag coefficient (Cd)/Doppler radar data and geometric dimensions of the projectile(s) of interest. For other bullets the Lapua Ballistics solver is limited to and based on G1 or G7 ballistic coefficients and the Mayevski/Siacci method.

Artillery software suites

Military organizations have developed ballistic models like the NATO Armament Ballistic Kernel (NABK) for fire-control systems for artillery like the SG2 Shareable (Fire Control) Software Suite (S4) from the NATO Army Armaments Group (NAAG). The NATO Armament Ballistic Kernel is a 4-DoF modified point mass model. This is a compromise between a simple point mass model and a computationally intensive 6-DoF model.[32] A six- and seven-degree-of-freedom standard called BALCO has also been developed within NATO working groups. BALCO is a trajectory simulation program based on the mathematical model defined by the NATO Standardization Recommendation 4618. The primary goal of BALCO is to compute high-fidelity trajectories for both conventional axisymmetric and precision-guided projectiles featuring control surfaces. The BALCO trajectory model is a FORTRAN 2003 program that implements the following features:

  • 6/7‐DoF equations of motion
  • 7th‐order Runge‐Kutta‐Fehlberg integration
  • Earth models
  • Atmosphere models
  • Aerodynamic models
  • Thrust and Base Burn models
  • Actuator models[33]

The predictions these models yield are subject to comparison study.[34]

Doppler radar-measurements

For the precise establishment of drag or air resistance effects on projectiles, Doppler radar measurements are required. Weibel 1000e or Infinition BR-1001 Doppler radars are used by governments, professional ballisticians, defence forces and a few ammunition manufacturers to obtain real-world data of the flight behavior of projectiles of their interest. Correctly established state of the art Doppler radar measurements can determine the flight behavior of projectiles as small as airgun pellets in three-dimensional space to within a few millimetres accuracy. The gathered data regarding the projectile deceleration can be derived and expressed in several ways, such as ballistic coefficients (BC) or drag coefficients (Cd). Because a spinning projectile experiences both precession and nutation about its center of gravity as it flies, further data reduction of doppler radar measurements is required to separate yaw induced drag and lift coefficients from the zero yaw drag coefficient, in order to make measurements fully applicable to 6-dof trajectory analysis.

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 projectile's 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.

Doppler radar measurement results for a Lapua GB528 Scenar 19.44 g (300 gr) 8.59 mm (0.338 in) calibre very-low-drag bullet look like this:

Mach number 0.000 0.400 0.500 0.600 0.700 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975 1.000 1.025 1.050 1.075 1.100 1.150 1.200 1.300 1.400 1.500 1.600 1.800 2.000 2.200 2.400
Drag coefficient 0.230 0.229 0.200 0.171 0.164 0.144 0.141 0.137 0.137 0.142 0.154 0.177 0.236 0.306 0.334 0.341 0.345 0.347 0.348 0.348 0.343 0.336 0.328 0.321 0.304 0.292 0.282 0.270

This tested bullet experiences its maximum drag coefficient when entering the transonic flight regime around Mach 1.200.

<graph>{"legends":[],"scales":[{"clamp":true,"type":"linear","name":"x","domainMax":2.401,"zero":false,"domain":{"data":"chart","field":"x"},"range":"width","nice":true},{"type":"linear","name":"y","domain":{"data":"chart","field":"y"},"zero":false,"range":"height","nice":true},{"domain":{"data":"chart","field":"series"},"type":"ordinal","name":"color","range":"category10"}],"version":2,"marks":[{"type":"line","properties":{"hover":{"stroke":{"value":"red"}},"update":{"stroke":{"scale":"color","field":"series"}},"enter":{"y":{"scale":"y","field":"y"},"x":{"scale":"x","field":"x"},"stroke":{"scale":"color","field":"series"},"strokeWidth":{"value":2.5}}},"from":{"data":"chart"}}],"height":300,"axes":[{"type":"x","title":"Mach number","scale":"x","properties":{"title":{"fill":{"value":"#54595d"}},"grid":{"stroke":{"value":"#54595d"}},"ticks":{"stroke":{"value":"#54595d"}},"axis":{"strokeWidth":{"value":2},"stroke":{"value":"#54595d"}},"labels":{"fill":{"value":"#54595d"}}},"grid":true},{"type":"y","title":"Drag coefficient","scale":"y","properties":{"title":{"fill":{"value":"#54595d"}},"grid":{"stroke":{"value":"#54595d"}},"ticks":{"stroke":{"value":"#54595d"}},"axis":{"strokeWidth":{"value":2},"stroke":{"value":"#54595d"}},"labels":{"fill":{"value":"#54595d"}}},"grid":true}],"data":[{"format":{"parse":{"y":"number","x":"number"},"type":"json"},"name":"chart","values":[{"y":0.23,"series":"y","x":0},{"y":0.229,"series":"y","x":0.4},{"y":0.2,"series":"y","x":0.5},{"y":0.171,"series":"y","x":0.6},{"y":0.164,"series":"y","x":0.7},{"y":0.144,"series":"y","x":0.8},{"y":0.141,"series":"y","x":0.825},{"y":0.137,"series":"y","x":0.85},{"y":0.137,"series":"y","x":0.875},{"y":0.142,"series":"y","x":0.9},{"y":0.154,"series":"y","x":0.925},{"y":0.177,"series":"y","x":0.95},{"y":0.236,"series":"y","x":0.975},{"y":0.306,"series":"y","x":1},{"y":0.334,"series":"y","x":1.025},{"y":0.341,"series":"y","x":1.05},{"y":0.345,"series":"y","x":1.075},{"y":0.347,"series":"y","x":1.1},{"y":0.348,"series":"y","x":1.15},{"y":0.348,"series":"y","x":1.2},{"y":0.343,"series":"y","x":1.3},{"y":0.336,"series":"y","x":1.4},{"y":0.328,"series":"y","x":1.5},{"y":0.321,"series":"y","x":1.6},{"y":0.304,"series":"y","x":1.8},{"y":0.292,"series":"y","x":2},{"y":0.282,"series":"y","x":2.2},{"y":0.27,"series":"y","x":2.4}]}],"width":700}</graph>
Graphics for drag coefficient from Doppler radar measurement for a Lapua GB528 Scenar 19.44 g (300 gr) 8.59 mm (0.338 in) bullet

With the help of Doppler radar measurements projectile specific drag models can be established that are most useful when shooting at extended ranges where the bullet speed slows to the transonic speed region near the speed of sound. This is where the projectile drag predicted by mathematic modeling can significantly depart from the actual drag experienced by the projectile. Further Doppler radar measurements are used to study subtle in-flight effects of various bullet constructions.[35]

Governments, professional ballisticians, defence forces and ammunition manufacturers can supplement Doppler radar measurements with measurements gathered by telemetry probes fitted to larger projectiles.

General trends in drag or ballistic coefficient

File:7.9 SS and SME Patrone.jpg
Left German 7.9 mm s.S. (FMJ) and right S.m.E. ammunition beside their boat-tailed projectiles with cannelures

In general, a pointed projectile will have a better drag coefficient (Cd) or ballistic coefficient (BC) than a round nosed bullet, and a round nosed bullet will have a better Cd or BC than a flat point bullet. 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. Projectiles designed for supersonic use often have a slightly tapered base at the rear, called a boat tail, which reduces air resistance in flight.[36] The usefulness of a "tapered rear" for long-range firing was well established already by early 1870s,[37] but technological difficulties prevented their wide adoption before well into 20th century. Cannelures, which are recessed rings around the projectile used to crimp the projectile securely into the case, will cause an increase in drag.

Analytical software was developed by the Ballistics Research Laboratory – later called the Army Research Laboratory – which reduced actual test range data to parametric relationships for projectile drag coefficient prediction.[38] Large caliber artillery also employ drag reduction mechanisms in addition to streamlining geometry. Rocket-assisted projectiles employ a small rocket motor that ignites upon muzzle exit providing additional thrust to overcome aerodynamic drag. Rocket assist is most effective with subsonic artillery projectiles. For supersonic long range artillery, where base drag dominates, base bleed is employed. Base bleed is a form of a gas generator that does not provide significant thrust, but rather fills the low-pressure area behind the projectile with gas, effectively reducing the base drag and the overall projectile drag coefficient.

Transonic problem

A projectile fired at supersonic muzzle velocity will at some point slow to approach the speed of sound. At the transonic region (about Mach 1.2–0.8) the centre of pressure (CP) of most non spherical projectiles shifts forward as the projectile decelerates. That CP shift affects the (dynamic) stability of the projectile. If the projectile is not well stabilized, it cannot remain pointing forward through the transonic region (the projectile starts to exhibit an unwanted precession or coning motion called limit cycle yaw that, if not damped out, can eventually end in uncontrollable tumbling along the length axis). However, even if the projectile has sufficient stability (static and dynamic) to be able to fly through the transonic region and stays pointing forward, it is still affected. The erratic and sudden CP shift and (temporary) decrease of dynamic stability can cause significant dispersion (and hence significant accuracy decay), even if the projectile's flight becomes well behaved again when it enters the subsonic region. This makes accurately predicting the ballistic behavior of projectiles in the transonic region very difficult.

Because of this, marksmen normally restrict themselves to engaging targets close enough that the projectile is still supersonic.[note 1] In 2015 the American ballistician Bryan Litz introduced the "Extended Long Range" concept to define rifle shooting at ranges where supersonic fired (rifle) bullets enter the transonic region. According to Litz, "Extended Long Range starts whenever the bullet slows to its transonic range. As the bullet slows down to approach Mach 1, it starts to encounter transonic effects, which are more complex and difficult to account for, compared to the supersonic range where the bullet is relatively well-behaved."[39]

The ambient air density has a significant effect on dynamic stability during transonic transition. Though the ambient air density is a variable environmental factor, adverse transonic transition effects can be negated better by a projectile traveling through less dense air, than when traveling through denser air. Projectile or bullet length also affects limit cycle yaw. Longer projectiles experience more limit cycle yaw than shorter projectiles of the same diameter. Another feature of projectile design that has been identified as having an effect on the unwanted limit cycle yaw motion is the chamfer at the base of the projectile. At the very base, or heel of a projectile or bullet, there is a 0.25 to 0.50 mm (0.01 to 0.02 in) chamfer, or radius. The presence of this radius causes the projectile to fly with greater limit cycle yaw angles.[40] Rifling can also have a subtle effect on limit cycle yaw.[41] In general faster spinning projectiles experience less limit cycle yaw.

Research into guided projectiles

To circumvent the transonic problems encountered by spin-stabilized projectiles, projectiles can theoretically be guided during flight. The Sandia National Laboratories announced in January 2012 it has researched and test-fired 4-inch (102 mm) long prototype dart-like, self-guided bullets for small-caliber, smooth-bore firearms that could hit laser-designated targets at distances of more than a mile (about 1,610 meters or 1760 yards). These projectiles are not spin stabilized and the flight path can steered within limits with an electromagnetic actuator 30 times per second. The researchers also claim they have video of the bullet radically pitching as it exits the barrel and pitching less as it flies down range, a disputed phenomenon known to long-range firearms experts as “going to sleep”. Because the bullet's motions settle the longer it is in flight, accuracy improves at longer ranges, Sandia researcher Red Jones said. “Nobody had ever seen that, but we’ve got high-speed video photography that shows that it’s true,” he said.[42] Recent testing indicates it may be approaching or already achieved initial operational capability.[43]

Testing the predictive qualities of software

Due to the practical inability to know in advance and compensate for all the variables of flight, no software simulation, however advanced, will yield predictions that will always perfectly match real world trajectories. It is however possible to obtain predictions that are very close to actual flight behavior.

Empirical measurement method

Ballistic prediction computer programs intended for (extreme) long ranges can be evaluated by conducting field tests at the supersonic to subsonic transition range (the last 10 to 20% of the supersonic range of the rifle/cartridge/bullet combination). For a typical .338 Lapua Magnum rifle for example, shooting standard 16.2 gram (250 gr) Lapua Scenar GB488 bullets at 905 m/s (2969 ft/s) muzzle velocity, field testing of the software should be done at ≈ 1200-1300 meters (1312-1422 yd) under International Standard Atmosphere sea level conditions (air density ρ = 1.225 kg/m³). To check how well the software predicts the trajectory at shorter to medium range, field tests at 20, 40 and 60% of the supersonic range have to be conducted. At those shorter to medium ranges, transonic problems and hence unbehaved bullet flight should not occur, and the BC is less likely to be transient. Testing the predictive qualities of software at (extreme) long ranges is expensive because it consumes ammunition; the actual muzzle velocity of all shots fired must be measured to be able to make statistically dependable statements. Sample groups of less than 24 shots may not obtain the desired statistically significant confidence interval.

Doppler radar measurement method

Governments, professional ballisticians, defence forces and a few ammunition manufacturers use Doppler radars and/or telemetry probes fitted to larger projectiles to obtain precise real world data regarding the flight behavior of the specific projectiles of their interest and thereupon compare the gathered real world data against the predictions calculated by ballistic computer programs. The normal shooting or aerodynamics enthusiast, however, has no access to such expensive professional measurement devices. Authorities and projectile manufacturers are generally reluctant to share the results of Doppler radar tests and the test derived drag coefficients (Cd) of projectiles with the general public. Around 2020 more affordable but less capable (amateur) Doppler rader equipment to determine free flight drag coefficients became available for the general public.[44]

In January 2009 the Scandinavian ammunition manufacturer Nammo/Lapua published Doppler radar test-derived drag coefficient data for most of their rifle projectiles.[45][46] In 2015 the US ammunition manufacturer Berger Bullets announced the use of Doppler radar in unison with PRODAS 6 DoF software to generate trajectory solutions.[47] In 2016 US ammunition manufacturer Hornady announced the use of Doppler radar derived drag data in software utilizing a modified point mass model to generate trajectory solutions.[48][49][50][51] With the measurement derived Cd data engineers can create algorithms that utilize both known mathematical ballistic models as well as test specific, tabular data in unison. When used by predictive software like QuickTARGET Unlimited, Lapua Edition,[52] Lapua Ballistics[53] or Hornady 4DOF the Doppler radar test-derived drag coefficient data can be used for more accurate external ballistic predictions.

Some of the Lapua-provided drag coefficient data shows drastic increases in the measured drag around or below the Mach 1 flight velocity region. This behavior was observed for most of the measured small calibre bullets, and not so much for the larger calibre bullets. This implies some (mostly smaller calibre) rifle bullets exhibited more limit cycle yaw (coning and/or tumbling) in the transonic/subsonic flight velocity regime. The information regarding unfavourable transonic/subsonic flight behavior for some of the tested projectiles is important. This is a limiting factor for extended range shooting use, because the effects of limit cycle yaw are not easily predictable and potentially catastrophic for the best ballistic prediction models and software.

Presented Cd data can not be simply used for every gun-ammunition combination, since it was measured for the barrels, rotational (spin) velocities and ammunition lots the Lapua testers used during their test firings. Variables like differences in rifling (number of grooves, depth, width and other dimensional properties), twist rates and/or muzzle velocities impart different rotational (spin) velocities and rifling marks on projectiles. Changes in such variables and projectile production lot variations can yield different downrange interaction with the air the projectile passes through that can result in (minor) changes in flight behavior. This particular field of external ballistics is currently (2009) not elaborately studied nor well understood.[54]

Predictions of several drag resistance modelling and measuring methods

The method employed to model and predict external ballistic behavior can yield differing results with increasing range and time of flight. To illustrate this several external ballistic behavior prediction methods for the Lapua Scenar GB528 19.44 g (300 gr) 8.59 mm (0.338 in) calibre very-low-drag rifle bullet with a manufacturer stated G1 ballistic coefficient (BC) of 0.785 fired at 830 m/s (2723 ft/s) muzzle velocity under International Standard Atmosphere sea level conditions (air density ρ = 1.225 kg/m³), Mach 1 = 340.3 m/s, Mach 1.2 = 408.4 m/s), predicted this for the projectile velocity and time of flight from 0 to 3,000 m (0 to 3,281 yd):[note 2]

Range (m) 0 300 600 900 1,200 1,500 1,800 2,100 2,400 2,700 3,000
Radar test derived drag coefficients method V (m/s) 830 711 604 507 422 349 311 288 267 247 227
Time of flight (s) 0.0000 0.3918 0.8507 1.3937 2.0435 2.8276 3.7480 4.7522 5.8354 7.0095 8.2909
Total drop (m) 0.000 0.715 3.203 8.146 16.571 30.035 50.715 80.529 121.023 173.998 241.735
6 DoF modeling method V (m/s) 830 711 604 506 420 347 310 287 266 244 222
Time of flight (s) 0.0000 0.3919 0.8511 1.3949 2.0467 2.8343 3.7575 4.7641 5.8508 7.0332 8.3346
Total drop (m) 0.000 0.714 3.195 8.132 16.561 30.060 50.836 80.794 121.498 174.796 243.191
G1 drag model method V (m/s) 830 718 615 522 440 374 328 299 278 261 248
Time of flight (s) 0.0000 0.3897 0.8423 1.3732 2.0009 2.7427 3.6029 4.5642 5.6086 6.7276 7.9183
Total drop (m) 0.000 0.710 3.157 7.971 16.073 28.779 47.810 75.205 112.136 160.739 222.430
Pejsa drag model method V (m/s) 830 712 603 504 413 339 297 270 247 227 208
Time of flight (s) 0.0000 0.3902 0.8479 1.3921 2.0501 2.8556 3.8057 4.8682 6.0294 7.2958 8.6769
Total drop (m) 0.000 0.719 3.198 8.129 16.580 30.271 51.582 82.873 126.870 185.318 260.968
G7 drag model method V (m/s) 830 713 606 508 418 339 303 283 265 249 235
Time of flight (s) 0.0000 0.3912 0.8487 1.3901 2.0415 2.8404 3.7850 4.8110 5.9099 7.0838 8.3369
Total drop (m) 0.000 0.714 3.191 8.109 16.503 30.039 51.165 81.863 123.639 178.082 246.968

The table shows the Doppler radar test derived drag coefficients (Cd) prediction method and the 2017 Lapua Ballistics 6 DoF App predictions produce similar results. The 6 DoF modeling estimates bullet stability ((Sd) and (Sg)) that gravitates to over-stabilization for ranges over 2,400 m (2,625 yd) for this bullet. At 2,400 m (2,625 yd) the total drop predictions deviate 47.5 cm (19.7 in) or 0.20 mil (0.68 moa) at 50° latitude and up to 2,700 m (2,953 yd) the total drop predictions are within 0.30 mil (1 moa) at 50° latitude. The 2016 Lapua Ballistics 6 DoF App version predictions were even closer to the Doppler radar test predictions.

The traditional Siacci/Mayevski G1 drag curve model prediction method generally yields more optimistic results compared to the modern Doppler radar test derived drag coefficients (Cd) prediction method.[note 3] At 300 m (328 yd) range the differences will be hardly noticeable, but at 600 m (656 yd) and beyond the differences grow over 10 m/s (32.8 ft/s) projectile velocity and gradually become significant. At 1,500 m (1,640 yd) range the projectile velocity predictions deviate 25 m/s (82.0 ft/s), which equates to a predicted total drop difference of 125.6 cm (49.4 in) or 0.83 mil (2.87 moa) at 50° latitude.

The Pejsa drag model closed-form solution prediction method, without slope constant factor fine tuning, yields very similar results in the supersonic flight regime compared to the Doppler radar test derived drag coefficients (Cd) prediction method. At 1,500 m (1,640 yd) range the projectile velocity predictions deviate 10 m/s (32.8 ft/s), which equates to a predicted total drop difference of 23.6 cm (9.3 in) or 0.16 mil (0.54 moa) at 50° latitude.

The G7 drag curve model prediction method (recommended by some manufacturers for very-low-drag shaped rifle bullets) when using a G7 ballistic coefficient (BC) of 0.377 yields very similar results in the supersonic flight regime compared to the Doppler radar test derived drag coefficients (Cd) prediction method. At 1,500 m (1,640 yd) range the projectile velocity predictions have their maximum deviation of 10 m/s (32.8 ft/s). The predicted total drop difference at 1,500 m (1,640 yd) is 0.4 cm (0.16 in) at 50° latitude. The predicted total drop difference at 1,800 m (1,969 yd) is 45.0 cm (17.7 in), which equates to 0.25 mil (0.86 moa).

Decent prediction models are expected to yield similar results in the supersonic flight regime. The five example models down to 1,200 m (1,312 yd) all predict supersonic Mach 1.2+ projectile velocities and total drop differences within a 51 cm (20.1 in) bandwidth. In the transonic flight regime at 1,500 m (1,640 yd) the models predict projectile velocities around Mach 1.0 to Mach 1.1 and total drop differences within a much larger 150 cm (59 in) bandwidth.

  1. ↑ "Maximum Point Blank Range and the Battlesight Zero". 30 April 2010. Retrieved 24 December 2022.
  2. ↑ "An Improved Battlesight Zero for the M4 Carbine and M16A2 Rifle". Retrieved 2007-09-11.
  3. ↑ "TM 9-1005-319-10 (2010) - Operator's Manual for Rifle, 5.56 MM, M16A2/M16A3/M4 (Battlesight Zero pages 48-55)" (PDF). Retrieved 2014-06-03.
  4. ↑ "Using a 400 meter zero with 5.45×39". Retrieved 16 November 2014.
  5. ↑ "Prediction of Projectile Performance, Stability, and Free-Flight Motion Using Computational Fluid Dynamics, Weinacht, US Army Research Laboratory, 2003" (PDF). Archived from the original (PDF) on 2021-01-21. Retrieved 2022-06-02.
  6. ↑ "Ballistic Coefficients Do Not Exist!". Retrieved 12 January 2017.
  7. ↑ "Weite Schüsse - drei (German)". Archived from the original on 24 May 2008. Retrieved 24 December 2022.
  8. ↑ "exterior ballistics". Archived from the original on 8 August 2003. Retrieved 12 January 2017.
  9. ↑ "LM Class Bullets, very high BC bullets for windy long Ranges". Archived from the original on 19 February 2008. Retrieved 24 December 2022.
  10. ↑ Litz, Bryan (8 March 2021) [Originally posted 17 June 2009, much extended later]. "A Better Ballistic Coefficient". Archived from the original on 2 August 2009.
  11. ↑ ".338 Lapua Magnum product brochure from Lapua" (PDF). Retrieved 24 December 2022.
  12. ↑ "300 grs Scenar HPBT brochure from Lapua" (PDF). Retrieved 24 December 2022.
  13. ↑ "Ballistic Coefficients - Explained" (PDF). Archived from the original (PDF) on 29 October 2013. Retrieved 12 January 2017.
  14. ↑ "The mathematical modelling of projectile trajectories under the influence of environmental effects, Ryan F. Hooke,∗University of New South Wales Canberra at the Australian Defence Force Academy, 2612, Australia". Archived from the original on 2018-02-04. Retrieved 2018-02-02.
  15. ↑ "Form Factors: A Useful Analysis Tool - Berger Bullets Blog". 16 May 2011. Retrieved 12 January 2017.
  16. ↑ About Art Pejsa Archived 2008-08-29 at the Wayback Machine
  17. ↑ Pejsa, Arthur. "Pejsa Rifle Ballistics: Art Pejsa's Rifle ballistics software and books". Pejsa Ballistics. Pejsa Ballistics. Archived from the original on 8 February 2012. Retrieved 31 July 2018.
  18. ↑ Arthur J Pejsa (2002). Pejsa's Handbook of New, Precision Ballistics. Kenwood Publishing. p. 3.
  19. ↑ 19.0 19.1 Arthur J Pejsa (2008). New Exact Small Arms Ballistics. Kenwood Publishing. pp. 65–76.
  20. ↑ Arthur J Pejsa (2008). New Exact Small Arms Ballistics. Kenwood Publishing. p. 63.
  21. ↑ Arthur J Pejsa (2002). Pejsa's Handbook of New, Precision Ballistics. Kenwood Publishing. p. 34.
  22. ↑ Arthur J Pejsa (2002). Pejsa's Handbook of New, Precision Ballistics. Kenwood Publishing. p. 4.
  23. ↑ Arthur J Pejsa (2008). New Exact Small Arms Ballistics. Kenwood Publishing. pp. 131–134.
  24. ↑ Meijer, Robert. "Pejsa and regular drag functions". Retrieved 12 January 2017.
  25. ↑ Arthur J Pejsa (2008). New Exact Small Arms Ballistics. Kenwood Publishing. pp. 33–35.
  26. ↑ "Test Options & Analysis Techniques:Aerodynamic Coefficients: What's Important & How Can I Measure Them? Jeff Siewert Systems Engineer Arrow Tech Associates, Inc. 2012 NDIA Joint Armaments" (PDF). Archived from the original (PDF) on 2016-12-13. Retrieved 2016-10-15.
  27. ↑ SPIN-73 An Updated Version of the Spinner Computer Program, White, 1973
  28. ↑ Aerodynamic Design Manual for Tactical Weapons, NSWC 1981
  29. ↑ Improved Solids Modeling for Axisymmetric Projectile Design, 1988
  30. ↑ "Six Degree of Freedom Digital Simulation Model for Unguided Fin-Stabilized Rockets" (PDF). Archived from the original (PDF) on 2017-02-21. Retrieved 24 December 2022.
  31. ↑ "Lapua Ballistics App - Resources - Nammo Lapua". Archived from the original on 20 December 2016. Retrieved 12 January 2017.
  32. ↑ "NABK BASED NEXT GENERATION BALLISTIC TABLE TOOLKIT, Sevsay Aytar Ortac, Umut Durak, Umit Kutluay, Koray Kucuk, Maj. Can Candan, 23RD INTERNATIONAL SYMPOSIUM ON BALLISTICS, TARRAGONA, SPAIN 16-20 APRIL 2007". Retrieved 24 December 2022.
  33. ↑ "BALCO 6/7-DoF Trajectory Model, Pierre Wey, Daniel Corriveau, Thomas A. Saitz, Wim de Ruijter, Peter Strömbäck, 29th International Symposium on Ballistics, Edinburgh, Scotland, May 9‐13, 2016". Retrieved 24 December 2022.
  34. ↑ Corriveau, D. (2017). "Validation of the NATO Armaments Ballistic Kernel for use in small-arms fire control systems". Defence Technology. 13 (3): 188–199. doi:10.1016/j.dt.2017.04.006.
  35. ↑ "A Technical Discussion of the ELD-X™ (Extremely Low Drag – eXpanding) & ELD™ Match (Extremely Low Drag Match) Bullets with Heat Shield™ Tip, Dave Emary, October 2015" (PDF). Archived from the original (PDF) on 2015-11-06. Retrieved 2015-11-01.
  36. ↑ Chuck Hawks. "The 8x50R Lebel (8mm Lebel)".
  37. ↑ Richardson v. United States, 72 Ct. Cl. 51 (1930)
  38. ↑ MC DRAG - A Computer Program for Estimating the Drag Coefficients of Projectiles, McCoy, US Army Ballistic Research Laboratory, 1981
  39. ↑ "New Video Series from Applied Ballistics « Daily Bulletin". Archived from the original on 21 October 2016. Retrieved 12 January 2017.
  40. ↑ "The Effect of Boattail Geometry on the Yaw Limit Cycle of Small Caliber Projectiles by Bradley E. Howell Data Matrix Solutions, Aberdeen Proving Ground, MD 21005-5066 and Sidra I. Silton and Paul Weinacht Weapons and Materials Research Directorate, ARL, Aberdeen Proving Ground, MD 21005-5066 27th AIAA Applied Aerodynamics Conference 22 - 25 June 2009, San Antonio, Texas" (PDF). Archived from the original (PDF) on 2016-04-06. Retrieved 2015-01-11.
  41. ↑ "EFFECT OF RIFLING GROOVES ON THE PERFORMANCE OF SMALL-CALIBER AMMUNITION Sidra I. Silton and Paul Weinacht US Army Research Laboratory Aberdeen Proving Ground, MD 21005-5066". Archived from the original on 2015-01-11. Retrieved 24 December 2022.
  42. ↑ "Sandia National Laboratories: News Releases : Sandia's self-guided bullet prototype can hit target a mile away". Archived from the original on 5 February 2012. Retrieved 12 January 2017.
  43. ↑ "Guided .50 Caliber Projectile — DARPA's Steerable Bullet". Retrieved 2020-09-27.
  44. ↑ Courtney, Elya, Collin Morris, and Michael Courtney. "Accurate Measurements of Free Flight Drag Coefficients with Amateur Doppler Radar." Cornell University Library (2016). arXiv:1608.06500
  45. ↑ "Lapua Bullets Drag Coefficient Data for QuickTARGET Unlimited" (PDF). Archived from the original (PDF) on 2010-12-29. Retrieved 24 December 2022.
  46. ↑ "Lapua bullets CD data (zip file)". Retrieved 24 December 2022.
  47. ↑ "Use of Doppler Radar to Generate Trajectory Solutions" (PDF). Archived from the original (PDF) on 2015-09-23. Retrieved 2015-07-22.
  48. ↑ "Hornady® 4DOF Ballistic Calculator Overview". Archived from the original on 2016-08-23. Retrieved 12 January 2017.
  49. ↑ "4DOF Ballistic Calculator". Retrieved 12 January 2017.
  50. ↑ "Drag Variability and the use of the "Axial Form Factor" in the Hornady 4DOF™ Trajectory Calculator" (PDF). Archived from the original (PDF) on 19 October 2016. Retrieved 14 October 2016.
  51. ↑ "Hornady® 4DOF™ (Four Degree of Freedom) Ballistic Calculator Technical Document" (PDF). Archived from the original (PDF) on 2016-10-19. Retrieved 2016-10-14.
  52. ↑ "QuickTARGET Unlimited, Lapua Edition". Archived from the original on 1 July 2012. Retrieved 24 December 2022.
  53. ↑ "Lapua Ballistics freeware exterior ballistic software for mobile phones". Archived from the original on 1 July 2012. Retrieved 24 December 2022.
  54. ↑ "EFFECT OF RIFLING GROOVES ON THE PERFORMANCE OF SMALL-CALIBER AMMUNITION Sidra I. Silton* and Paul Weinacht US Army Research Laboratory Aberdeen Proving Ground, MD 21005-5066". Archived from the original on 2012-10-06. Retrieved 24 December 2022.


Cite error: <ref> tags exist for a group named "note", but no corresponding <references group="note"/> tag was found

Retrieved from "https://www.ooda.wiki/index.php?title=Bullet_drop&oldid=1978"