713:(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, 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. The FINNER aeroprediction code calculates 6-dof inputs for fin stabilized projectiles. 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. Finally, algorithms for 6-dof numerical integration suitable to a 4th order Runge-Kutta are readily available. All that is required for the amateur ballistician to investigate the finer analytical details of projectile trajectories, along with bullet nutation and
2112:. The Rifleman's rule and the slightly more complex and less well known Improved Rifleman's rule models produce sufficiently accurate predictions for many small arms applications. Simple prediction models however ignore minor gravity effects when shooting uphill or downhill. The only practical way to compensate for this is to use a ballistic computer program. Besides gravity at very steep angles over long distances, the effect of air density changes the projectile encounters during flight become problematic. The mathematical prediction models available for inclined fire scenarios, depending on the amount and direction (uphill or downhill) of the inclination angle and range, yield varying accuracy expectation levels. Less advanced ballistic computer programs predict the same trajectory for uphill and downhill shots at the same vertical angle and range. The more advanced programs factor in the small effect of gravity on uphill and on downhill shots resulting in slightly differing trajectories at the same vertical angle and range. No publicly available ballistic computer program currently (2017) accounts for the complicated phenomena of differing air densities the projectile encounters during flight.
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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.
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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.
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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. The second equation is identical to the one used to find the weighted average at R / 4; add N Ă (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. 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 Ă (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
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Since dynamic stability is mostly governed by transonic aerodynamics, it is very hard to predict when a projectile will have sufficient dynamic stability (these are the hardest aerodynamic coefficients to calculate accurately at the most difficult speed regime to predict (transonic)). The aerodynamic coefficients that govern dynamic stability: pitching moment, Magnus moment and the sum of the pitch and angle of attack dynamic moment coefficient (a very hard quantity to predict). In the end, there is little that modeling and simulation can do to accurately predict the level of dynamic stability that a bullet will have downrange. If a projectile has a very high or low level of dynamic stability, modeling may get the answer right. However, if a situation is borderline (dynamic stability near 0 or 2) modeling cannot be relied upon to produce the right answer. This is one of those things that have to be field tested and carefully documented.
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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.
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because the difference in the point of impact between 400 and 500 yd (460 m) is 25â32 in (depending on zero), in other words if the shooter estimates that the target is 400 yd away when it is in fact 500 yd away the shot will impact 25â32 in (635â813 mm) below where it was aimed, possibly missing the target completely. Secondly, the rifle should be zeroed to a distance appropriate to the typical range of targets, because the shooter might have to aim so far above the target to compensate for a large bullet drop that he may lose sight of the target completely (for instance being outside the field of view of a telescopic sight). In the example of the rifle zeroed at 200 yd (180 m), the shooter would have to aim 49 in or more than 4 ft (1.2 m) above the point of impact for a target at 500 yd.
2108:) of a shot will also affect the trajectory of the shot. Ballistic tables for small calibre projectiles (fired from pistols or rifles) assume a horizontal line of sight between the shooter and target with gravity acting perpendicular to the earth. Therefore, if the shooter-to-target angle is up or down, (the direction of the gravity component does not change with slope direction), then the trajectory curving acceleration due to gravity will actually be less, in proportion to the cosine of the slant angle. As a result, a projectile fired upward or downward, on a so-called "slant range," will over-shoot the same target distance on flat ground. The effect is of sufficient magnitude that hunters must adjust their target hold off accordingly in mountainous terrain. A well known formula for slant range adjustment to horizontal range hold off is known as the
658:Ï = 1.2209 kg/m). 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. 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.
596:). 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.
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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.
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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 âgoing to sleepâ. Because the bullet's motions settle the longer it is in flight, accuracy improves at longer ranges, Sandia researcher Red Jones said. âNobody had ever seen that, but weâve got high-speed video photography that shows that itâs true,â he said. Recent testing indicates it may be approaching or already achieved initial operational capability.
2340:, whereby the spin of the bullet creates a force acting either up or down, perpendicular to the sideways vector of the wind. In the simple case of horizontal wind, and a right hand (clockwise) direction of rotation, the Magnus effect induced pressure differences around the bullet cause a downward (wind from the right) or upward (wind from the left) force viewed from the point of firing to act on the projectile, affecting its point of impact. The vertical deflection value tends to be small in comparison with the horizontal wind induced deflection component, but it may nevertheless be significant in winds that exceed 4 m/s (14.4 km/h or 9 mph).
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segments fits the form V / C and the retardation coefficient curve segments fits the form V / (V / C) = C Ă V 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,
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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. Rifling can also have a subtle effect on limit cycle yaw. In general faster spinning projectiles experience less limit cycle yaw.
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1330:Ï = 1.225 kg/mÂł). 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
325:
562:). 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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accounts for aerodynamic behavior along the three axial directionsâelevation, range, and deflectionâand the three rotational directionsâpitch, 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.
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horizontal plane; for right-handed (clockwise) spinning bullets, the bullet's axis of symmetry deflects to the right and a little bit upward with respect to the direction of the velocity vector, as the projectile moves along its ballistic arc. As the result of this small inclination, there is a continuous air stream, which tends to deflect the bullet to the right. Thus the occurrence of the yaw of repose is the reason for the bullet drifting to the right (for right-handed spin) or to the left (for left-handed spin). This means that the bullet is "skidding" sideways at any given moment, and thus experiencing a sideways component.
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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.
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622:
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
2046:) prediction method. 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.
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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.
75:. However, exterior ballistics analysis also deals with the trajectories of rocket-assisted gun-launched projectiles and gun-launched rockets; and rockets that acquire all their trajectory velocity from the interior ballistics of their on-board propulsion system, either a rocket motor or air-breathing engine, both during their boost phase and after motor burnout. External ballistics is also concerned with the free-flight of other projectiles, such as
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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.
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2167:"yaw of repose." For a right hand (clockwise) direction of rotation this component will always be to the right. For a left hand (counterclockwise) direction of rotation this component will always be to the left. This is because the projectile's longitudinal axis (its axis of rotation) and the direction of the velocity vector of the center of gravity (CG) deviate by a small angle, which is said to be the equilibrium
637:
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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:
2062:) 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).
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417:. 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
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been completely missed. When the same target was set up at a less challenging 1000 m distance it could be hit between 987 m and 1013 m, meaning a 1.3% ranging error would just be acceptable to be able to hit a 2 MOA tall target with a .338 Lapua Magnum sniper round. This makes it obvious that with increasing distance apparently minor measuring and judgment errors become a major problem.
3933:"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"
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359:
350:, 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. The most reliable method, therefore, of establishing the necessary projectile aerodynamic properties to properly describe flight trajectories is by empirical measurement.
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projectile turn into the wind, much like a weather vane, keeping the centre of air pressure on its nose. From the shooterâs perspective, this causes the nose of the projectile to turn into the wind and the tail to turn away from the wind. The result of this turning effect is that the drag pushes the projectile downwind in a nose-to-tail direction.
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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.
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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
814:). 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.
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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
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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."
122:, 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.
2973:) prediction method predictions calculated with QuickTARGET Unlimited, Lapua Edition. Pejsa predictions calculated with Lex Talus Corporation Pejsa based ballistic software with the slope constant factor set at the 0.5 default value. 6 DoF modeling predictions calculated with Lapua Ballistics 1.0 app for Android.
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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.
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changes slightly with latitude) to hit a human torso sized target dead centre at 1400 m. The ballistic curve plot showed that between 1392 m and 1408 m the bullets would have hit a 60 cm (2 ft) tall target. This means that if only a 0.6% ranging error was made a 60 cm tall target at 1400 m would have
3007:
website defines effective range as: The range in which a competent and trained individual using the firearm has the ability to hit a target sixty to eighty percent of the time. In reality, most firearms have a true range much greater than this but the likelihood of hitting a target is poor at greater
2446:
Viewed from a non-rotating reference frame (i.e. not one rotating with the Earth) and ignoring the forces of gravity and air resistance, a projectile moves in a straight line. When viewed from a reference frame fixed with respect to the Earth, that straight trajectory appears to curve sideways. The
2319:
The table shows that the gyroscopic drift cannot be predicted on weight and diameter alone. In order to make accurate predictions on gyroscopic drift several details about both the external and internal ballistics must be considered. Factors such as the twist rate of the barrel, the velocity of the
2166:
Gyroscopic drift is an interaction of the bullet's mass and aerodynamics with the atmosphere that it is flying in. Even in completely calm air, with no sideways air movement at all, a spin-stabilized projectile will experience a spin-induced sideways component, due to a gyroscopic phenomenon known as
1370:
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
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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
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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
616:
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 Ă 3.0 = 0.75, in some places 0.8 rather than 0.75
557:
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
301:
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
271:
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
2959:
Most spin-stabilized projectiles that suffer from lack of dynamic stability have the problem near the speed of sound where the aerodynamic forces and moments exhibit great changes. It is less common (but possible) for bullets to display significant lack of dynamic stability at supersonic velocities.
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causes
Coriolis drift in a direction perpendicular to the Earth's axis; for most locations on Earth and firing directions, this deflection includes horizontal and vertical components. The deflection is to the right of the trajectory in the northern hemisphere, to the left in the southern hemisphere,
2148:
Precipitation can cause significant yaw and accompanying deflection when a bullet collides with a raindrop. The further downrange such a coincidental collision occurs, the less the deflection on target will be. The weight of the raindrop and bullet also influences how much yaw is induced during such
2065:
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
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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
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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
805:
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
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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
678:
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
553:
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
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or the yaw of repose. The magnitude of the yaw of repose angle is typically less than 0.5 degree. Since rotating objects react with an angular velocity vector 90 degrees from the applied torque vector, the bullet's axis of symmetry moves with a component in the vertical plane and a component in the
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Because of this, marksmen normally restrict themselves to engaging targets close enough that the projectile is still supersonic. 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
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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
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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
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trajectory for that particular target. 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
2881:
This table demonstrates that, even with a fairly aerodynamic bullet fired at high velocity, the "bullet drop" or change in the point of impact is significant. This change in point of impact has two important implications. Firstly, estimating the distance to the target is critical at longer ranges,
2462:
The Eötvös effect is largest at the equator and decreases to zero at the poles. It causes eastward-traveling projectiles to deflect upward, and westward-traveling projectiles to deflect downward. The effect is less pronounced for trajectories in other directions, and is zero for trajectories aimed
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of the projectile, and increase drag and the corresponding drop. A tailwind will reduce the drag and the projectile/bullet drop. In the real world, pure head or tailwinds are rare, since wind is seldom constant in force and direction and normally interacts with the terrain it is blowing over. This
2083:
Wind also causes aerodynamic jump which is the vertical component of cross wind deflection caused by lateral (wind) impulses activated during free flight of a projectile or at or very near the muzzle leading to dynamic imbalance. The amount of aerodynamic jump is dependent on cross wind speed, the
1342:
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
934:
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.
310:
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,
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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
745:
only and available for Android and iOS devices. 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
644:
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
2510:
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 at point targets. The data to calculate these fire
762:
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. A six- and seven-degree-of-freedom standard called BALCO has also been developed
666:
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
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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
2389:
Another minor cause of drift, which depends on the nose of the projectile being above the trajectory, is the Poisson Effect. This, if it occurs at all, acts in the same direction as the gyroscopic drift and is even less important than the Magnus effect. It supposes that the uptilted nose of the
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the center of gravity. The location of the center of pressure depends on the flow field structure, in other words, depending on whether the bullet is in supersonic, transonic or subsonic flight. What this means in practice depends on the shape and other attributes of the bullet, in any case the
1312:
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
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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
293:
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
280:
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
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Wind has a range of effects, the first being the effect of making the projectile deviate to the side (horizontal deflection). From a scientific perspective, the "wind pushing on the side of the projectile" is not what causes horizontal wind drift. What causes wind drift is drag. Drag makes the
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In January 2009, the Scandinavian ammunition manufacturer Nammo/Lapua published Doppler radar test-derived drag coefficient data for most of their rifle projectiles. In 2015 the US ammunition manufacturer Berger Bullets announced the use of Doppler radar in unison with PRODAS 6 DoF software to
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at three different points. 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
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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
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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
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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 200 yards
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1.2â0.8) 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
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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.
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Projectiles like small arms bullets and artillery shells must deal with their CP being in front of their CM, which destabilizes these projectiles during flight. To stabilize such projectiles the projectile is spun around its longitudinal (leading to trailing) axis. The spinning mass creates
674:
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
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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. The model allows the drag curve to change slopes
2057:
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
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due to their length have a tendency to exhibit greater Magnus destabilizing errors because they have a greater surface area to present to the oncoming air they are travelling through, thereby reducing their aerodynamic efficiency. This subtle effect is one of the reasons why a calculated
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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
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due north or south. In the case of large changes of momentum, such as a spacecraft being launched into Earth orbit, the effect becomes significant. It contributes to the fastest and most fuel-efficient path to orbit: a launch from the equator that curves to a directly eastward heading.
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This simple explanation is quite popular. There is, however, no evidence to show that increased pressure means increased friction and unless this is so, there can be no effect. Even if it does exist it must be quite insignificant compared with the gyroscopic and Coriolis drifts.
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paragraph become important and have to be taken into account. The practical effects of these minor variables are generally irrelevant for most firearms users, since normal group scatter at short and medium ranges prevails over the influence these effects exert on projectile
579:. 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. The Pejsa model is a
2422:, the magnitude of the Coriolis effect is generally insignificant (for high powered rifles in the order of about 10 cm (3.9 in) at 1,000 m (1,094 yd)), but for ballistic projectiles with long flight times, such as extreme long-range rifle projectiles,
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ballistic software (requires free registration to download) - Supports the G1, G2, G5, G6, G7, G8, GL, GS Spherical 9/16"SAAMI, GS Spherical Don Miller, RA4, Soviet 1943, British 1909 Hatches Notebook and for some Lapua projectiles doppler radar-test derived (Cd) drag
449:) 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
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The ability to hit a point 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
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direction of this horizontal curvature is to the right in the northern hemisphere and to the left in the southern hemisphere, and does not depend on the azimuth of the shot. The horizontal curvature is largest at the poles and decreases to zero at the equator.
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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
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when a projectile leaves a gun barrel off axis leading to static imbalance. If present it causes dispersion. The effect is unpredictable, since it is generally small and varies from projectile to projectile, round to round and/or gun barrel to gun barrel.
2054:) 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.
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9â11, 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 665â674. Originally conceived to model projectile drag for
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paragraph have to be taken into account for small arms. Meso variables can become significant for firearms users that have to deal with angled shot scenarios or extended ranges, but are seldom relevant at common hunting and target shooting distances.
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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,
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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.
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Though not forces acting on projectile trajectories there are some equipment related factors that influence trajectories. Since these factors can cause otherwise unexplainable external ballistic flight behavior they have to be briefly mentioned.
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projectile causes an air cushion to build up underneath it. It further supposes that there is an increase of friction between this cushion and the projectile so that the latter, with its spin, will tend to roll off the cushion and move sideways.
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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
2039:) 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.
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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
1224:, which reduces air resistance in flight. The usefulness of a "tapered rear" for long-range firing was well established already by early 1870s, but technological difficulties prevented their wide adoption before well into 20th century.
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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
4895:
Set of MS Excel add-ins functions - Supports the G1, G2, G5, G6, G7 G8 and RA4 and Pejsa drag models as well as one for air rifle pellets. Able to handle user supplied models, e.g. Lapua projectiles doppler radar-test derived (Cd)
31:
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Lateral jump is caused by a slight lateral and rotational movement of a gun barrel at the instant of firing. It has the effect of a small error in bearing. The effect is ignored, since it is small and varies from round to round.
63:
in flight. The projectile may be powered or un-powered, guided or unguided, spin or fin stabilized, flying through an atmosphere or in the vacuum of space, but most certainly flying under the influence of a gravitational field.
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Both the Poisson and Magnus Effects will reverse their directions of drift if the nose falls below the trajectory. When the nose is off to one side, as in equilibrium yaw, these effects will make minute alterations in range.
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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
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An example of how accurate a long-range shooter has to establish sighting parameters to calculate a correct ballistic solution is explained by these test shoot results. A .338 Lapua Magnum rifle sighted in at 300 m shot 250
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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 â 1200-1300 meters (1312-1422 yd) under
3416:"The mathematical modelling of projectile trajectories under the influence of environmental effects, Ryan F. Hooke,âUniversity of New South Wales Canberra at the Australian Defence Force Academy, 2612, Australia"
1239:â which reduced actual test range data to parametric relationships for projectile drag coefficient prediction. Large caliber artillery also employ drag reduction mechanisms in addition to streamlining geometry.
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data is used by engineers to create algorithms that utilize both known mathematical ballistic models as well as test specific, tabular data in unison to obtain predictions that are very close to actual flight
510:
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:
3764:"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"
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and meteorological measuring equipment as aids to determine ballistic solutions, long-range shooting beyond 1000 m (1100 yd) at unknown ranges becomes guesswork for even the most expert long-range marksmen.
2031:)) 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
1351:) 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.
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gyroscopic stability of the bullet at the muzzle and if the barrel twist is clockwise or anti-clockwise. Like the wind direction reversing the twist direction will reverse the aerodynamic jump direction.
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Governments, professional ballisticians, defence forces and ammunition manufacturers can supplement Doppler radar measurements with measurements gathered by telemetry probes fitted to larger projectiles.
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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
176:, the altitudes involved have a significant effect as well, with part of the flight taking place in a near-vacuum well above a rotating Earth, steadily moving the target from where it was at launch time.
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Online trajectory calculators - Supports the G1, G2, G5, G6, G7 (for some projectiles experimentally measured G7 ballistic coefficients), G8, GI, GL and for some projectiles doppler radar-test derived (C
626:. Dr. Pejsa states that the retardation coefficient can be modeled by C Ă V where C is a fitting coefficient which disappears during the derivation of the drop formula and N the slope constant factor.
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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.
3642:"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"
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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.)
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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 (
3785:"BALCO 6/7-DoF Trajectory Model, Pierre Wey, Daniel Corriveau, Thomas A. Saitz, Wim de Ruijter, Peter StrömbĂ€ck, 29th International Symposium on Ballistics, Edinburgh, Scotland, May 9â13, 2016"
208:(CM) with tail surfaces. The CP behind the CM condition yields stable projectile flight, meaning the projectile will not overturn during flight through the atmosphere due to aerodynamic forces.
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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.
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aerodynamic jump (the vertical component of cross wind deflection caused by lateral (wind) impulses activated during free flight or at or very near the muzzle leading to dynamic imbalance)
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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.
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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.
671:, 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.
2140:
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.
439:
with BC's â„ 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.
2042:
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 (C
153:, although air resistance affects this. Extreme long range projectiles are subject to significant deflections, depending on circumstances, from the line toward the target; and all
1363:
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
2348:
The Magnus effect has a significant role in bullet stability because the Magnus force does not act upon the bullet's center of gravity, but the center of pressure affecting the
4300:"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"
2459:
changes the perceived gravitational pull on a moving object based on the relationship between the direction and velocity of movement and the direction of the Earth's rotation.
35:
34:
4774:
The Production of Firing Tables for Cannon Artillery, BRL rapport no. 1371 by Elizabeth R. Dickinson, U.S. Army Materiel Command Ballistic Research Laboratories, November 1967
3964:"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"
36:
4837:
1404:Ï = 1.225 kg/mÂł), 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):
445:
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
421:
in 1881. The G1 model standard projectile has a BC of 1. The French GĂąvre Commission decided to use this projectile as their first reference projectile, giving the G1 name.
4035:
Courtney, Elya, Collin Morris, and Michael Courtney. "Accurate Measurements of Free Flight Drag Coefficients with Amateur Doppler Radar." Cornell University Library (2016).
3839:"A Technical Discussion of the ELD-Xâą (Extremely Low Drag â eXpanding) & ELDâą Match (Extremely Low Drag Match) Bullets with Heat Shieldâą Tip, Dave Emary, October 2015"
1359:
announced the use of Doppler radar derived drag data in software utilizing a modified point mass model to generate trajectory solutions. With the measurement derived C
759:
4880:
4273:
1367:, Lapua Edition, Lapua Ballistics or Hornady 4DOF the Doppler radar test-derived drag coefficient data can be used for more accurate external ballistic predictions.
4816:- An open source 3DOF ballistics computer for Windows, Linux, and Mac - Supports the G1, G2, G5, G6, G7, and G8 drag models. Created and maintained by Derek Yates.
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3894:
3693:
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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
4869:
4247:
3383:
4863:
4779:
NABK (NATO Armament Ballistic Kernel) Based Next Generation Ballistic Table Tookit, 23rd International Symposium on Ballistics, Tarragona, Spain 16-20 April 2007
4752:
Virtual Wind Tunnel Experiments for Small Caliber Ammunition Aerodynamic Characterization - Paul Weinacht US Army Research Laboratory Aberdeen Proving Ground, MD
3838:
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1216:
or BC than a flat point bullet. Large radius curves, resulting in a shallower point angle, will produce lower drags, particularly at supersonic velocities.
33:
1220:
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
4883:
Exterior ballistic software for Java or Android mobile phones. Based on doppler radar-test derived (Cd) drag models for Lapua projectiles and cartridges.
3187:"Prediction of Projectile Performance, Stability, and Free-Flight Motion Using Computational Fluid Dynamics, Weinacht, US Army Research Laboratory, 2003"
4712:
4721:
4216:
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lateral throw-off (dispersion that is caused by mass imbalance in the applied projectile or it leaving the barrel off axis leading to static imbalance)
3238:
721:
Though 6 DoF modeling and software applications are used by professional well equipped organizations for decades, the computing power restrictions of
686:
The Proceedings of the 11th International Ballistic Symposium are available through the National Defense Industrial Association (NDIA) at the website
4927:
Free online ballistic calculatoy by SAKO. Calculator also available as an android app (maybe on iOS also, I don't know) under "SAKO Ballistics" name.
4113:
4589:
The US Army Research Laboratory did a study in 1999 on the practical limits of several sniper weapon systems and different methods of fire control.
2023:) prediction method and the 2017 Lapua Ballistics 6 DoF App predictions produce similar results. The 6 DoF modeling estimates bullet stability ((S
3415:
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3290:
2179:
Projectile or bullet length: longer projectiles experience more gyroscopic drift because they produce more lateral "lift" for a given yaw angle.
125:
For medium to longer ranges and flight times, besides gravity, air resistance and wind, several intermediate or meso variables described in the
2153:
and a light bullet will yield maximal yaw effect. A heavy bullet colliding with an equal raindrop will experience significant less yaw effect.
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velocities), a BC provided by a bullet manufacturer will be an average BC that represents the common range of velocities for that bullet. For
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so that the point of aim intersects with the trajectory at a given distance, allowing the user to consistently hit the target being aimed at.
2927:
1299:
To circumvent the transonic problems encountered by spin-stabilized projectiles, projectiles can theoretically be guided during flight. The
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251:
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
201:
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690:
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gyroscopic forces that keep the bullet's length axis resistant to the destabilizing overturning torque of the CP being in front of the CM.
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324:
114:
In small arms external ballistics applications, gravity imparts a downward acceleration on the projectile, causing it to drop from the
4137:
3264:
1228:, which are recessed rings around the projectile used to crimp the projectile securely into the case, will cause an increase in drag.
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3895:
MC DRAG - A Computer Program for Estimating the Drag Coefficients of Projectiles, McCoy, US Army Ballistic Research Laboratory, 1981
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the point of aim does not necessarily need to be adjusted over that range; the projectile is considered to have a sufficiently flat
3481:
3186:
32:
2434:, it is a significant factor in calculating the trajectory. The magnitude of the drift depends on the firing and target location,
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3082:, Engineering Design Handbook: Ballistics Series, United States Army Materiel Command, pp. 1â2, AMCP 706-150, archived from
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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
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with a manufacturer stated G1 ballistic coefficient (BC) of 0.785 fired at 830 m/s (2723 ft/s) muzzle velocity under
3932:
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projectile as it exits the muzzle, barrel harmonics, and atmospheric conditions, all contribute to the path of a projectile.
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the inherent potential accuracy of the computer program and other firing control components used to calculate the trajectory
133:
For long to very long small arms target ranges and flight times, minor effects and forces such as the ones described in the
4704:
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2900:- The behavior of the projectile from the time it leaves the muzzle until the pressure behind the projectile is equalized.
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released a 6 DoF calculation model based ballistic free software named Lapua Ballistics. The software is distributed as a
737:
impaired field use as calculations generally have to be done on the fly. In 2016 the Scandinavian ammunition manufacturer
4510:
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1129:
This tested bullet experiences its maximum drag coefficient when entering the transonic flight regime around Mach 1.200.
592:
differences in the retardation rate of different bullet shapes and sizes. It ranges from 0.1 (flat-nose bullets) to 0.9 (
435:), 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.
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region. This makes accurately predicting the ballistic behavior of projectiles in the transonic region very difficult.
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Lateral throw-off is caused by mass imbalance in applied spin stabilized projectiles or pressure imbalances during the
374:, introduced in 1881, are the most common method used to work with external ballistics. Projectiles are described by a
3845:
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Spin rate: faster spin rates will produce more gyroscopic drift because the nose ends up pointing farther to the side.
938:
Doppler radar measurement results for a Lapua GB528 Scenar 19.44 g (300 gr) 8.59 mm (0.338 in) calibre
3990:"Sandia National Laboratories: News Releases : Sandia's self-guided bullet prototype can hit target a mile away"
4636:(Simplified calculation of the motion of a projectile under a drag force proportional to the square of the velocity)
495:. During the flight of the projectile the M will decrease, and therefore (in most cases) the BC will also decrease.
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upward for eastward shots, and downward for westward shots. The vertical Coriolis deflection is also known as the
1240:
726:
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115:
4828:- Gavre exterior ballistics (zip file) - Supports the G1, G2, G5, G6, G7, G8, GS, GL, GI, GB and RA4 drag models
4807:
3143:"TM 9-1005-319-10 (2010) - Operator's Manual for Rifle, 5.56 MM, M16A2/M16A3/M4 (Battlesight Zero pages 48-55)"
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G7 (long 7.5° boat-tail, 10 calibers tangent ogive, preferred by some manufacturers for very-low-drag bullets)
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The Effects of Aerodynamic Jump Caused by a Uniform Sequence of Lateral Impulses - Gene R. Cooper, July 2004
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the sighting components of a gun. To plan for projectile drop and compensate properly, one must understand
4810:. (MS Excel spreadsheet)] - A substantial enhancement & modification of the Pejsa spreadsheet (below).
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than effective range. There seems to be no good formula for the effective ranges of the various firearms.
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754:
Military organizations have developed ballistic models like the NATO Armament Ballistic Kernel (NABK) for
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is meant, as velocity is a vector quantity and speed is the magnitude of the velocity vector. Because the
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507:, etc. bullet types or shapes. They assume one invariable drag function as indicated by the published BC.
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data can not be simply used for every gun-ammunition combination, since it was measured for the barrels,
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employs custom macroinstructions to calculate the trajectory variables of interest. A modified 4th order
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4734:
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Tan, A.; Frick, C.H. & Castillo, O. (1987). "The fly ball trajectory: An older approach revisited".
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and subsonic flight regimes BC is not well approximated by a single constant, but is considered to be a
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Since different projectile shapes will respond differently to changes in velocity (particularly between
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must fire projectiles along trajectories that are not even approximately straight; they are closer to
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Design for Control of Projectile Flight Characteristics, AMCP 706â242, US Army Materiel Command, 1966
2918:- Procedures or "rules" for a rifleman for aiming at targets at a distance either uphill or downhill.
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Magnus force greatly affects stability because it tries to "twist" the bullet along its flight path.
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A somewhat less obvious effect is caused by head or tailwinds. A headwind will slightly increase the
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4185:"Drag Variability and the use of the "Axial Form Factor" in the Hornady 4DOFâą Trajectory Calculator"
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Range, time of flight and trajectory height: gyroscopic drift increases with all of these variables.
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of a bullet travelling in free-flight demonstrates the air-pressure dynamics surrounding the bullet.
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velocity, for pistol bullets it will probably be subsonic. For projectiles that travel through the
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2415:. Coriolis drift is not an aerodynamic effect; it is a consequence of the rotation of the Earth.
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3993:
2915:
2575:
2109:
289:
Projectile/bullet path analysis is of great use to shooters because it allows them to establish
169:, very subtle effects that are not covered in this article can further refine aiming solutions.
4591:
Sniper Weapon Fire Control Error Budget Analysis - Raymond Von Wahlde, Dennis Metz, August 1999
4191:
687:
4567:
4370:
4343:
3807:"Validation of the NATO Armaments Ballistic Kernel for use in small-arms fire control systems"
3741:
2738:
2647:
2101:
2088:
453:
of the G1 projectile will have a BC of 0.5, and a quarter scale model will have a BC of 0.25.
442:
383:
312:
239:
173:
4578:
4085:
2997:
821:(Lost River J40 .510-773 grain monolithic solid bullet / twist rate 1:15 in) look like this:
4627:
3818:
3335:
3317:
2192:
Doppler radar measurement results for the gyroscopic drift of several US military and other
1379:
1344:
1201:
807:
722:
613:
537:
530:
379:
193:
111:; if in powered flight, thrust; and if guided, the forces imparted by the control surfaces.
4141:
3268:
2616:(interrelated with the Coriolis effect, latitude and direction of fire dictate this effect)
2531:(powder temperature affects muzzle velocity, primer ignition is also temperature dependent)
4924:
4746:
4716:
4694:
4687:
4590:
4327:
3468:
3142:
3019:
3004:
2711:
2599:
2528:
2407:
2168:
730:
694:
290:
197:
4739:
4034:
2515:
ballistic coefficient or test derived drag coefficients (Cd)/behavior of the bullets used
575:
Another attempt at building a ballistic calculator is the model presented in 1980 by Dr.
4623:
746:
limited to and based on G1 or G7 ballistic coefficients and the Mayevski/Siacci method.
645:
retardation rate function at a projectile velocity of 2600 fps (792.5 m/s) using a
4768:
4763:
4710:
Probabalistic Weapon Employment Zone (WEZ) Analysis A Conceptual Overview by Bryan Litz
3605:
3193:
3035:
2936:
1276:
1184:
655:
650:
605:
492:
461:
390:
336:
205:
119:
104:
4914:
Free online web-based ballistics calculator, with data export capability and charting.
4651:
3461:
636:
4970:
4682:
4498:
4487:
4455:
4417:
3784:
3763:
3712:
3083:
3031:
2940:
2593:
2521:
the zero range at which the sighting components and rifle combination were sighted in
2337:
2150:
2036:
802:
798:
4729:
4476:
3706:"Six Degree of Freedom Digital Simulation Model for Unguided Fin-Stabilized Rockets"
523:
G1 or Ingalls (flatbase with 2 caliber (blunt) nose ogive - by far the most popular)
4535:
3939:
2507:
618:
260:
139:
3967:
1196:) and right S.m.E. ammunition beside their boat-tailed projectiles with cannelures
1150:
184:
Two methods can be employed to stabilize non-spherical projectiles during flight:
4561:
4217:"HornadyÂź 4DOFâą (Four Degree of Freedom) Ballistic Calculator Technical Document"
3023:
2932:
2656:
2569:
2557:
2137:
2133:
2125:
2092:
often makes ultra long range shooting in head or tailwind conditions difficult.
2032:
1401:
1327:
1287:
1266:
948:
797:
For the precise establishment of drag or air resistance effects on projectiles,
488:
450:
332:
252:
235:
67:
Gun-launched projectiles may be unpowered, deriving all their velocity from the
17:
4930:
3122:
2638:
The ambient air density is at its maximum at Arctic sea level conditions. Cold
2625:
the inherent potential accuracy and adjustment range of the sighting components
1261:
muzzle velocity will at some point slow to approach the speed of sound. At the
4514:
4418:
Nenstiel The yaw of repose angle of a M80 bullet (7.62 x 51 Nato) fired at 32°
2589:
2503:
2419:
1271:
1258:
1245:
817:
Doppler radar measurement results for a lathe-turned monolithic solid .50 BMG
742:
734:
714:
473:
469:
457:
340:
231:
96:
72:
68:
60:
56:
4902:"GunSim" free browser-based ballistics simulator program for Windows and Mac.
4784:
Trajectory Calculator in C++ that can deduce drag function from firing tables
3823:
3806:
3217:
1212:(BC) than a round nosed bullet, and a round nosed bullet will have a better C
4917:
4873:
4801:
4722:
Weite SchĂŒsse - part 4, Basic explanation of the Pejsa model by Lutz Möller
3022:(16.2 g) Lapua Scenar bullets at a measured muzzle velocity of 905 m/s. The
2894:- The behavior of the projectile and propellant before it leaves the barrel.
2639:
2423:
1364:
1262:
609:
588:
500:
477:
166:
146:
4813:
4340:"Understanding Uphill and Downhill Shots in Long Range Shooting: A Primer"
3870:
4795:
4274:"Lapua Ballistics freeware exterior ballistic software for mobile phones"
3482:"Pejsa Rifle Ballistics: Art Pejsa's Rifle ballistics software and books"
3165:
2603:
2583:
2579:
2565:
2561:
2553:
2129:
2121:
256:
150:
4825:
2592:
drift (horizontal and vertical plane gyroscopic effect â often known as
4819:
4699:
3672:
SPIN-73 An Updated Version of the Spinner Computer Program, White, 1973
2435:
2427:
1356:
432:
425:
248:
100:
4841:
2537:
supersonic range of the employed gun, cartridge and bullet combination
1387:
Predictions of several drag resistance modelling and measuring methods
1149:
Graphs are unavailable due to technical issues. There is more info on
3883:
2188:
density of the atmosphere: denser air will increase gyroscopic drift.
518:
G7 shape standard projectile. All measurements in calibers/diameters.
362:
G1 shape standard projectile. All measurements in calibers/diameters.
4908:"Ballistic Simulator" free ballistics simulator program for Windows.
4905:
4783:
4631:
789:
The predictions these models yield are subject to comparison study.
378:, or BC, which combines the air resistance of the bullet shape (the
4041:
2019:
The table shows the Doppler radar test derived drag coefficients (C
4840:
A ballistic calculator for Remington factory ammunition (based on
4822:
links to / hosts 4 freeware external ballistics computer programs.
2327:
2175:
The following variables affect the magnitude of gyroscopic drift:
1355:
generate trajectory solutions. In 2016 US ammunition manufacturer
1183:
738:
635:
601:
513:
465:
418:
357:
323:
225:
189:
92:
80:
29:
3694:
Improved Solids Modeling for Axisymmetric Projectile Design, 1988
3123:"An Improved Battlesight Zero for the M4 Carbine and M16A2 Rifle"
354:
Fixed drag curve models generated for standard-shaped projectiles
4566:(illustrated ed.). Cambridge University Press. p. 45.
3907:"New Video Series from Applied Ballistics « Daily Bulletin"
2546:
623:
108:
76:
4054:"Lapua Bullets Drag Coefficient Data for QuickTARGET Unlimited"
3291:"LM Class Bullets, very high BC bullets for windy long Ranges"
2668:(183 m) and 300 yards (274 m) respectively.
2381:
or BC based on shape and sectional density is of limited use.
1135:
801:
measurements are required. Weibel 1000e or Infinition BR-1001
198:
M829 Armor-Piercing, Fin-Stabilized, Discarding Sabot (APFSDS)
4683:
How do bullets fly? by Ruprecht Nennstiel, Wiesbaden, Germany
2906:- The behavior of the projectile upon impact with the target.
366:
Use of ballistics tables or ballistics software based on the
4735:
JBM Small Arms Ballistics with online ballistics calculators
4730:
Patagonia Ballistics ballistics mathematical software engine
3441:"Form Factors: A Useful Analysis Tool - Berger Bullets Blog"
2511:
control corrections has a long list of variables including:
688:
http://www.ndia.org/Resources/Pages/Publication_Catalog.aspx
4933:
LGPL Python library for point-mass ballistic calculations .
2518:
height of the sighting components above the rifle bore axis
705:
There are also advanced professional ballistic models like
446:
4511:"Gyroscopic Drift and Coreolis Acceleration by Bryan Litz"
4016:"Guided .50 Caliber Projectile â DARPA's Steerable Bullet"
2969:
G1, G7 and Doppler radar test derived drag coefficients (C
4889:
6 DoF model limited to Lapua bullets for Android and iOS.
3683:
Aerodynamic Design Manual for Tactical Weapons, NSWC 1981
2596:- induced by the barrel's twist direction and twist rate)
706:
4899:
4740:
Bison Ballistics Point Mass Online Ballistics Calculator
2364:
force on any bullet with the center of pressure located
491:
M; here M equals the projectile velocity divided by the
4911:
4107:"Use of Doppler Radar to Generate Trajectory Solutions"
3489:
2360:
of the center of gravity, while conversely acting as a
4892:
4764:
British Artillery Fire Control - Ballistics & Data
2356:
force on any bullet with a center of pressure located
4798:
FREE ballistics app. iOS, Android, OSX & Windows.
3034:) elevation correction from a 300 m zero range at 61
4769:
Field Artillery, Volume 6, Ballistics and Ammunition
3101:"Maximum Point Blank Range and the Battlesight Zero"
1448:
Radar test derived drag coefficients method V (m/s)
1200:
In general, a pointed projectile will have a better
4342:. backcountrymaven.com. 16 May 2013. Archived from
2540:
inclination angle in case of uphill/downhill firing
2438:of firing, projectile velocity and time of flight.
2418:The magnitude of the Coriolis effect is small. For
339:around a bullet in supersonic flight, published by
180:
Stabilizing non-spherical projectiles during flight
647:.30-06 Springfield Cartridge, Ball, Caliber .30 M2
3884:Richardson v. United States, 72 Ct. Cl. 51 (1930)
2631:the inherent potential accuracy of the ammunition
2136:. Humidity has a counter intuitive impact. Since
4790:Freeware small arms external ballistics software
3738:"Lapua Ballistics App - Resources - Nammo Lapua"
2924:- Early scientific study of external ballistics.
2912:- Basic external ballistics mathematic formulas.
2336:Spin stabilized projectiles are affected by the
760:SG2 Shareable (Fire Control) Software Suite (S4)
431:ranging from 0.177 to 0.50 inches (4.50 to
161:must be taken into account when aiming. In very
4323:
4321:
3336:".338 Lapua Magnum product brochure from Lapua"
2352:of the bullet. The Magnus effect will act as a
2208:US military M118 Special Ball (7.62Ă51mm NATO)
1180:General trends in drag or ballistic coefficient
4804:free ballistics for rim fire and pellet guns.
4705:Articles on long range shooting by Bryan Litz
4138:"HornadyÂź 4DOF Ballistic Calculator Overview"
3576:Pejsa's Handbook of New, Precision Ballistics
3561:Pejsa's Handbook of New, Precision Ballistics
3513:Pejsa's Handbook of New, Precision Ballistics
529:G5 (short 7.5° boat-tail, 6.19 calibers long
8:
3524:
3522:
2628:the inherent potential accuracy of the rifle
1308:Testing the predictive qualities of software
546:G8 (flatbase, 10 calibers long secant ogive)
405:will experience is proportional to 1/BC, 1/
71:'s ignition until the projectile exits the
770:7thâorder RungeâKuttaâFehlberg integration
386:(a function of mass and bullet diameter).
222:Projectile/bullet drop and projectile path
4369:. Exterior Ballistics.com. Archived from
3822:
3360:"300 grs Scenar HPBT brochure from Lapua"
3030:. The test rifle needed 13.2 mils (45.38
1270:projectile starts to exhibit an unwanted
1231:Analytical software was developed by the
306:Maximum point-blank range and battle zero
4678:Software for calculating ball ballistics
2670:
2198:
2196:at 1000 yards (914.4 m) look like this:
1406:
944:
823:
4937:
4864:Sharpshooter Friend (Palm PDA software)
4858:Pejsa Ballistics (MS Excel spreadsheet)
4848:JBM's small-arms ballistics calculators
4814:GNU Exterior Ballistics Computer (GEBC)
3593:. Kenwood Publishing. pp. 131â134.
3054:
2952:
2260:Projectile diameter (in inches and mm)
2205:US military M193 Ball (5.56Ă45mm NATO)
158:
134:
4394:"Effects of Rain on Bullet Trajectory"
4248:"QuickTARGET Unlimited, Lapua Edition"
3218:"Ballistic Coefficients Do Not Exist!"
2568:variations (these make up the ambient
4870:Quick Target Unlimited, Lapua Edition
4700:A Short Course in External Ballistics
3630:. Kenwood Publishing. pp. 33â35.
3533:. Kenwood Publishing. pp. 65â76.
3166:"Using a 400 meter zero with 5.45Ă39"
2928:Table of handgun and rifle cartridges
2673:
154:
126:
7:
3384:"Ballistic Coefficients - Explained"
2289:Gyroscopic drift (in inches and mm)
4844:). - Siacci/Mayevski G1 drag model.
4452:"Articles - Applied Ballistics LLC"
4392:Beckstrand, Tom (6 February 2020).
3324:from the original on 2 August 2009.
3026:Ï during the test shoot was 1.2588
2502:The maximum practical range of all
2432:intercontinental ballistic missiles
217:Main effects in external ballistics
200:achieve stability by forcing their
4086:"Lapua bullets CD data (zip file)"
3606:"Pejsa and regular drag functions"
3488:. Pejsa Ballistics. Archived from
2498:Maximum effective small arms range
2344:Magnus effect and bullet stability
2231:Projectile mass (in grains and g)
624:average of any integrable function
503:, flat-based, spitzer, boat-tail,
91:When in flight, the main or major
59:that deals with the behavior of a
25:
4365:William T. McDonald (June 2003).
3563:. Kenwood Publishing. p. 34.
3548:. Kenwood Publishing. p. 63.
1398:International Standard Atmosphere
1324:International Standard Atmosphere
681:RungeâKutta integration algorithm
468:bullets, this will probably be a
27:Behavior of projectiles in flight
4834:- Siacci/Mayevski G1 drag model.
4832:PointBlank Ballistics (zip file)
4688:Exterior Ballistics.com articles
3578:. Kenwood Publishing. p. 4.
3515:. Kenwood Publishing. p. 3.
3318:"A Better Ballistic Coefficient"
1790:Pejsa drag model method V (m/s)
1338:Doppler radar measurement method
1295:Research into guided projectiles
1140:
3628:New Exact Small Arms Ballistics
3591:New Exact Small Arms Ballistics
3546:New Exact Small Arms Ballistics
3531:New Exact Small Arms Ballistics
3239:"Weite SchĂŒsse - drei (German)"
2608:northern or southern hemisphere
2132:variations make up the ambient
337:detached shock or bow shockwave
230:Typical trajectory graph for a
87:Forces acting on the projectile
4672:Small arms external ballistics
2298:12.75 inches (323.85 mm)
2295:11.50 inches (292.10 mm)
2292:23.00 inches (584.20 mm)
1562:6 DoF modeling method V (m/s)
1233:Ballistics Research Laboratory
709:available. These are based on
649:152 grains (9.8 g) rifle
633:within a given flight regime.
536:G6 (flatbase, 6 calibers long
1:
4758:Artillery external ballistics
4641:"The Perfect Basketball Shot"
3871:"The 8x50R Lebel (8mm Lebel)"
3316:Litz, Bryan (8 March 2021) .
2655:Interesting further reading:
2307:6.50 inches (165.10 mm)
2304:7.75 inches (196.85 mm)
2162:Gyroscopic drift (spin drift)
1904:G7 drag model method V (m/s)
1676:G1 drag model method V (m/s)
4820:6mmbr.com ballistics section
2506:and especially high-powered
2313:1.90 inches (48.26 mm)
2310:0.87 inches (22.10 mm)
2301:3.00 inches (76.20 mm)
2284:.408 inches (10.36 mm)
1380:rotational (spin) velocities
1317:Empirical measurement method
1301:Sandia National Laboratories
701:Six degrees of freedom model
393:that a projectile with mass
4612:American Journal of Physics
4604:General external ballistics
4164:"4DOF Ballistic Calculator"
3076:Interior Ballistics of Guns
2822:Zeroed for 300 yards/274 m
2764:Zeroed for 200 yards/184 m
2281:.375 inches (9.53 mm)
2278:.338 inches (8.59 mm)
2275:.308 inches (7.82 mm)
2272:.308 inches (7.82 mm)
2269:.308 inches (7.82 mm)
2266:.308 inches (7.82 mm)
2263:.224 inches (5.69 mm)
1241:Rocket-assisted projectiles
782:Thrust and Base Burn models
767:6/7âDoF equations of motion
727:personal digital assistants
5003:
4893:BfX - Ballistics for Excel
4826:2DOF & 3DOF R.L. McCoy
4563:Fundamentals of Geophysics
4499:Nenstiel The Magnus moment
4477:Nenstiel The Magnus effect
2910:Trajectory of a projectile
2543:target speed and direction
2255:419 grains (27.15 g)
2252:350 grains (22.68 g)
2249:300 grains (19.44 g)
2246:220 grains (14.26 g)
2243:190 grains (12.31 g)
2240:155 grains (10.04 g)
2237:173 grains (11.21 g)
1394:very-low-drag rifle bullet
793:Doppler radar measurements
669:120 mm tank gun ammunition
526:G2 (Aberdeen J projectile)
145:At extremely long ranges,
4842:Pinsoft's Shoot! software
4488:Nenstiel The Magnus force
2826:
2821:
2768:
2763:
2706:
2610:data dictate this effect)
2491:transitional flight phase
2149:a collision. A big heavy
750:Artillery software suites
566:More advanced drag models
424:Sporting bullets, with a
4667:- basketball ballistics.
4560:Lowrie, William (1997).
3824:10.1016/j.dt.2017.04.006
2699:
2694:
2689:
2684:
2679:
2676:
2606:, direction of fire and
2451:Vertical (Eötvös) effect
2234:55 grains (3.56 g)
1237:Army Research Laboratory
723:mobile computing devices
389:The deceleration due to
3626:Arthur J Pejsa (2008).
3589:Arthur J Pejsa (2008).
3574:Arthur J Pejsa (2002).
3559:Arthur J Pejsa (2002).
3544:Arthur J Pejsa (2008).
3529:Arthur J Pejsa (2008).
3511:Arthur J Pejsa (2002).
2898:Transitional ballistics
2578:(changes slightly with
2211:Palma Sierra MatchKing
758:for artillery like the
608:does not have constant
163:large-calibre artillery
3805:Corriveau, D. (2017).
3073:Army (February 1965),
2333:
1400:sea level conditions (
1326:sea level conditions (
1257:A projectile fired at
1197:
880:Ballistic coefficient
711:six degrees of freedom
641:
631:closed-form expression
519:
363:
344:
276:Projectile/bullet path
267:Projectile/bullet drop
244:
45:
4874:QuickTARGET Unlimited
4429:Nennstiel, Ruprecht.
3265:"exterior ballistics"
2663:Using ballistics data
2657:Marksmanship Wikibook
2374:very-low-drag bullets
2331:
2194:very-low-drag bullets
1365:QuickTARGET Unlimited
1210:ballistic coefficient
1187:
639:
594:very-low-drag bullets
517:
437:Very-low-drag bullets
376:ballistic coefficient
361:
327:
229:
39:
4946:"JBM Bullet Library"
4887:Lapua Ballistics App
1218:Hollow point bullets
940:very-low-drag bullet
819:very-low-drag bullet
756:fire-control systems
587:(true/calibrate) or
581:closed-form solution
549:GL (blunt lead nose)
283:ballistic trajectory
4624:1987AmJPh..55...37T
4536:"The Eötvös effect"
4373:on 25 November 2014
4346:on 25 November 2014
3744:on 20 December 2016
3297:on 19 February 2008
2904:Terminal ballistics
2892:Internal ballistics
2116:Ambient air density
1942:Time of flight (s)
1828:Time of flight (s)
1714:Time of flight (s)
1600:Time of flight (s)
1486:Time of flight (s)
1332:confidence interval
1235:â later called the
348:Mathematical models
53:exterior ballistics
49:External ballistics
4923:2016-03-15 at the
4906:BallisticSimulator
4745:2011-05-15 at the
4715:2015-09-23 at the
4693:2013-03-06 at the
4579:Extract of page 45
4197:on 19 October 2016
3996:on 5 February 2012
3913:on 21 October 2016
3811:Defence Technology
3492:on 8 February 2012
3467:2008-08-29 at the
3396:on 29 October 2013
3089:on January 8, 2016
3003:2007-11-07 at the
2935:- Calibrating the
2922:Franklin Ware Mann
2648:laser rangefinders
2334:
2157:Long range factors
1198:
779:Aerodynamic models
725:like (ruggedized)
693:2012-01-26 at the
642:
520:
364:
345:
245:
204:(CP) behind their
202:center of pressure
174:ballistic missiles
159:long range factors
135:long range factors
107:, and if present,
46:
4458:on 7 January 2016
2879:
2878:
2485:Lateral throw-off
2467:Equipment factors
2442:Horizontal effect
2332:The Magnus effect
2317:
2316:
2220:Sierra MatchKing
2217:Sierra MatchKing
2089:relative velocity
2017:
2016:
1345:drag coefficients
1313:flight behavior.
1253:Transonic problem
1162:
1161:
1127:
1126:
1039:Drag coefficient
932:
931:
808:drag coefficients
776:Atmosphere models
443:Sectional density
384:sectional density
313:point-blank range
188:Projectiles like
37:
16:(Redirected from
4994:
4962:
4961:
4959:
4957:
4948:. Archived from
4942:
4931:py-ballisticcalc
4881:Lapua Ballistics
4838:Remington Shoot!
4725:
4666:
4664:
4662:
4657:on March 5, 2006
4656:
4650:. Archived from
4645:
4635:
4593:
4587:
4581:
4577:
4557:
4551:
4550:
4548:
4546:
4532:
4526:
4525:
4523:
4522:
4513:. Archived from
4507:
4501:
4496:
4490:
4485:
4479:
4474:
4468:
4467:
4465:
4463:
4454:. Archived from
4448:
4442:
4441:
4439:
4437:
4426:
4420:
4415:
4409:
4408:
4406:
4404:
4389:
4383:
4382:
4380:
4378:
4362:
4356:
4355:
4353:
4351:
4336:
4330:
4325:
4316:
4315:
4313:
4311:
4302:. Archived from
4296:
4290:
4289:
4287:
4285:
4276:. Archived from
4270:
4264:
4263:
4261:
4259:
4250:. Archived from
4244:
4238:
4237:
4235:
4234:
4228:
4222:. Archived from
4221:
4213:
4207:
4206:
4204:
4202:
4196:
4190:. Archived from
4189:
4181:
4175:
4174:
4172:
4170:
4160:
4154:
4153:
4151:
4149:
4140:. Archived from
4134:
4128:
4127:
4125:
4124:
4118:
4112:. Archived from
4111:
4103:
4097:
4096:
4094:
4092:
4082:
4076:
4075:
4073:
4071:
4065:
4059:. Archived from
4058:
4050:
4044:
4032:
4026:
4025:
4023:
4022:
4012:
4006:
4005:
4003:
4001:
3992:. Archived from
3986:
3980:
3979:
3977:
3975:
3966:. Archived from
3960:
3954:
3953:
3951:
3950:
3944:
3938:. Archived from
3937:
3929:
3923:
3922:
3920:
3918:
3909:. Archived from
3903:
3897:
3892:
3886:
3881:
3875:
3874:
3866:
3860:
3859:
3857:
3856:
3850:
3844:. Archived from
3843:
3835:
3829:
3828:
3826:
3802:
3796:
3795:
3793:
3791:
3781:
3775:
3774:
3772:
3770:
3760:
3754:
3753:
3751:
3749:
3740:. Archived from
3734:
3728:
3727:
3725:
3723:
3717:
3711:. Archived from
3710:
3702:
3696:
3691:
3685:
3680:
3674:
3669:
3663:
3662:
3660:
3659:
3653:
3647:. Archived from
3646:
3638:
3632:
3631:
3623:
3617:
3616:
3614:
3612:
3604:Meijer, Robert.
3601:
3595:
3594:
3586:
3580:
3579:
3571:
3565:
3564:
3556:
3550:
3549:
3541:
3535:
3534:
3526:
3517:
3516:
3508:
3502:
3501:
3499:
3497:
3486:Pejsa Ballistics
3477:
3471:
3459:
3453:
3452:
3450:
3448:
3437:
3431:
3430:
3428:
3427:
3418:. Archived from
3412:
3406:
3405:
3403:
3401:
3395:
3389:. Archived from
3388:
3380:
3374:
3373:
3371:
3369:
3364:
3356:
3350:
3349:
3347:
3345:
3340:
3332:
3326:
3325:
3313:
3307:
3306:
3304:
3302:
3293:. Archived from
3287:
3281:
3280:
3278:
3276:
3271:on 8 August 2003
3267:. Archived from
3261:
3255:
3254:
3252:
3250:
3241:. Archived from
3235:
3229:
3228:
3226:
3224:
3214:
3208:
3207:
3205:
3204:
3198:
3192:. Archived from
3191:
3183:
3177:
3176:
3174:
3172:
3162:
3156:
3155:
3153:
3152:
3147:
3139:
3133:
3132:
3130:
3129:
3119:
3113:
3112:
3110:
3108:
3097:
3091:
3090:
3088:
3081:
3070:
3064:
3059:
3044:
3015:
3009:
2994:
2988:
2980:
2974:
2967:
2961:
2957:
2671:
2199:
2070:External factors
1407:
1202:drag coefficient
1144:
1143:
1136:
945:
942:look like this:
824:
731:tablet computers
380:drag coefficient
291:ballistic tables
238:using identical
165:cases, like the
155:external factors
127:external factors
38:
21:
18:Boat-tail bullet
5002:
5001:
4997:
4996:
4995:
4993:
4992:
4991:
4967:
4966:
4965:
4955:
4953:
4944:
4943:
4939:
4925:Wayback Machine
4918:SAKO Ballistics
4872:- A version of
4853:
4796:Hawke X-ACT Pro
4792:
4747:Wayback Machine
4723:
4717:Wayback Machine
4695:Wayback Machine
4660:
4658:
4654:
4643:
4639:
4632:10.1119/1.14968
4609:
4601:
4596:
4588:
4584:
4574:
4559:
4558:
4554:
4544:
4542:
4534:
4533:
4529:
4520:
4518:
4509:
4508:
4504:
4497:
4493:
4486:
4482:
4475:
4471:
4461:
4459:
4450:
4449:
4445:
4435:
4433:
4428:
4427:
4423:
4416:
4412:
4402:
4400:
4391:
4390:
4386:
4376:
4374:
4367:"INCLINED FIRE"
4364:
4363:
4359:
4349:
4347:
4338:
4337:
4333:
4326:
4319:
4309:
4307:
4298:
4297:
4293:
4283:
4281:
4272:
4271:
4267:
4257:
4255:
4246:
4245:
4241:
4232:
4230:
4226:
4219:
4215:
4214:
4210:
4200:
4198:
4194:
4187:
4183:
4182:
4178:
4168:
4166:
4162:
4161:
4157:
4147:
4145:
4136:
4135:
4131:
4122:
4120:
4116:
4109:
4105:
4104:
4100:
4090:
4088:
4084:
4083:
4079:
4069:
4067:
4063:
4056:
4052:
4051:
4047:
4033:
4029:
4020:
4018:
4014:
4013:
4009:
3999:
3997:
3988:
3987:
3983:
3973:
3971:
3962:
3961:
3957:
3948:
3946:
3942:
3935:
3931:
3930:
3926:
3916:
3914:
3905:
3904:
3900:
3893:
3889:
3882:
3878:
3868:
3867:
3863:
3854:
3852:
3848:
3841:
3837:
3836:
3832:
3804:
3803:
3799:
3789:
3787:
3783:
3782:
3778:
3768:
3766:
3762:
3761:
3757:
3747:
3745:
3736:
3735:
3731:
3721:
3719:
3715:
3708:
3704:
3703:
3699:
3692:
3688:
3681:
3677:
3670:
3666:
3657:
3655:
3651:
3644:
3640:
3639:
3635:
3625:
3624:
3620:
3610:
3608:
3603:
3602:
3598:
3588:
3587:
3583:
3573:
3572:
3568:
3558:
3557:
3553:
3543:
3542:
3538:
3528:
3527:
3520:
3510:
3509:
3505:
3495:
3493:
3480:Pejsa, Arthur.
3479:
3478:
3474:
3469:Wayback Machine
3462:About Art Pejsa
3460:
3456:
3446:
3444:
3439:
3438:
3434:
3425:
3423:
3414:
3413:
3409:
3399:
3397:
3393:
3386:
3382:
3381:
3377:
3367:
3365:
3362:
3358:
3357:
3353:
3343:
3341:
3338:
3334:
3333:
3329:
3315:
3314:
3310:
3300:
3298:
3289:
3288:
3284:
3274:
3272:
3263:
3262:
3258:
3248:
3246:
3237:
3236:
3232:
3222:
3220:
3216:
3215:
3211:
3202:
3200:
3196:
3189:
3185:
3184:
3180:
3170:
3168:
3164:
3163:
3159:
3150:
3148:
3145:
3141:
3140:
3136:
3127:
3125:
3121:
3120:
3116:
3106:
3104:
3103:. 30 April 2010
3099:
3098:
3094:
3086:
3079:
3072:
3071:
3067:
3060:
3056:
3052:
3047:
3016:
3012:
3005:Wayback Machine
2995:
2991:
2985:
2981:
2977:
2972:
2968:
2964:
2958:
2954:
2950:
2916:Rifleman's rule
2888:
2701:
2696:
2691:
2686:
2681:
2665:
2600:Coriolis effect
2576:Earth's gravity
2534:range to target
2529:muzzle velocity
2500:
2487:
2478:
2469:
2453:
2444:
2408:Coriolis effect
2404:
2387:
2380:
2372:Paradoxically,
2346:
2326:
2226:LRBT J40 Match
2223:LRBT J40 Match
2214:LRBT J40 Match
2164:
2159:
2146:
2118:
2110:Rifleman's rule
2098:
2096:Vertical angles
2077:
2072:
2061:
2053:
2045:
2030:
2026:
2022:
1980:Total drop (m)
1866:Total drop (m)
1752:Total drop (m)
1638:Total drop (m)
1524:Total drop (m)
1389:
1377:
1362:
1350:
1340:
1319:
1310:
1297:
1255:
1215:
1207:
1182:
1169:
1168:
1167:
1164:
1163:
1158:
1145:
1141:
813:
795:
785:Actuator models
752:
703:
695:Wayback Machine
664:
577:Arthur J. Pejsa
573:
568:
401:, and diameter
368:Mayevski/Siacci
356:
329:Schlieren photo
322:
320:Drag resistance
308:
278:
269:
224:
219:
182:
172:In the case of
89:
55:is the part of
42:schlieren image
30:
28:
23:
22:
15:
12:
11:
5:
5000:
4998:
4990:
4989:
4984:
4979:
4969:
4968:
4964:
4963:
4936:
4935:
4934:
4928:
4915:
4909:
4903:
4897:
4890:
4884:
4878:
4867:
4866:- Pejsa model.
4861:
4860:- Pejsa model.
4855:
4854:) drag models.
4851:
4845:
4835:
4829:
4823:
4817:
4811:
4805:
4799:
4791:
4788:
4787:
4786:
4781:
4776:
4771:
4766:
4755:
4754:
4749:
4737:
4732:
4727:
4719:
4707:
4702:
4697:
4685:
4680:
4669:
4668:
4637:
4600:
4599:External links
4597:
4595:
4594:
4582:
4572:
4552:
4540:www.cleonis.nl
4527:
4502:
4491:
4480:
4469:
4443:
4421:
4410:
4384:
4357:
4331:
4317:
4291:
4280:on 1 July 2012
4265:
4254:on 1 July 2012
4239:
4208:
4176:
4155:
4129:
4098:
4077:
4045:
4027:
4007:
3981:
3955:
3924:
3898:
3887:
3876:
3861:
3830:
3817:(3): 188â199.
3797:
3776:
3755:
3729:
3697:
3686:
3675:
3664:
3633:
3618:
3596:
3581:
3566:
3551:
3536:
3518:
3503:
3472:
3454:
3432:
3407:
3375:
3351:
3327:
3308:
3282:
3256:
3245:on 24 May 2008
3230:
3209:
3178:
3157:
3134:
3114:
3092:
3065:
3053:
3051:
3048:
3046:
3045:
3010:
2989:
2983:
2975:
2970:
2962:
2951:
2949:
2946:
2945:
2944:
2930:
2925:
2919:
2913:
2907:
2901:
2895:
2887:
2884:
2877:
2876:
2873:
2870:
2865:
2862:
2859:
2856:
2852:
2851:
2848:
2845:
2840:
2837:
2834:
2831:
2828:
2824:
2823:
2819:
2818:
2815:
2812:
2809:
2804:
2801:
2798:
2794:
2793:
2790:
2787:
2784:
2779:
2776:
2773:
2770:
2766:
2765:
2761:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2734:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2708:
2704:
2703:
2698:
2693:
2688:
2683:
2678:
2675:
2664:
2661:
2636:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2611:
2597:
2587:
2573:
2550:
2544:
2541:
2538:
2535:
2532:
2525:
2522:
2519:
2516:
2499:
2496:
2486:
2483:
2477:
2474:
2468:
2465:
2452:
2449:
2443:
2440:
2403:
2402:Coriolis drift
2400:
2386:
2385:Poisson effect
2383:
2378:
2345:
2342:
2325:
2322:
2315:
2314:
2311:
2308:
2305:
2302:
2299:
2296:
2293:
2290:
2286:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2257:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2228:
2227:
2224:
2221:
2218:
2215:
2212:
2209:
2206:
2203:
2190:
2189:
2186:
2183:
2180:
2163:
2160:
2158:
2155:
2145:
2142:
2117:
2114:
2102:vertical angle
2097:
2094:
2076:
2073:
2071:
2068:
2059:
2051:
2043:
2028:
2024:
2020:
2015:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1977:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1939:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1901:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1863:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1825:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1787:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1749:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1711:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1677:
1673:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1635:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1597:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1559:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1528:
1525:
1521:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1483:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1449:
1445:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1388:
1385:
1375:
1360:
1348:
1339:
1336:
1318:
1315:
1309:
1306:
1296:
1293:
1265:region (about
1254:
1251:
1213:
1205:
1181:
1178:
1165:
1160:
1159:
1148:
1146:
1139:
1134:
1133:
1132:
1131:
1125:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1036:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
930:
929:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
881:
877:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
846:
843:
840:
837:
834:
831:
828:
811:
803:Doppler radars
794:
791:
787:
786:
783:
780:
777:
774:
771:
768:
751:
748:
739:Nammo Lapua Oy
702:
699:
663:
660:
651:spitzer bullet
606:power function
572:
569:
567:
564:
551:
550:
547:
544:
541:
534:
527:
524:
493:speed of sound
355:
352:
321:
318:
307:
304:
277:
274:
268:
265:
247:The effect of
223:
220:
218:
215:
214:
213:
209:
206:center of mass
192:or arrow like
181:
178:
95:acting on the
88:
85:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4999:
4988:
4985:
4983:
4980:
4978:
4975:
4974:
4972:
4952:on 3 May 2010
4951:
4947:
4941:
4938:
4932:
4929:
4926:
4922:
4919:
4916:
4913:
4910:
4907:
4904:
4901:
4898:
4894:
4891:
4888:
4885:
4882:
4879:
4875:
4871:
4868:
4865:
4862:
4859:
4856:
4849:
4846:
4843:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4808:Ballistic_XLR
4806:
4803:
4800:
4797:
4794:
4793:
4789:
4785:
4782:
4780:
4777:
4775:
4772:
4770:
4767:
4765:
4762:
4761:
4760:
4759:
4753:
4750:
4748:
4744:
4741:
4738:
4736:
4733:
4731:
4728:
4726:
4720:
4718:
4714:
4711:
4708:
4706:
4703:
4701:
4698:
4696:
4692:
4689:
4686:
4684:
4681:
4679:
4676:
4675:
4674:
4673:
4661:September 26,
4653:
4649:
4642:
4638:
4633:
4629:
4625:
4621:
4617:
4613:
4608:
4607:
4606:
4605:
4598:
4592:
4586:
4583:
4580:
4575:
4573:0-521-46728-4
4569:
4565:
4564:
4556:
4553:
4541:
4537:
4531:
4528:
4517:on 2007-11-14
4516:
4512:
4506:
4503:
4500:
4495:
4492:
4489:
4484:
4481:
4478:
4473:
4470:
4457:
4453:
4447:
4444:
4432:
4425:
4422:
4419:
4414:
4411:
4399:
4398:Guns and Ammo
4395:
4388:
4385:
4372:
4368:
4361:
4358:
4345:
4341:
4335:
4332:
4329:
4324:
4322:
4318:
4306:on 2012-10-06
4305:
4301:
4295:
4292:
4279:
4275:
4269:
4266:
4253:
4249:
4243:
4240:
4229:on 2016-10-19
4225:
4218:
4212:
4209:
4193:
4186:
4180:
4177:
4165:
4159:
4156:
4144:on 2016-08-23
4143:
4139:
4133:
4130:
4119:on 2015-09-23
4115:
4108:
4102:
4099:
4087:
4081:
4078:
4066:on 2010-12-29
4062:
4055:
4049:
4046:
4043:
4039:
4036:
4031:
4028:
4017:
4011:
4008:
3995:
3991:
3985:
3982:
3970:on 2015-01-11
3969:
3965:
3959:
3956:
3945:on 2016-04-06
3941:
3934:
3928:
3925:
3912:
3908:
3902:
3899:
3896:
3891:
3888:
3885:
3880:
3877:
3872:
3869:Chuck Hawks.
3865:
3862:
3851:on 2015-11-06
3847:
3840:
3834:
3831:
3825:
3820:
3816:
3812:
3808:
3801:
3798:
3786:
3780:
3777:
3765:
3759:
3756:
3743:
3739:
3733:
3730:
3718:on 2017-02-21
3714:
3707:
3701:
3698:
3695:
3690:
3687:
3684:
3679:
3676:
3673:
3668:
3665:
3654:on 2016-12-13
3650:
3643:
3637:
3634:
3629:
3622:
3619:
3607:
3600:
3597:
3592:
3585:
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3525:
3523:
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3514:
3507:
3504:
3491:
3487:
3483:
3476:
3473:
3470:
3466:
3463:
3458:
3455:
3443:. 16 May 2011
3442:
3436:
3433:
3422:on 2018-02-04
3421:
3417:
3411:
3408:
3392:
3385:
3379:
3376:
3361:
3355:
3352:
3337:
3331:
3328:
3323:
3319:
3312:
3309:
3296:
3292:
3286:
3283:
3270:
3266:
3260:
3257:
3244:
3240:
3234:
3231:
3219:
3213:
3210:
3199:on 2021-01-21
3195:
3188:
3182:
3179:
3167:
3161:
3158:
3144:
3138:
3135:
3124:
3118:
3115:
3102:
3096:
3093:
3085:
3078:
3077:
3069:
3066:
3063:
3058:
3055:
3049:
3041:
3037:
3033:
3029:
3025:
3021:
3014:
3011:
3006:
3002:
2999:
2993:
2990:
2979:
2976:
2966:
2963:
2956:
2953:
2947:
2942:
2941:ranged weapon
2938:
2934:
2931:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
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2672:
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2658:
2652:
2649:
2643:
2641:
2633:
2630:
2627:
2624:
2621:
2618:
2615:
2614:Eötvös effect
2612:
2609:
2605:
2601:
2598:
2595:
2591:
2588:
2585:
2581:
2577:
2574:
2571:
2567:
2563:
2559:
2555:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2526:
2523:
2520:
2517:
2514:
2513:
2512:
2509:
2508:sniper rifles
2505:
2497:
2495:
2492:
2484:
2482:
2475:
2473:
2466:
2464:
2460:
2458:
2457:Eötvös effect
2450:
2448:
2441:
2439:
2437:
2433:
2429:
2425:
2421:
2416:
2414:
2413:Eötvös effect
2409:
2401:
2399:
2395:
2391:
2384:
2382:
2375:
2370:
2367:
2363:
2359:
2355:
2354:destabilizing
2351:
2343:
2341:
2339:
2338:Magnus effect
2330:
2324:Magnus effect
2323:
2321:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2287:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2258:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2229:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2200:
2197:
2195:
2187:
2184:
2181:
2178:
2177:
2176:
2173:
2170:
2161:
2156:
2154:
2152:
2144:Precipitation
2143:
2141:
2139:
2135:
2131:
2127:
2123:
2115:
2113:
2111:
2107:
2103:
2095:
2093:
2090:
2085:
2081:
2074:
2069:
2067:
2063:
2055:
2047:
2040:
2038:
2034:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1978:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1940:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1902:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1864:
1860:
1857:
1854:
1851:
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1845:
1842:
1839:
1836:
1833:
1830:
1827:
1826:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
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1788:
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1778:
1775:
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1769:
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1760:
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1708:
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1702:
1699:
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1690:
1687:
1684:
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1678:
1675:
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1670:
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1664:
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1614:
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1602:
1599:
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1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
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1547:
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1538:
1535:
1532:
1529:
1526:
1523:
1522:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
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1485:
1484:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
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1450:
1447:
1446:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1408:
1405:
1403:
1399:
1395:
1386:
1384:
1381:
1372:
1368:
1366:
1358:
1352:
1346:
1337:
1335:
1333:
1329:
1325:
1316:
1314:
1307:
1305:
1302:
1294:
1292:
1289:
1284:
1280:
1278:
1273:
1268:
1264:
1260:
1252:
1250:
1247:
1242:
1238:
1234:
1229:
1227:
1223:
1219:
1211:
1203:
1195:
1191:
1190:German 7.9 mm
1186:
1179:
1177:
1173:
1156:
1155:MediaWiki.org
1152:
1147:
1138:
1137:
1130:
1122:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1037:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
955:
952:
950:
947:
946:
943:
941:
936:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
878:
874:
871:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
835:
832:
829:
826:
825:
822:
820:
815:
809:
804:
800:
799:Doppler radar
792:
790:
784:
781:
778:
775:
772:
769:
766:
765:
764:
761:
757:
749:
747:
744:
740:
736:
732:
728:
724:
719:
716:
712:
708:
700:
698:
696:
692:
689:
684:
682:
676:
672:
670:
661:
659:
657:
652:
648:
638:
634:
632:
627:
625:
620:
615:
611:
607:
603:
597:
595:
590:
584:
582:
578:
570:
565:
563:
561:
555:
548:
545:
542:
539:
535:
532:
531:tangent ogive
528:
525:
522:
521:
516:
512:
508:
506:
505:very-low-drag
502:
496:
494:
490:
486:
483:
479:
475:
471:
467:
463:
459:
454:
452:
448:
444:
440:
438:
434:
430:
427:
422:
420:
416:
412:
408:
404:
400:
396:
392:
387:
385:
381:
377:
373:
372:G1 drag model
369:
360:
353:
351:
349:
342:
338:
334:
330:
326:
319:
317:
314:
305:
303:
299:
295:
292:
287:
284:
275:
273:
266:
264:
262:
258:
254:
250:
241:
237:
233:
228:
221:
216:
210:
207:
203:
199:
195:
191:
187:
186:
185:
179:
177:
175:
170:
168:
164:
160:
156:
152:
148:
143:
141:
136:
131:
128:
123:
121:
117:
116:line-of-sight
112:
110:
106:
102:
98:
94:
86:
84:
82:
78:
74:
70:
65:
62:
58:
54:
50:
43:
19:
4987:Aerodynamics
4954:. Retrieved
4950:the original
4940:
4802:ChairGun Pro
4757:
4756:
4671:
4670:
4659:. Retrieved
4652:the original
4647:
4615:
4611:
4603:
4602:
4585:
4562:
4555:
4543:. Retrieved
4539:
4530:
4519:. Retrieved
4515:the original
4505:
4494:
4483:
4472:
4460:. Retrieved
4456:the original
4446:
4434:. Retrieved
4424:
4413:
4401:. Retrieved
4397:
4387:
4375:. Retrieved
4371:the original
4360:
4348:. Retrieved
4344:the original
4334:
4308:. Retrieved
4304:the original
4294:
4282:. Retrieved
4278:the original
4268:
4256:. Retrieved
4252:the original
4242:
4231:. Retrieved
4224:the original
4211:
4199:. Retrieved
4192:the original
4179:
4167:. Retrieved
4158:
4146:. Retrieved
4142:the original
4132:
4121:. Retrieved
4114:the original
4101:
4089:. Retrieved
4080:
4068:. Retrieved
4061:the original
4048:
4030:
4019:. Retrieved
4010:
3998:. Retrieved
3994:the original
3984:
3972:. Retrieved
3968:the original
3958:
3947:. Retrieved
3940:the original
3927:
3915:. Retrieved
3911:the original
3901:
3890:
3879:
3864:
3853:. Retrieved
3846:the original
3833:
3814:
3810:
3800:
3788:. Retrieved
3779:
3767:. Retrieved
3758:
3746:. Retrieved
3742:the original
3732:
3720:. Retrieved
3713:the original
3700:
3689:
3678:
3667:
3656:. Retrieved
3649:the original
3636:
3627:
3621:
3609:. Retrieved
3599:
3590:
3584:
3575:
3569:
3560:
3554:
3545:
3539:
3530:
3512:
3506:
3494:. Retrieved
3490:the original
3485:
3475:
3457:
3445:. Retrieved
3435:
3424:. Retrieved
3420:the original
3410:
3398:. Retrieved
3391:the original
3378:
3366:. Retrieved
3354:
3342:. Retrieved
3330:
3311:
3299:. Retrieved
3295:the original
3285:
3273:. Retrieved
3269:the original
3259:
3247:. Retrieved
3243:the original
3233:
3221:. Retrieved
3212:
3201:. Retrieved
3194:the original
3181:
3169:. Retrieved
3160:
3149:. Retrieved
3137:
3126:. Retrieved
3117:
3105:. Retrieved
3095:
3084:the original
3075:
3068:
3057:
3013:
2992:
2978:
2965:
2955:
2880:
2867:
2842:
2806:
2781:
2702:457 m
2700:500 yd
2697:366 m
2695:400 yd
2692:274 m
2690:300 yd
2687:183 m
2685:200 yd
2680:100 yd
2666:
2654:
2653:
2644:
2637:
2501:
2488:
2479:
2476:Lateral jump
2470:
2461:
2454:
2445:
2417:
2405:
2396:
2392:
2388:
2371:
2365:
2361:
2357:
2353:
2349:
2347:
2335:
2318:
2202:Bullet type
2191:
2174:
2165:
2147:
2119:
2099:
2086:
2082:
2078:
2064:
2056:
2048:
2041:
2018:
1390:
1373:
1369:
1353:
1341:
1320:
1311:
1298:
1286:The ambient
1285:
1281:
1256:
1230:
1225:
1221:
1199:
1174:
1170:
1128:
937:
933:
816:
796:
788:
773:Earth models
753:
720:
704:
685:
677:
673:
665:
662:Manges model
643:
628:
619:power series
598:
585:
574:
559:
556:
552:
538:secant ogive
509:
497:
484:
455:
441:
428:
423:
414:
410:
406:
402:
398:
394:
388:
365:
346:
309:
300:
296:
288:
279:
270:
261:trajectories
246:
196:such as the
183:
171:
144:
140:trajectories
132:
124:
113:
90:
66:
52:
48:
47:
4982:Projectiles
4956:24 December
4724:(in German)
4545:24 December
4403:24 December
4310:24 December
4284:24 December
4258:24 December
4091:24 December
4070:24 December
3974:24 December
3790:24 December
3769:24 December
3722:24 December
3368:24 December
3344:24 December
3301:24 December
3249:24 December
3171:16 November
3107:24 December
3024:air density
2998:snipershide
2933:Sighting in
2682:91 m
2570:air density
2558:temperature
2524:bullet mass
2362:stabilizing
2138:water vapor
2134:air density
2126:temperature
1402:air density
1374:Presented C
1328:air density
1288:air density
1151:Phabricator
949:Mach number
735:smartphones
656:air density
571:Pejsa model
489:Mach number
451:scale model
397:, velocity
370:method and
333:Shadowgraph
236:M16A2 rifle
4977:Ballistics
4971:Categories
4521:2008-06-24
4462:12 January
4436:12 January
4377:12 January
4233:2016-10-14
4201:14 October
4169:12 January
4148:12 January
4123:2015-07-22
4042:1608.06500
4021:2020-09-27
4000:12 January
3949:2015-01-11
3917:12 January
3855:2015-11-01
3748:12 January
3658:2016-10-15
3611:12 January
3447:12 January
3426:2018-02-02
3400:12 January
3275:12 January
3223:12 January
3203:2022-06-02
3151:2014-06-03
3128:2007-09-11
3050:References
3038:latitude (
2594:spin drift
2590:gyroscopic
2504:small arms
2420:small arms
1410:Range (m)
1272:precession
1259:supersonic
1246:base bleed
1226:Cannelures
827:Range (m)
743:mobile app
715:precession
474:supersonic
470:supersonic
458:supersonic
382:) and its
341:Ernst Mach
232:M4 carbine
97:projectile
73:gun barrel
69:propellant
61:projectile
57:ballistics
4618:(1): 37.
2987:behavior.
2707:Velocity
2640:gunpowder
2424:artillery
2106:elevation
1263:transonic
1222:boat tail
612:a simple
610:curvature
589:curvature
501:wadcutter
478:transonic
257:parabolic
167:Paris Gun
151:parabolic
147:artillery
4921:Archived
4743:Archived
4713:Archived
4691:Archived
4350:28 March
3465:Archived
3322:Archived
3001:Archived
2886:See also
2604:latitude
2584:altitude
2580:latitude
2566:humidity
2562:altitude
2554:pressure
2151:raindrop
2130:humidity
2122:pressure
2027:) and (S
2013:246.968
2010:178.082
2007:123.639
1899:260.968
1896:185.318
1893:126.870
1785:222.430
1782:160.739
1779:112.136
1671:243.191
1668:174.796
1665:121.498
1557:241.735
1554:173.998
1551:121.023
1277:subsonic
691:Archived
482:function
462:subsonic
4877:models.
4620:Bibcode
4431:"longr"
3496:31 July
3040:gravity
3036:degrees
2827:Height
2769:Height
2602:drift (
2527:actual
2436:azimuth
2428:rockets
2004:81.863
2001:51.165
1998:30.039
1995:16.503
1975:8.3369
1972:7.0838
1969:5.9099
1966:4.8110
1963:3.7850
1960:2.8404
1957:2.0415
1954:1.3901
1951:0.8487
1948:0.3912
1945:0.0000
1890:82.873
1887:51.582
1884:30.271
1881:16.580
1861:8.6769
1858:7.2958
1855:6.0294
1852:4.8682
1849:3.8057
1846:2.8556
1843:2.0501
1840:1.3921
1837:0.8479
1834:0.3902
1831:0.0000
1776:75.205
1773:47.810
1770:28.779
1767:16.073
1747:7.9183
1744:6.7276
1741:5.6086
1738:4.5642
1735:3.6029
1732:2.7427
1729:2.0009
1726:1.3732
1723:0.8423
1720:0.3897
1717:0.0000
1662:80.794
1659:50.836
1656:30.060
1653:16.561
1633:8.3346
1630:7.0332
1627:5.8508
1624:4.7641
1621:3.7575
1618:2.8343
1615:2.0467
1612:1.3949
1609:0.8511
1606:0.3919
1603:0.0000
1548:80.529
1545:50.715
1542:30.035
1539:16.571
1519:8.2909
1516:7.0095
1513:5.8354
1510:4.7522
1507:3.7480
1504:2.8276
1501:2.0435
1498:1.3937
1495:0.8507
1492:0.3918
1489:0.0000
1357:Hornady
1153:and on
487:of the
433:12.7 mm
426:calibre
343:in 1888
335:of the
259:shaped
253:zeroing
249:gravity
101:gravity
4900:GunSim
4570:
2937:sights
2850:â35.0
2817:â1245
2792:â49.0
2732:1,834
2674:Range
2426:, and
2366:behind
2128:, and
2035:(0.68
1992:8.109
1989:3.191
1986:0.714
1983:0.000
1878:8.129
1875:3.198
1872:0.719
1869:0.000
1764:7.971
1761:3.157
1758:0.710
1755:0.000
1650:8.132
1647:3.195
1644:0.714
1641:0.000
1536:8.146
1533:3.203
1530:0.715
1527:0.000
1443:3,000
1440:2,700
1437:2,400
1434:2,100
1431:1,800
1428:1,500
1425:1,200
1192:s.S. (
1123:0.270
1120:0.282
1117:0.292
1114:0.304
1111:0.321
1108:0.328
1105:0.336
1102:0.343
1099:0.348
1096:0.348
1093:0.347
1090:0.345
1087:0.341
1084:0.334
1081:0.306
1078:0.236
1075:0.177
1072:0.154
1069:0.142
1066:0.137
1063:0.137
1060:0.141
1057:0.144
1054:0.164
1051:0.171
1048:0.200
1045:0.229
1042:0.230
1034:2.400
1031:2.200
1028:2.000
1025:1.800
1022:1.600
1019:1.500
1016:1.400
1013:1.300
1010:1.200
1007:1.150
1004:1.100
1001:1.075
998:1.050
995:1.025
992:1.000
989:0.975
986:0.950
983:0.925
980:0.900
977:0.875
974:0.850
971:0.825
968:0.800
965:0.700
962:0.600
959:0.500
956:0.400
953:0.000
928:1.032
925:1.042
922:1.050
919:1.056
916:1.060
913:1.064
910:1.066
907:1.068
904:1.068
901:1.068
898:1.067
895:1.064
892:1.063
889:1.057
886:1.051
883:1.040
707:PRODAS
243:plane.
194:sabots
190:arrows
93:forces
81:arrows
4896:ones.
4655:(PDF)
4648:(PDF)
4644:(PDF)
4227:(PDF)
4220:(PDF)
4195:(PDF)
4188:(PDF)
4117:(PDF)
4110:(PDF)
4064:(PDF)
4057:(PDF)
4038:arXiv
3943:(PDF)
3936:(PDF)
3849:(PDF)
3842:(PDF)
3716:(PDF)
3709:(PDF)
3652:(PDF)
3645:(PDF)
3394:(PDF)
3387:(PDF)
3363:(PDF)
3339:(PDF)
3197:(PDF)
3190:(PDF)
3146:(PDF)
3087:(PDF)
3080:(PDF)
3028:kg/mÂł
3020:grain
2982:The C
2948:Notes
2939:on a
2875:â889
2855:(mm)
2847:â13.1
2830:(in)
2797:(mm)
2789:â24.3
2772:(in)
2729:1,992
2726:2,158
2723:2,331
2720:2,512
2717:2,700
2430:like
2358:ahead
1208:) or
1188:Left
875:2000
872:1900
869:1800
866:1700
863:1600
860:1500
857:1400
854:1300
851:1200
848:1100
845:1000
614:chord
602:speed
485:BC(M)
466:rifle
419:Krupp
83:etc.
77:balls
40:This
4958:2022
4912:5H0T
4663:2005
4568:ISBN
4547:2022
4464:2017
4438:2017
4405:2022
4379:2017
4352:2017
4312:2022
4286:2022
4260:2022
4203:2016
4171:2017
4150:2017
4093:2022
4072:2022
4002:2017
3976:2022
3919:2017
3792:2022
3771:2022
3750:2017
3724:2022
3613:2017
3498:2018
3449:2017
3402:2017
3370:2022
3346:2022
3303:2022
3277:2017
3251:2022
3225:2017
3173:2014
3109:2022
2996:The
2872:â333
2833:â1.5
2814:â617
2811:â213
2786:â8.4
2775:â1.5
2759:559
2712:ft/s
2582:and
2564:and
2552:air
2547:wind
2455:The
2406:The
2120:Air
2104:(or
2100:The
2075:Wind
1937:235
1934:249
1931:265
1928:283
1925:303
1922:339
1919:418
1916:508
1913:606
1910:713
1907:830
1823:208
1820:227
1817:247
1814:270
1811:297
1808:339
1805:413
1802:504
1799:603
1796:712
1793:830
1709:248
1706:261
1703:278
1700:299
1697:328
1694:374
1691:440
1688:522
1685:615
1682:718
1679:830
1595:222
1592:244
1589:266
1586:287
1583:310
1580:347
1577:420
1574:506
1571:604
1568:711
1565:830
1481:227
1478:247
1475:267
1472:288
1469:311
1466:349
1463:422
1460:507
1457:604
1454:711
1451:830
1422:900
1419:600
1416:300
1267:Mach
842:900
839:800
836:700
833:600
830:500
460:and
413:and
391:drag
240:M855
234:and
157:and
120:Drag
109:wind
105:drag
99:are
4628:doi
3819:doi
3032:MOA
2864:142
2861:122
2858:â38
2839:5.6
2836:4.8
2800:â38
2778:2.0
2756:607
2753:658
2750:710
2747:766
2744:823
2739:m/s
2350:yaw
2169:yaw
2037:moa
2033:mil
1194:FMJ
733:or
51:or
4973::
4646:.
4626:.
4616:55
4614:.
4538:.
4396:.
4320:^
3815:13
3813:.
3809:.
3521:^
3484:.
3320:.
2803:51
2741:)
2714:)
2677:0
2560:,
2556:,
2124:,
2058:(C
2050:(C
1413:0
1347:(C
1334:.
1204:(C
810:(C
729:,
697:.
583:.
476:,
447:pi
415:dÂČ
411:vÂČ
409:,
263:.
142:.
118:.
103:,
79:,
4960:.
4852:d
4665:.
4634:.
4630::
4622::
4576:.
4549:.
4524:.
4466:.
4440:.
4407:.
4381:.
4354:.
4314:.
4288:.
4262:.
4236:.
4205:.
4173:.
4152:.
4126:.
4095:.
4074:.
4040::
4024:.
4004:.
3978:.
3952:.
3921:.
3873:.
3858:.
3827:.
3821::
3794:.
3773:.
3752:.
3726:.
3661:.
3615:.
3500:.
3451:.
3429:.
3404:.
3372:.
3348:.
3305:.
3279:.
3253:.
3227:.
3206:.
3175:.
3154:.
3131:.
3111:.
2984:d
2971:d
2868:0
2843:0
2807:0
2782:0
2737:(
2710:(
2586:)
2572:)
2379:d
2377:C
2060:d
2052:d
2044:d
2029:g
2025:d
2021:d
1376:d
1361:d
1349:d
1214:d
1206:d
1157:.
812:d
560:i
540:)
533:)
429:d
407:m
403:d
399:v
395:m
331:/
20:)
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