External Ballistics
External Ballistics
External Ballistics
External ballistics
External ballistics is the part of the science of ballistics that deals with the behaviour of a non-powered projectile in flight. External ballistics is frequently associated with firearms, and deals with the behaviour of the bullet after it exits the barrel and before it hits the target, so it lies between transitional ballistics and terminal ballistics..
For medium to longer ranges and flight times, besides gravity, air resistance and wind, several meso variables described in the external factors paragraph have to be taken into account. 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. For long to very long ranges and flight times, minor effects and forces such as the ones described in the long range factors paragraph become important and have to be taken into account. The practical effects of these 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 firearms projectiles trajectories. At extremely long ranges, artillery must fire projectiles along trajectories that are not even approximately straight; they are closer to parabolic, although air resistance affects this. In the case of ballistic missiles, the altitudes involved have a significant effect as well, with part of the flight taking place in a near-vacuum.
Schlieren image of a bullet travelling in free-flight demonstrating the air pressure dynamics surrounding the bullet.
External ballistics
External ballistics
The deceleration due to drag that a projectile with mass m, velocity v, and diameter d will experience is proportional to 1/BC, 1/m, v and d. The BC gives the ratio of ballistic efficiency compared to the standard G1 projectile, which is a 1 pound (454 g), 1inch (25.4mm) diameter bullet with a flat base, a length of 3inches (76.2mm), and a 2inch (50.8mm) radius tangential curve for the point. The G1 standard projectile originates from the "C" standard reference projectile defined by the German steel, ammunition and armaments manufacturer Krupp in 1881. The G1 model standard projectile has a BC of 1.[1] The French Gvre Commission decided to use this projectile as their first reference projectile, giving the G1 name.[2][3] Sporting bullets, with a calibre d ranging from 0.177 to 0.50inches (4.50 to 12.7 mm), have G1 BCs in the range 0.12 to slightly over 1.00, with 1.00 being the most aerodynamic, and 0.12 being the least. Very-low-drag bullets with BC's 1.10 can be designed and produced on CNC precision lathes out of mono-metal rods, but they often have to be fired from custom made full bore rifles with special barrels.[4] Sectional density is a very important aspect of a bullet, and is the ratio of frontal surface area (half the bullet diameter squared, times pi) to bullet mass. Since, for a given bullet shape, frontal surface increases as the square of the calibre, and mass increases as the cube of the diameter, then sectional density grows linearly with bore diameter. Since BC combines shape and sectional density, a half scale model of the G1 projectile will have a BC of 0.5, and a quarter scale model will have a BC of 0.25. Since different projectile shapes will respond differently to changes in velocity (particularly between supersonic and subsonic velocities), a BC provided by a bullet manufacturer will be an average BC that represents the common range of velocities for that bullet. For rifle bullets, this will probably be a supersonic velocity, for pistol bullets it will be probably be subsonic. For projectiles that travel through the supersonic, transonic and subsonic flight regimes BC is not well approximated by a single constant, but is considered to be a function BC(M) of the Mach number M; here M equals the projectile velocity divided by the speed of sound. During the flight of the projectile the M will decrease, and therefore (in most cases) the BC will also decrease. Most ballistic tables or software takes for granted that one specific drag function correctly describes the drag and hence the flight characteristics of a bullet related to its ballistics coefficient. Those models do not differentiate between wadcutter, flat-based, spitzer, boat-tail, very-low-drag, etc. bullet types or shapes. They assume one invariable drag function as indicated by the published BC. Several drag curve models optimized for several standard projectile shapes are however available. The resulting fixed drag curve models for several standard projectile shapes or types are referred to as the:
Schlieren photo/Shadowgraph of the detached shock or bow shokwave around a bullet in supersonic flight, published by Ernst Mach in 1888.
External ballistics G1 or Ingalls (flatbase with 2 caliber (blunt) nose ogive - by far the most popular) G2 (Aberdeen J projectile) G5 (short 7.5 boat-tail, 6.19 calibers long tangent ogive) G6 (flatbase, 6 calibers long secant ogive) G7 ((long 7.5 boat-tail, 10 calibers tangent ogive, preferred by some manufacturers for very-low-drag bullets[5]) G8 (flatbase, 10 calibers long secant ogive) GL (blunt lead nose)
How different speed regimes affect .338 calibre rifle bullets can be seen in the .338 Lapua Magnum product brochure which states Doppler radar established G1 BC data.[6][7] The reason for publishing data like in this brochure is that the Siacci/Mayevski G1 model can not be tuned for the drag behaviour of a specific projectile 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 behaviour at longer ranges compared to calculations that use only one BC constant. The above example illustrates the central problem fixed drag curve models have. These models will only yield satisfactory accurate predictions as long as the projectile of interest has the same shape as the reference projectile or a shape that closely resembles the reference projectile. Any deviation from the reference projectile shape will result in less accurate predictions.[8] How much a projectile deviates from the applied reference projectile is mathematically expressed by the form factor (i).[9]
External ballistics beyond 1,000 m (1,094 yd) where high powered very-low-drag rifle bullets tend to go transonic and eventually subsonic.[13][14] Manges model Although not as well known as the Pejsa model, an additional alternative ballistic model was presented in 1989 by Colonel Duff Manges (U S Army Retired) at the American Defense Preparedness (ADPA) 11th International Ballistic Symposium held at the Brussels Congress Center, Brussels, Belgium, May 911, 1989. A paper titled "Closed Form Trajectory Solutions for Direct Fire Weapons Systems" appears in the proceedings, Volume 1, Propulsion Dynamics, Launch Dynamics, Flight Dynamics, pages 665-674. Originally conceived to model projectile drag for 120mm tank ammunition, the novel drag coefficient formula has been applied subsequently to ballistic trajectories of center-fired rifle ammunition with results comparable to those claimed for the Pejsa model. The Manges model uses a first principles theoretical approach that eschews "G" curves and "ballistic coefficients" based on the standard G1 and other similarity curves. The theoretical description has three main parts. The first is to develop and solve a formulation of the two dimensional differential equations of motion governing flat trajectories of point mass projectiles by defining mathematically a set of quadratures that permit closed form solutions for the trajectory differential equations of motion. A sequence of successive approximation drag coefficient functions is generated that converge rapidly to actual observed drag data. The vacuum trajectory, simplified aerodynamic, d'Antonio, and Euler drag law models are special cases. The Manges drag law thereby provides a unifying influence with respect to earlier models used to obtain two dimensional closed form solutions to the point-mass equations of motion. The third purpose of this paper is to describe a least squares fitting procedure for obtaining the new drag functions from observed experimental data. The author claims that results show excellent agreement with six degree of freedom numerical calculations for modern tank ammunition and available published firing tables for center-fired rifle ammunition having a wide variety of shapes and sizes. A Microsoft Excel application has been authored that uses least squares fits of wind tunnel acquired tabular drag coefficients. Alternatively, manufacturer supplied ballistic trajectory data, or Doppler acquired velocity data can be fitted as well to calibrate the model. The Excel application then employs custom macroinstructions to calculate the trajectory variables of interest. A modified 4th order Runge-Kutta integration algorithm is used. Like Pejsa, Colonel Manges claims center-fired rifle accuracies to the nearest one tenth of an inch for bullet position, and nearest foot per second for the projectile velocity. The Proceedings of the 11th International Ballistic Symposium are available through the National Defense Industrial Association (NDIA) at the website http://www.ndia.org/Resources/Pages/Publication_Catalog.aspx. 6 Degrees of Freedom (6 DoF) model There are also advanced professional ballistic models like PRODAS [15] available. These are based on 6 Degrees of Freedom (6 DoF) calculations. 6 DoF modelling needs such elaborate input, knowledge of the employed projectiles and long calculation time on computers that it is impractical for non-professional ballisticians and field use where calculations generally have to be done on the fly on mobile computing devices like (ruggedized) PDAs or smartphones with relatively modest computing power. 6 DoF is generally used by military organizations that study the ballistic behaviour of a limited number of (intended) military issue projectiles. Calculated 6 DoF trends can be incorporated as correction tables in more conventional ballistic software applications.
External ballistics
Doppler radar-measurements
For the precise establishment of drag or air resistance effects on projectiles, Doppler radar-measurements are required. Weibel 1000e Doppler radars are used by governments, professional ballisticians, defence forces and a few ammunition manufacturers to obtain real world data of the flight behaviour of projectiles of their interest. Correctly established state of the art Doppler radar measurements can determine the flight behaviour of projectiles as small as airgun pellets in three-dimensional space to within a few millimetres accuracy. The gathered data regarding the projectile deceleration can be derived and expressed in several ways, such as ballistic coefficients (BC) or drag coefficients (Cd). Doppler radar measurement results for a lathe-turned monolithic solid .50 BMG very-low-drag bullet (Lost River J40 .510-773 grain monolithic solid bullet / twist rate 1:15 in) look like this:
Range (m) 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
Ballistic coefficient 1.040 1.051 1.057 1.063 1.064 1.067 1.068 1.068 1.068 1.066 1.064 1.060 1.056 1.050 1.042 1.032
The initial rise in the BC value is attributed to a projectile's always present yaw and precession out of the bore. The test results were obtained from many shots not just a single shot. The bullet was assigned 1.062 for its BC number by the bullet's manufacturer Lost River Ballistic Technologies. Doppler radar measurement results for a Lapua GB528 Scenar 19.44 g (300 gr) 8.59mm (0.338in) calibre very-low-drag bullet look like this:
Mach number Drag coefficient 0.230 0.229 0.200 0.171 0.164 0.144 0.141 0.137 0.137 0.142 0.154 0.177 0.236 0.306 0.334 0.341 0.345 0.347 0.348 0.348 0.343 0.336 0.328 0.321 0.304 0.292 0.282 0.270 0.000 0.400 0.500 0.600 0.700 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975 1.000 1.025 1.050 1.075 1.100 1.150 1.200 1.300 1.400 1.500 1.600 1.800 2.000 2.200 2.400
This tested bullet experiences its maximum drag coefficient when entering the transonic flight regime around Mach 1.200.
External ballistics
External ballistics Research into guided projectiles To circumvent the transonic problems encountered by spin stabilized projectiles small arms projectiles can theoretically be guided during flight. The Sandia National Laboratories announced in January 2012 it has researched and test fired 4 inch (102mm) 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 2,000 meters). These projectiles are not spin stabilized and the flight path can be course adjusted with an electromagnetic actuator 30 times per second. The researchers also claim they have video of the bullet radically pitching as it exited the barrel and pitching less as it flies down range, a disputed phenomenon known to long-range firearms experts as going to sleep. Because the bullets motions settle the longer it is in flight, accuracy improves at longer ranges, Sandia researcher Red Jones said. Nobody had ever seen that, but weve got high-speed video photography that shows that its true, he said.[18] Since Sandia is seeking a private company partner to complete testing of the prototype and bring a guided bullet to the marketplace the future of this technology remains uncertain.
External ballistics tumbling in the transonic/subsonic flight velocity regime. The information regarding unfavourable transonic/subsonic flight behaviour for some of the tested projectiles is important. This is a limiting factor for extended range shooting use, because the effects of coning and tumbling are not easily predictable and potentially catastrophic for the best ballistic prediction models and software. Presented Cd data can not be simply used for every gun-ammunition combination, since it was measured for the barrels, rotational (spin) velocities and ammunition lots the Lapua testers used during their test firings. Variables like differences in rifling (number of grooves, depth, width and other dimensional properties), twist rates and/or muzzle velocities impart different rotational (spin) velocities and rifling marks on projectiles. Changes in such variables and projectile production lot variations can yield different downrange interaction with the air the projectile passes through that can result in (minor) changes in flight behaviour. This particular field of external ballistics is currently (2009) not elaborately studied nor well understood.[23]
0.0000 0.3918 0.8507 1.3937 2.0435 2.8276 3.7480 4.7522 5.8354 0.000 830 0.715 718 3.203 615 8.146 522
7.0095
8.2909
16.571 30.035 50.715 80.529 121.023 173.998 241.735 440 374 328 299 278 261 6.7276 248 7.9183
0.0000 0.3897 0.8423 1.3732 2.0009 2.7427 3.6029 4.5642 5.6086 0.000 830 0.710 712 3.157 603 7.971 504
16.073 28.779 47.810 75.205 112.136 160.739 222.430 413 339 297 270 247 227 7.2958 208 8.6769
0.0000 0.3902 0.8479 1.3921 2.0501 2.8556 3.8057 4.8682 6.0294 0.000 830 0.719 713 3.198 606 8.129 508
16.580 30.271 51.582 82.873 126.870 185.318 260.968 418 339 303 283 265 249 7.0838 235 8.3369
0.0000 0.3912 0.8487 1.3901 2.0415 2.8404 3.7850 4.8110 5.9099 0.000 0.714 3.191 8.109
The table shows that the traditional Siacci/Mayevski G1 drag curve model prediction method generally yields more optimistic results compared to the modern Doppler radar test derived drag coefficients (Cd) prediction method.[25] At 300 m (328 yd) range the differences will be hardly noticeable, but at 600 m (656 yd) and beyond the differences grow over 10m/s (32.8ft/s) projectile velocity and gradually become significant. At 1,500 m (1,640 yd) range the projectile velocity predictions deviate 25m/s (82.0ft/s), which equates to a predicted total drop difference of 125.6cm (49.4in) or 0.83 mrad (2.87 MOA) at 50 latitude. The Pejsa drag analytic closed-form solution prediction method, without slope constant factor fine tuning, yields very similar results in the supersonic flight regime compared to the Doppler radar test derived drag coefficients (Cd) prediction method. At 1,500 m (1,640 yd) range the projectile velocity predictions deviate 10m/s (32.8ft/s), which equates to a predicted total drop difference of 23.6cm (9.3in) or 0.16 mrad (0.54 MOA) at 50 latitude.
External ballistics The G7 drag curve model prediction method (recommended by some manufacturers for very-low-drag shaped rifle bullets) when using a G7 ballistic coefficient (BC) of 0.377 yields very similar results in the supersonic flight regime compared to the Doppler radar test derived drag coefficients (Cd) prediction method. At 1,500 m (1,640 yd) range the projectile velocity predictions have their maximum deviation of 10m/s (32.8ft/s). The predicted total drop difference at 1,500 m (1,640 yd) is 0.4cm (0.16in) at 50 latitude. The predicted total drop difference at 1,800 m (1,969 yd) is 45.0cm (17.7in), which equates to 0.25 mrad (0.86 MOA).
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External factors
Wind
Wind has a range of effects, the first being the effect of making the bullet deviate to the side. From a scientific perspective, the "wind pushing on the side of the bullet" is not what causes wind drift. What causes wind drift is drag. Drag makes the bullet turn into the wind, keeping the centre of air pressure on its nose. This causes the nose to be cocked (from your perspective) into the wind, the base is cocked (from your perspective) "downwind." So, (again from your perspective), the drag is pushing the bullet downwind making bullets follow the wind. A somewhat less obvious effect is caused by head or tailwinds. A headwind will slightly increase the relative velocity of the projectile, and increase drag and the corresponding drop. A tailwind will reduce the drag and the bullet drop. In the real world pure head or tailwinds are rare, since wind seldom is constant in force and direction and normally interacts with the terrain it is blowing over. This often makes ultra long range shooting in head or tailwind conditions difficult.
Vertical angles
The vertical angle (or elevation) of a shot will also affect the trajectory of the shot. Ballistic tables for small calibre projectiles (fired from pistols or rifles) assume that gravity is acting nearly perpendicular to the bullet path. If the angle is up or down, then the perpendicular acceleration will actually be less. The effect of the path wise acceleration component will be negligible, so shooting up or downhill will both result in a similar decrease in bullet drop. Often mathematical ballistic prediction models are limited to "flat fire" scenarios based on the rifleman's rule, meaning they can not produce adequately accurate predictions when combined with steep elevation angles over -15 to 15 degrees and longer ranges. There are however several mathematical prediction models for inclined fire scenarios available which yield rather varying accuracy expectation levels.[26]
External ballistics effect 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 bullet drift 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.[27][28] The following variables affect the magnitude of gyroscopic drift: Projectile or bullet length: longer projectiles experience more gyroscopic drift because they produce more lateral "lift" for a given yaw angle. Spin rate: faster spin rates will produce more gyroscopic drift because the nose ends up pointing farther to the side. Range, time of flight and trajectory height: gyroscopic drift increases with all of these variables. Doppler radar measurement results for the gyroscopic drift of several US military and other very-low-drag bullets at 1000 yards (914.4 m) look like this:
Bullet type US military M193 Ball US military M118 Special Ball 173 gr Palma Sierra MatchKing LRBT J40 Match Sierra MatchKing Sierra MatchKing LRBT J40 Match LRBT J40 Match
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Projectile weight (in 55 gr grain) Projectile diameter (in inches and mm) Gyroscopic drift (in inches and mm) .223 in / 5.56mm 23.00 in / 584mm
155 gr
190 gr
220 gr
300 gr
350 gr
419 gr
The table shows that the gyroscopic drift is rather variable and no clear trend is easily distinguishable.
Magnus effect
Spin stabilized projectiles are affected by the Magnus effect, 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.[29] 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 4m/s (14.4km/h or 9mph). Magnus effect and bullet stability
The Magnus effect. V represents the wind, the arrow F is the resulting Magnus force towards the side of lower pressure.
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 yaw of the bullet. The Magnus effect will act as a destabilizing force on any bullet with a center of pressure located ahead of the center of gravity, while conversely acting as a stabilizing force on any bullet with the center of pressure located behind 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 Magnus force greatly affects stability because it tries to "twist" the bullet along its flight path.[30][31]
External ballistics Paradoxically, very-low-drag bullets 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 Cd or BC based on shape and sectional density is of limited use.
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Poisson effect
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 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. 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. 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.
Coriolis drift
Coriolis drift is caused by the Coriolis effect and the Etvs effect. These effects cause drift related to the spin of the Earth, known as Coriolis drift. Coriolis drift can be up, down, left or right. Coriolis drift is not an aerodynamic effect; it is a consequence of flying from one point to another across the surface of a rotating planet (Earth). The direction of Coriolis drift depends on the firer's and target's location or latitude on the planet Earth, and the azimuth of firing. The magnitude of the drift depends on the firing and target location, azimuth, and time of flight. Coriolis effect The Coriolis effect causes subtle trajectory variations caused by a rotating reference frame. The coordinate system that is used to specify the location of the point of firing and the location of the target is the system of latitudes and longitudes, which is in fact a rotating coordinate system, since the planet Earth is a rotating sphere. During its flight, the projectile moves in a straight line (not counting gravitation and air resistance for now). Since the target is co-rotating with the Earth, it is in fact a moving target, relative to the projectile, so in order to hit it the gun must be aimed to the point where the projectile and the target will arrive simultaneously. When the straight path of the projectile is plotted in the rotating coordinate system that is used, then this path appears as curvilinear. The fact that the coordinate system is rotating must be taken into account, and this is achieved by adding terms for a "centrifugal force" and a "Coriolis effect" to the equations of motion. When the appropriate Coriolis term is added to the equation of motion the predicted path with respect to the rotating coordinate system is curvilinear, corresponding to the actual straight line motion of the projectile. For an observer with his frame of reference in the northern hemisphere Coriolis makes the projectile appear to curve over to the right. Actually it is not the projectile swinging to the right but the earth (frame of reference) rotating to the left which produces this result. The opposite will seem to happen in the southern hemisphere. The Coriolis effect is at its maximum at the poles and negligible at the equator of the Earth. The reason for this is that the Coriolis effect depends on the vector of the angular velocity of the Earth's rotation with respect to xyz coordinate system (frame of reference).[32] For small arms, the Coriolis effect is generally insignificant, but for ballistic projectiles with long flight times, such as extreme long-range rifle projectiles, artillery and intercontinental ballistic missiles, it is a significant factor in calculating the trajectory.
External ballistics Etvs effect The Etvs effect changes the apparent gravitational pull on a moving object based on the relationship between the direction of movement and the direction of the Earth's rotation. It causes subtle trajectory variations. It is not an aerodynamic effect and is latitude dependent, being at its most significant at equatorial latitude. Eastward-traveling objects will be deflected upwards (feel lighter), while westward-traveling objects will be deflected downwards (feel heavier). In addition, objects traveling upwards or downwards will be deflected to the west or east respectively. The principle behind these counterintuitive variations during flight are explained in more detail in the equivalence principle article dealing with the physics of general relativity. For small arms, the Etvs effect is generally insignificant, but for long range ballistic projectiles with long flight times it can become a significant factor in accurately calculating the trajectory.
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Equipment factors
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 behaviour they have to be briefly mentioned.
Lateral jump
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.
Lateral throw-off
Lateral throw-off is caused by mass imbalance in applied spin stabilized projectiles or pressure imbalances during the transitional flight phase when a projectile leaves a gun barrel. 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.
wind speed and direction (main cause for horizontal projectile deflection and generally the hardest ballistic variable to measure and judge correctly. Wind effects can also cause vertical deflection.) air temperature, pressure, altitude and humidity variations (these make up the ambient air density) Earth's gravity (changes slightly with latitude and altitude)
External ballistics gyroscopic drift (horizontal and vertical plane gyroscopic effect often known as spin drift - induced by the barrels twist direction and twist rate) Coriolis effect drift (latitude, direction of fire and northern or southern hemisphere data dictate this effect) Etvs effect (interrelated with the Coriolis effect, latitude and direction of fire dictate this effect) lateral throw-off (dispersion that is caused by mass imbalance in the applied projectile) aerodynamic jump (dispersion that is caused by lateral (wind) impulses activated during free flight at or very near the muzzle)[35] the inherent potential accuracy and adjustment range of the sighting components the inherent potential accuracy of the rifle the inherent potential accuracy of the ammunition the inherent potential accuracy of the computer program and other firing control components used to calculate the trajectory The ambient air density is at its maximum at Arctic sea level conditions. Cold gunpowder also produces lower pressures and hence lower muzzle velocities than warm powder. This means that the maximum practical range of rifles will be at it shortest at Arctic sea level conditions. 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 laser rangefinders 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.[36] Interesting further reading: Marksmanship Wikibook
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Zeroed for 200 yards (184 m) Height in -1.5 2.0 51 0 0 -8.4 -213 -24.3 -617 -49.0 -1245
mm -38
Zeroed for 300 yards (274 m) Height in -1.5 4.8 122 5.6 142 0 0 -13.1 -333 -35.0 -889
mm -38
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, because the difference in the point of impact between 400 and 500yd (460m) is 2532 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 2532 in (635813 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
External ballistics 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 200yd (180m), 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.
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References
[1] [2] [3] [4] [5] [6] [7] [8] [9] Ballistic Coefficients Do Not Exist! By Randy Wakeman (http:/ / www. chuckhawks. com/ bc_not_exist. htm) Weite Schsse - drei (German) (http:/ / www. lima-wiederladetechnik. de/ Weite-Schuesse/ Weite-Schuesse-3. htm) Historical Summary (http:/ / www. exteriorballistics. com/ ebexplained/ 4th/ 30. cfm) LM Class Bullets, very high BC bullets for windy long Ranges (http:/ / www. lima-wiederladetechnik. de/ Englisch/ LM-Class-Bullets. htm) A Better Ballistic Coefficient (http:/ / 02b0516. netsolhost. com/ blog1/ ?p=62) .338 Lapua Magnum product brochure from Lapua (http:/ / www. lapua. com/ uploads/ media/ 338LapuaMagnum2009. pdf) 300 grs Scenar HPBT brochure from Lapua (http:/ / www. lapua. com/ uploads/ media/ 338LapuaMagnum300grScenar2009. pdf) Ballistic Coefficients - Explained (http:/ / www. precisionshooting. com. au/ downloads/ ballisticcoefficients-explained(4). pdf) Form Factors: A Useful Analysis Tool by Bryan Litz, Chief Ballistician Berger Bullets (http:/ / www. bergerbullets. com/ form-factors-a-useful-analysis-tool/ ) [10] http:/ / home. sprintmail. com/ ~pejsa/ aboutartpejsa. htm [11] Pejsa Ballistics (http:/ / www. pejsa. com/ ) [12] Super-Accurate Prediction At Extreme Ranges: Modeling Drag And Calculating Trajectories by Arthur Pejsa (http:/ / www. longrangehunting. com/ articles/ long-range-drag-trajectories-1. php) [13] Lex Talus Corporation Pejsa based ballistic software (http:/ / www. precisionworkbench. com/ index. htm) [14] Patagonia Ballistics Pejsa based ballistic software (http:/ / www. patagoniaballistics. com/ index. html) [15] http:/ / www. prodas. com/
External ballistics
[17] 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. 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 modelling and simulation can do to accurately predict the level of dynamic stability that a bullet will have downrange. If a bullet has a very high or low level of dynamic stability, modelling may get the answer right. However, if a situation is borderline (dynamic stability near 0 or 2) modelling cannot be relied upon to produce the right answer. This is one of those things that have to be field tested and carefully documented. [18] Sandias self-guided bullet prototype can hit target a mile away (https:/ / share. sandia. gov/ news/ resources/ news_releases/ bullet/ ) [19] Lapua Bullets Drag Coefficient Data for QuickTARGET Unlimited (http:/ / www. lapua. com/ uploads/ media/ LapuaBulletsDragCoefficient. pdf) [20] Lapua bullets CD data (zip file) (http:/ / www. lapua. com/ uploads/ media/ LapuaBulletsCD-Data. zip) [21] QuickTARGET Unlimited, Lapua Edition (http:/ / www. lapua. com/ en/ customer-center/ lapua-ballistics/ download-lapua-edition. html) [22] Lapua Ballistics freeware exterior ballistic software for mobile phones (http:/ / www. lapua. com/ en/ customer-center/ lapua-ballistics/ lapua-ballistics-software. html) [23] 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 (http:/ / www. dtic. mil/ cgi-bin/ GetTRDoc?Location=U2& doc=GetTRDoc. pdf& AD=ADA505719) [24] G1, G7 and Doppler radar test derived drag coefficients (Cd) 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. [25] The Cd 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 behaviour. [26] Inclined fire - 3 methods for aiming adjustment - by William T. McDonald, June 2003 (http:/ / www. exteriorballistics. com/ ebexplained/ article1. html) [27] Nenstiel Yaw of repose (http:/ / www. nennstiel-ruprecht. de/ bullfly/ longr. htm#header_longranges) [28] Gyroscopic Drift and Coreolis Acceleration by Brian Litz (http:/ / bryanlitz. bravehost. com/ GyroCor. html) [29] Nenstiel The Magnus effect (http:/ / www. nennstiel-ruprecht. de/ bullfly/ fig9. htm) [30] Nenstiel The Magnus force (http:/ / www. nennstiel-ruprecht. de/ bullfly/ fig10. htm) [31] Nenstiel The Magnus moment (http:/ / www. nennstiel-ruprecht. de/ bullfly/ fig11. htm) [32] Gyroscopic Drift and Coreolis Acceleration by Bryan Litz (http:/ / bryanlitz. bravehost. com/ GyroCor. html) [33] The snipershide (http:/ / www. snipersparadise. com/ ballistics/ bulletbasics. htm) 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 than effective range. There seems to be no good formula for the effective ranges of the various firearms. [34] 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. Sniper Weapon Fire Control Error Budget Analysis - Raymond Von Wahlde, Dennis Metz, August 1999 (http:/ / www. arl. army. mil/ arlreports/ 1999/ ARL-TR-2065. pdf) [35] The Effects of Aerodynamic Jump Caused by a Uniform Sequence of Lateral Impulses - Gene R. Cooper, July 2004 (http:/ / www. arl. army. mil/ arlreports/ 2004/ ARL-TR-3206. pdf) [36] 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 grain (16.2 g) Lapua Scenar bullets at a measured muzzle velocity of 905 m/s. The air density during the test shoot was 1.2588 kg/m. The test rifle needed 13.2 mils (45.38 MOA) elevation correction from a 300 m zero range at 61 degrees latitude (gravity 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 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. [37] http:/ / sourceforge. net/ projects/ balcomp [38] http:/ / www. 6mmbr. com/ ballistics. html [39] http:/ / www. cronander. net/ CRONXR1A. zip [40] http:/ / www. huntingnut. com/ files/ pointblank/ PointBlankCRBSv18a. zip [41] http:/ / www. remington. com/ products/ ammunition/ ballistics/ remington_shoot_ballistics_software. asp [42] http:/ / www. ballistics-software. com/ [43] http:/ / www. jbmballistics. com/ ballistics/ calculators/ calculators. shtml [44] JBM Bullet Library (http:/ / www. jbmballistics. com/ calculations/ common/ bclibrary. shtml) [45] http:/ / accurateshooter. net/ Downloads/ pejsajacksonballistics. xls
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External ballistics
[46] [47] [48] [49] [50] [51] [52] [53] http:/ / www. bestpalmsoftware. com/ mathematic/ misc/ pocketssf_sharpshooterfriend344. htm http:/ / www. lapua. com/ en/ customer-center/ lapua-ballistics/ download-lapua-edition. html http:/ / www. lapua. com/ en/ customer-center/ lapua-ballistics/ lapua-ballistics-software. html http:/ / www. bfxyz. nl http:/ / www. gunsim. com http:/ / www. ballisticsimulator. com http:/ / www. chairgun. com/ http:/ / www. 5h0t. com/
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External links
General external ballistics Tan, A., Frick, C.H., and Castillo, O. (1987). "The fly ball trajectory: An older approach revisited". American Journal of Physics 55 (1): 37. Bibcode: 1987AmJPh..55...37T (http://adsabs.harvard.edu/abs/1987AmJPh.. 55...37T). doi: 10.1119/1.14968 (http://dx.doi.org/10.1119/1.14968). (Simplified calculation of the motion of a projectile under a drag force proportional to the square of the velocity) "The Perfect Basketball Shot" (http://web.archive.org/web/20060305151025/http://www.wooster.edu/ physics/jris/Files/Satti.pdf). (PDF). Archived from the original (http://www.wooster.edu/physics/jris/Files/ Satti.pdf) on March 5, 2006. Retrieved September 26, 2005. - basketball ballistics. Small arms external ballistics Software for calculating ball ballistics (http://www.ctmuzzleloaders.com/ctml_experiments/rbballistics/ rbballistics.html) How do bullets fly? by Ruprecht Nennstiel, Wiesbaden, Germany (http://www.nennstiel-ruprecht.de/bullfly/ index.htm) Exterior Ballistics.com articles (http://www.exteriorballistics.com/ebexplained/index.cfm) A Short Course in External Ballistics (http://www.frfrogspad.com/extbal.htm) Articles on long range shooting by Bryan Litz (http://www.appliedballisticsllc.com/Articles.htm) Weite Schsse - part 4, Basic explanation of the Pejsa model by Lutz Mller (German)[[Category:Articles with German-language external links (http://lutz-moeller-jagd.de/Schiessen/Weite-Schuesse/Weite-Schuesse-4. htm)]] Patagonia Ballistics ballistics mathematical software engine (http://www.patagoniaballistics.com/balengine. html) JBM Small Arms Ballistics with online ballistics calculators (http://www.jbmballistics.com/ballistics/ calculators/calculators.shtml) Bison Ballistics Point Mass Online Ballistics Calculator (http://bisonballistics.com/point_mass_calculations/ new) Virtual Wind Tunnel Experiments for Small Caliber Ammunition Aerodynamic Characterization - Paul Weinacht US Army Research Laboratory Aberdeen Proving Ground, MD (http://www.dtic.mil/ndia/2006smallarms/ weinacht.pdf) Artillery external ballistics BRITISH ARTILLERY FIRE CONTROL - BALLISTICS & DATA (http://members.tripod.com/~nigelef/ fc_ballistics.htm) FIELD ARTILLERY, VOLUME 6, BALLISTICS AND AMMUNITION (http://www.nvbmb.nl/downloads/ b-gl-306-006fp-001.pdf)
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License
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