Chapter 3
BALLISTICS
Ballistics is the study of the firing, flight, and effect of ammunition. A fundamental understanding of ballistics is necessary to comprehend the factors that influence precision and accuracy and how to account for them in the determination of firing data. Gunnery is the practical application of ballistics so that the desired ejects are obtained by fire. To ensure accurate predicted fire, we must strive to account for and minimize those factors that cause roundtoround variations, particularly muzzle velocity. Ballistics can be broken down into four areas: interior, transitional, exterior, and terminal. Interior, transitional, and exterior ballistics directly affect the accuracy of artillery fire and are discussed in this chapter. Terminal ballistics are discussed in Appendix B.
31. Interior Ballistics
Interior ballistics is the science that deals with the factors that affect the motion of the projectile within the tube. The total effect of all interior ballistic factors determines the velocity at which the projectile leaves the muzzle of the tube, which directly influences the range achieved by the projectile. This velocity, called muzzle velocity (MV), is expressed in meters per second (m/s). Actual measurements of the muzzle velocities of a sample of rounds corrected for the effects of nonstandard projectile weight and propellant temperature show the performance of a specific weapon for that projectile familypropellant typecharge combination. The resulting measurement(s) are compared to the standard muzzle velocity shown in the firing table(s). This comparison gives the variation from standard, called muzzle velocity variation (MVV), for that weapon and projectile familypropellant typecharge combination. Application of corrections to compensate for the effects of nonstandard muzzle velocity is an important element in computing accurate firing data. (For further discussion of muzzle velocity, see Chapter 4.) The following equation for muzzle velocity is valid for our purposes:
MVV (m/s) = SHOOTING STRENGTH OF WPN + AMMUNITION EFFICIENCY
Tube wear, propellant efficiency, and projectile weight are the items normally accounted for in determination of a muzzle velocity. Other elements in the equation above generally have an effect not exceeding 1.5 m/s. As a matter of convenience, the other elements listed below are not individually measured, but their effects are realized to exist under the broader headings of shooting strength and ammunition efficiency.
SHOOTING STRENGTH OF WEAPON 
AMMUNITION EFFICIENCY 
1. Tube wear 2. Manufacturer tolerances 3. Reaction to recoil 
1. Propellant efficiency 2. Projectile efficiency a. Projectile weight (fuzed) b. Construction of (1) Rotating band (2) Bourrelet (3) Obturating band 
a. Nature of Propellant and Projectile Movement.
(1) A propellant is a loworder explosive that burns rather than detonates. In artillery weapons using separateloading ammunition, the propellant burns within a chamber formed by the obturator spindle assembly, powder chamber, rotating band, and base of the projectile. For cannons using semifixed ammunition, the chamber is formed by the shell casing and the base of the projectile. When the propellant is ignited by the primer, the burning propellant generates gases. When these gases develop enough pressure to overcome initial bore resistance, the projectile begins its forward motion.
(2) Several parts of the cannon tube affect interior ballistics. (See Figure 31.)
(a) The caliber of a tube is the inside diameter of the tube as measured between opposite lands.
(b) The breech recess receives the breechblock. The breech permits loading the howitzer from the rear.
(c) The powder chamber receives the complete round of ammunition. It is the portion of the tube between the gas check seat and the centering slope.
 The gas check seat is the tapered surface in the rear interior of the tube on weapons firing separateloading ammunition. It seats the split rings of the obturating mechanism when they expand under pressure in firing. This expansion creates a metaltometal seal and prevents the escape of gases through the rear or the breech. Weapons firing semifixed ammunition do not have gas check seats since the expansion of the ease against the walls of the chamber provides a gas seal for the breech.
 The centering slope is the tapered portion at or near the forward end of the chamber that causes the projectile to center itself in the bore during loading.
(d) The forcing cone is the tapered portion near the rear of the bore that allows the rotating band to be gradually engaged by the rifling, thereby centering the projectile in the bore.
(e) The bore is the rifled portion of the tube (lands and grooves). It extends from the forcing cone to the muzzle. The rifled portion of the tube imparts spin to the projectile increasing stability in flight. The grooves are the depressions in the rifling. The lands are the raised portions. These parts engrave the rotating band. All United States (US) howitzers have a righthand twist in rifling.
(f) The bore evacuator is located on enclosed, selfpropelled howitzers with semiautomatic breech mechanisms. It prevents contamination of the crew compartment by removing propellant gases from the bore after firing. The bore evacuator forces the gases to flow outward through the bore from a series of valves enclosed on the tube.
(g) The counterbore is the portion at the front of the bore from which the lands have been removed to relieve stress and prevents the tube from cracking.
(h) The muzzle brake is located at the end of the tube on some howitzers. As the projectile leaves the muzzle, the highvelocity gases strike the baffles of the muzzle brake and are deflected rearward and sideways. When striking the baffles, the gases exert a forward force on the baffles that partially counteracts and reduces the force of recoil.
(3) The projectile body has several components that affect ballistics. (See Figure 32.) Three of these affect interior ballisticsthe bourrelet the rotating band and the obturating band.
(a) The bourrelet is the widest part of the projectile and is located immediately to the rear of the ogive. The bourrelet centers the forward part of the projectile in the tube and bears on the lands of the tube. When the projectile is fired, only the bourrelet and rotating band bear on the lands of the tube.
(b) The rotating band is a band of soft metal (copper alloy) that is securely seated around the body of the projectile. It provides forward obturation (the forward gastight seal required to develop pressure inside the tube). The rotating band prevents the escape of gas pressure from around the projectile. When the weapon is fired, the rotating band contacts the lands and grooves and is pressed between them. As the projectile travels the length of the cannon tube, over the lands and grooves, spin is imparted. The rifling for the entire length of the tube must be smooth and free of burrs and scars. This permits uniform seating of the projectile and gives a more uniform muzzle velocity.
(c) The obturating band is a plastic band on certain projectiles. It provides forward obturation by preventing the escape of gas pressure from around the projectile.
(4) The sequence that occurs within the cannon tube is described below.
(a) The projectile is rammed into the cannon tube and rests on the bourrelet. The rotating band contacts the lands and grooves at the forcing cone.
(b) The propellant is inserted into the chamber.
(c) The propellant explosive train is initiated by the ignition of the primer. This causes the primer, consisting of hot gases and incandescent particles, to be injected into the igniter. The igniter burns and creates hot gases that flow between the propellant granules and ignite the granule surfaces; the igniter and propellant combustion products then act together, perpetuating the flame spread until all the propellant granules are ignited.
(d) The chamber is sealed, in the rear by the breech and obturator spindle group and forward by the projectile, so the gases and energy created by the primer, igniter, and propellant cannot escape. This results in a dramatic increase in the pressure and temperature within the chamber. The burning rate of the propellant is roughly proportional to the pressure, so the increase in pressure is accompanied by an increase in the rate at which further gas is produced.
(e) The rising pressure is moderated by the motion of the projectile along the barrel. The pressure at which this motion begins is the shotstart pressure. The projectile will then almost immediately encounter the rifling, and the projectile will slow or stop again until the pressure has increased enough to overcome the resistance in the bore. The rotating band and obturating band (if present) or the surface of the projectile itself, depending on design, will be engraved to the shape of the rifling. The resistance decreases, thereby allowing the rapidly increasing pressure to accelerate the projectile.
(f) As the projectile moves forward, it leaves behind an increasing volume to be filled by the highpressure propellant gases. the propellant is still burning, producing highpressure gases so rapidly that the motion of the projectile cannot fully compensate. As a result, the pressure continues to rise until the peak pressure is reached. The peak pressure is attained when the projectile has traveled about onetenth of the total length of a full length howitzer tube.
(g) The rate at which extra space is being created behind the rapidly accelerating projectile then exceeds the rate at which highpressure gas is being produced; thus the pressure begins to fall. The next stage is the allburnt position at which the burning of the propellant is completed. However, there is still considerable pressure in the tube; therefore, for the remaining motion along the bore, the projectile continues to accelerate. As it approaches the muzzle, the propellant gases expand, the pressure falls, and so the acceleration lessens. At the moment the projectile leaves the howitzer, the pressure will have been reduced to about one sixth of the peak pressure. Only about onethird of the energy developed pushes the projectile. The other twothirds is absorbed by the recoiling parts or it is lost because of heat and metal expansion.
(h) The flow of gases following the projectile out of the muzzle provides additional acceleration for a short distance (transitional ballistics), so that the full muzzle velocity is not reached until the projectile is some distance beyond the muzzle. The noise and shock of firing are caused by the jet action of the projectile as it escapes the flow of gases and encounters the atmosphere. After this, the projectile breaks away from the influence of the gun and begins independent flight.
(i) This entire sequence, from primer firing to muzzle exit, typically occurs within 15 milliseconds but perhaps as much as 25 milliseconds for a large artillery howitzer.
(5) Pressure travel curves are discussed below.
(a) Once the propellant ignites, gases are generated that develop enough pressure to overcome initial bore resistance, thereby moving the projectile. Two opposing forces act on a projectile within the howitzer. The first is a propelling force caused by the highpressure propellant gases pushing on the base of the projectile. The second is a frictional force between the projectile and bore, which includes the high resistance during the engraving process, that opposes the motion of the projectile. The peak pressure, together with the travel of the projectile in the bore (pressure travel curve), determines the velocity at which the projectile leaves the tube.
(b) To analyze the desired development of pressure within the tube, we identify three types of pressure travel curves:
 An elastic strength pressure travel curve represents the greatest interior pressure that the construction of the tube (thickness of the wall of the powder chamber, thickness of the tube, composition of the tube or chamber, and so on) will allow. It decreases as the projectile travels toward the muzzle because the thickness of the tube decreases.
 A permissible pressure travel curve mirrors the elastic strength pressure travel curve and accounts for a certain factor of safety. It also decreases as the projectile travels through the tube because tube thickness decreases.
 An actual pressure travel curve represents the actual pressure developed during firing within the tube. Initially, pressure increases dramatically as the repelling charge explosive train initiated and the initial resistance of the rammed projectile is overcome. After that resistance is overcome, the actual pressure gradually decreases because of the concepts explained by Boyle's Law. (Generally, as volume increases, pressure decreases.) The actual pressure should never exceed the permissible pressure.
Figure 33 depicts different actual pressure travel curves that are discussed below.
 Initial Excessive Pressure. This is undesirable pressure travel curve. It exceeds the elastic strength pressure and permissible pressure. Causes of this travel curve would be an obstruction in the tube, a dirty tube, an "extra" propellant placed in the chamber, an unfuzed projectile, or a cracked projectile.
 Delayed Excessive Pressure. This is an undesirable pressure travel curve. It exceeds the elastic strength pressure and remissible pressure. Causes that would result in this travel curve would be using wet powder or powder reversed.
 Desirable Pressure Travel Curve. This curve does not exceed permissible pressure. It develops peak pressure at about onetenth the length of the tube.
(6) The following general rules show how various factors affect the velocity performance of a weapon projectile familypropellant typecharge combination:
(a) An increase in the rate of propellant burning increases the resulting gas pressure developed within the chamber. An example of this is the performance of the multiperforated propellant grains used in white bag (WB) propellants. The result is that more gases are produced, gas pressure is increased, and the projectile develops a greater muzzle velocity. Damage to propellant grains, such as cracking and splitting from improper handling, also affect the rate of burn and thus the muzzle velocity.
(b) An increase in the size of the chamber without a corresponding increase in the amount of propellant decreases gas pressure; as a result, muzzle velocity will be less (Boyles Law).
(c) Gas escaping around the projectile decreases chamber pressure.
(d) An increase in bore resistance to projectile movement before peak pressure increases the pressure developed within the tube. Generally, this results in a dragging effect on the projectile, with a corresponding decrease in the developed muzzle velocity. Temporary variations in bore resistance can be caused by excessive deposits of residue within the cannon tube and on projectiles and by temperature differences between the inner and outer surfaces of the cannon tube.
b. Standard Muzzle Velocity.
(1) Applicable firing tables list the standard value of muzzle velocity for each charge. These standard values are based on an assumed set of standard conditions. These values are points of departure and not absolute standards. Essentially, we cannot assume that a given weapon projectile familypropellant typecharge combination when fired will produce the standard muzzle velocity.
(2) Velocities for each charge are indirectly established by the characteristics of the weapons. Cannons capable of highangle fire (howitzers) require a greater choice in the number of charges than cannons capable of only lowangle fire (guns). This choice is necessary to achieve range overlap between charges in highangle fire and the desired rangetrajectory combination in lowangle fire. Other factors considered are the maximum range specified for the weapon, the maximum elevation and charge, and the maximum permissible pressure that the weapon can accommodate.
(3) Manufacturing specifications for ammunition include a requirement for velocity performance to meet certain tolerances. Ammunition lots are subjected to test firings, which include measuring the performance of a tested lot and comparing it to the performance of a control (reference) lot that is tested concurrently with the same weapon. An assumption built into the testing procedure is that both lots of ammunition will be influenced in the same manner by the performance of the tube. This assumption, although accurate in most instances, allows some error to be introduced in the assessment of the performance of the tested lot of propellant. In field conditions, variations in the performance of different projectile or propellant lots can be expected even though quality control has been exercised during manufacturing and testing of lots. In other words, although a howitzer develops a muzzle velocity that is 3 meters per second greater (or less) than standard with propellant lot G, it will not necessarily be the same with any other propellant lot. The optimum method for determining ammunition performance is to measure the performance of a particular projectile familypropellant lotcharge combination (calibration). However, predictions of the performance of a projectile familypropellant lotcharge group combination may be inferred with the understanding that they will not be as accurate as actual performance measurements.
c. Factors Causing Nonstandard Velocities. Nonstandard muzzle velocity is expressed as a variation (plus or minus so many meters per second) from the accepted standard. Roundtoround corrections for dispersion cannot be made. Each of the following factors that cause nonstandard conditions is treated as a single entity assuming no influence from related factors.
(1) Velocity trends. Not all rounds of a series fired from the same weapon and using the same ammunition lot will develop the same muzzle velocity. Under most conditions, the first few rounds follow a somewhat regular pattern rather than the random pattern associated with normal dispersion. This phenomenon is called velocity trends (or velocity dispersion), and the magnitude varies with the cannon, charge, and tube condition at the time each round is fired. Velocity trends cannot be accurately predicted; thus, any attempt to correct for the effects of velocity trends is impractical. Generally, the magnitude and duration of velocity trends can be minimized when firing is started with a tube that is clean and completely free of oil. (See Figure 34.)
(2) Ammunition lots. Each ammunition, projectile, and propellant lot has its own mean performance level in relation to a common weapon. Although the roundtoround variations within a given lot of the same ammunition (ammo) types are similar, the mean velocity developed by one lot may differ significantly in comparison to that of another lot. With separateloading ammunition, both the projectile and propellant lots must be identified. Projectile lots allow for rapid identification of weight differences. Although other projectile factors affect achieved muzzle velocity (such as, diameter and hardness of rotating band), the cumulative effect of these elements generally does not exceed 1.5 m/s. As a matter of convenience and speed, they are ignored in the computation of firing data.
(3) Tolerances in new weapons. All new cannons of a given caliber and model will not necessarily develop the same muzzle velocity. In a new tube, the mean factors affecting muzzle velocity are variations in the size of the powder chamber and the interior dimensions of the bore. If a battalion equipped with new cannons fired all of them with a common lot of ammunition a variation of 4 meters per second between the cannon developing the greatest muzzle velocity and the cannon developing the lowest muzzle velocity would not be unusual. Calibration of all cannons allows the firing unit to compensate for small variations in the manufacture of cannon tubes and the resulting variation in developed muzzle velocity. The MVV caused by inconsistencies in tube manufacture remains constant and is valid for the life of the tube.
(4) Tube wear. Continued firing of a cannon wears away portions of the bore by the actions of hot gases and chemicals and movement of the projectile within the tube. These erosive actions are more pronounced when higher charges are fired. The greater the tube wear, the more the muzzle velocity decreases. Normal wear can be minimized by careful selection of the charge and by proper cleaning of both the tube and the ammunition.
(5) Nonuniform ramming. Weak ramming decreases the volume of the chamber and thereby theoretically increases the pressure imparted to the projectile. This occurs because the pressure of a gas varies inversely with volume. Therefore, only a partial gain in muzzle velocity might be achieved. Of greater note is the improper seating of the projectile within the tube. Improper seating can allow some of the expanding gases to escape around the rotating band of the projectile and thus result in decreased muzzle velocity. The combined effects of a smaller chamber and escaping gases are difficult to predict. Weak, nonuniform ramming results in an unnecessary and preventable increase in the size of the dispersion pattern. Hard, uniform ramming is desired for all rounds. When semifixed ammunition is fired, the principles of varying the size of the chamber and escape of gases still apply, particularly when ammunition is fired through worn tubes. When firing semifixed ammunition, rearward obturation is obtained by the expansion of the cartridge case against the walls of the powder chamber. Proper seating of the cartridge case is important in reducing the escape of gases.
(6) Rotating bands. The ideal rotating band permits proper seating of the projectile within the cannon tube. Proper seating of the projectile allows forward obturation, uniform pressure buildup, and initial resistance to projectile movement within the tube. The rotating band is also designed to provide a minimum drag effect on the projectile once the projectile overcomes the resistance to movement and starts to move. Dirt or burrs on the rotating band may cause improper seating. This increases tube wear and contributes to velocity dispersion. If excessively worn, the lands may not engage the rotating band well enough to impart the proper spin to the projectile. Insufficient spin reduces projectile stability in flight and can result in dangerously erratic round performance. When erratic rounds occur or excessive tube wear is noted, ordnance teams should be requested to determine the serviceability of the tube.
(7) Propellant and projectile temperatures. Any combustible material burns more rapidly when heated before ignition. When a propellant burns more rapidly than would be expected under standard conditions, gases are produced more rapidly and the pressure imparted to the projectile is greater. As a result, the muzzle velocity will be greater than standard and the projectile will travel farther. Table E in the tabular firing tables lists the magnitude of change in muzzle velocity resulting from a propellant temperature that is greater or less than standard. Appropriate corrections can be extracted from that table; however, such corrections are valid only if they are determined relative to the true propellant temperature. The temperature of propellant in sealed containers remains fairly uniform though not necessarily at the standard propellant temperature (70 degrees Fahrenheit [F]). Once propellant has been unpacked, its temperature more rapidly approaches the air temperature. The time and type of exposure to the weather result in temperature variations from round to round and within the firing unit. It is currently impractical to measure propellant temperature and apply corrections for each round fired by each cannon. Positive action must be taken to maintain uniform projectile and propellant temperatures. Failure to do this results in erratic firing. The effect of an extreme change in projectile or propellant temperature can invalidate even the most recent corrections determined from a registration.
(a) Ready ammunition should be kept off the ground and protected from dirt, moisture, and direct rays of the sun. At least 6 inches of airspace should be between the ammunition and protective covering on the sides, 6 inches of dunnage should be on the bottom, and the roof should be 18 inches from the top of the stack. These precautions will allow propellant and projectile temperatures to approach the air temperature at a uniform rate throughout the firing unit.
(b) Propellant should be prepared in advance so that it is never necessary to fire freshly unpacked ammunition with ammunition that has been exposed to weather during a fire mission.
(c) Ammunition should be fired in the order in which it was unpacked.
(d) Propellant temperature should be determined from ready ammunition on a periodic basis, particularly if there has been a change in the air temperature.
(8) Moisture content of propellant. Changes in the moisture content of propellant are caused by improper protection from the elements or improper handling of the propellant. These changes can affect muzzle velocity. Since the moisture content cannot be measured or corrected for, the propellant must be provided maximum protection from the elements and improper handling.
(9) Position of propellant in the chamber. In fixed and semifixed ammunition the propellant has a relatively fixed position with respect to the chamber, which is formed by the cartridge case. In separateloading ammunition, however, the rate at which the propellant burns and the developed muzzle velocity depends on how the cannoneer inserts the charge. To ensure proper ignition of the propellant he must insert the charge so that the base of the propellant bag is flush against the obturator spindle when the breech is closed. The cannoneer ensures this by placing the propellant flush against the Swiss groove (the cutaway portion in the powder chamber). The farther forward the charge is inserted, the slower the burning rate and the lower the subsequent muzzle velocity. An increase in the diameter of the propellant charge can also cause an increase in muzzle velocity. Loose tie straps or wrappings have the effect of increasing the diameter of the propellant charge. Propellant charge wrappings should always be checked for tightness, even when the full propellant charge is used.
(10) Weight of projectile. The weights of like projectiles vary within certain zones (normally termed square weight). The appropriate weight zone is stenciled on the projectile (in terms of so many squares). Some projectiles are marked with the weight in pounds. In general terms, a heavierthanstandard projectile normally experiences a decrease in muzzle velocity. This is because more of the force generated by the gases is used to overcome the initial resistance to movement. A lighterthanstandard projectile generally experiences an increase in velocity.
NOTE: Copperhead projectiles are not marked with weight in pounds. The precision manufacturing process used guarantees a weight of 137.6 pounds. 
(11) Coppering. When the projectile velocity within the bore is great, sufficient friction and heat are developed to remove the outer surface of the rotating band. Material left is a thin film of copper within the bore and is known as coppering. This phenomenon occurs in weapons that develop a high muzzle velocity and when high charges are fired. The amount of copper deposited varies with velocity. Firing higher charges increases the amount of copper deposited on the bore surfaces, whereas firing lower charges reduces the effects of coppering. Slight coppering resulting from firing a small sample of rounds at higher charges tends to increase muzzle velocity. Erratic velocity performance is a result of excessive coppering whereby the resistance of the bore to projectile movement is affected. Excessive coppering must be removed by ordnance personnel.
(12) Propellant residue. Residue from burned propellant and certain chemical agents mixed with the expanding gases are deposited on the bore surface in a manner similar to coppering. Unless the tube is properly cleaned and cared for, this residue will accelerate tube wear by causing pitting and augmenting the abrasive action of the projectile.
(13) Tube conditioning. The temperature of the tube has a direct bearing on the developed muzzle velocity. A cold tube offers a different resistance to projectile movement and is less susceptible to coppering, even at high velocities. In general, a cold tube yields more range dispersion; a hot tube, less range dispersion.
(14) Additional effects in interior ballistics. The additional effects include tube memory and tube jump.
(a) Tube memory is a physical phenomenon of the cannon tube tending to react to the firing stress in the same manner for each round, even after changing charges. It seems to "remember" the muzzle velocity of the last charge fired. For example, if a fire mission with charge 6 M4A2 is followed by a fire mission with charge 4 M4A2, the muzzle velocity of the first round of charge 4 may be unpredictably higher. The inverse is also true.
(b) Tube jump occurs as the projectile tries to maintain a straight line when exiting the muzzle. This phenomenon causes the tube to jump up when fired and may cause tube displacement.
32. Transitional Ballistics
Sometimes referred to as intermediate ballistics, this is the study of the transition from interior to exterior ballistics. Transitional ballistics is a complex science that involves a number of variables that are not fully understood; therefore, it is not an exact science. What is understood is that when the projectile leaves the muzzle, it receives a slight increase in MV from the escaping gases. Immediately after that, its MV begins to decrease because of drag.
33. Exterior Ballistics
Exterior ballistics is the science that deals with the factors affecting the motion of a projectile after it leaves the muzzle of a piece. At that instant, the total effects of interior ballistics in terms of developed muzzle velocity and spin have been imparted to the projectile. Were it not for gravity and the effects of the atmosphere, the projectile would continue indefinitely at a constant velocity along the infinite extension of the cannon tube. The discussion of exterior ballistics in the following paragraphs addresses elements of the trajectory, the trajectory in a vacuum, the trajectory within a standard atmosphere, and the factors that affect the flight of the projectile.
a. Trajectory Elements. The trajectory is the path traced by the center of gravity of the projectile from the origin to the level point. The elements of a trajectory are classified into three groupsintrinsic, initial, and terminal elements.
(1) Intrinsic elements. Elements that are characteristic of any trajectory, by definition, are intrinsic elements. (See Figure 35.)
(a) The origin is the location of the center of gravity of the projectile when it leaves the muzzle. It also denotes the center of the muzzle when the piece has been laid.
(b) The ascending branch is the part of the trajectory that is traced as the projectile rises from the origin.
(c) The summit is the highest point of the trajectory.
(d) The maximum ordinate is the difference in altitude (alt) between the origin and the summit.
(e) The descending branch is the part of the trajectory that is traced as the projectile is falling.
(f) The level point is the point on the descending branch that is the same altitude as the origin.
(g) The base of the trajectory is the straight line from the origin to the level point.
(2) Initial elements. Elements that are characteristic at the origin of the trajectory are initial elements. (See Figure 36.)
(a) When the piece is laid, the line of elevation is the axis of the tube extended.
(b) The line of departure is a line tangent to the trajectory at the instant the projectile leaves the tube.
(c) Jump is the displacement of the line of departure from the line of elevation that exists at the instant the projectile leaves the tube.
(d) The angle of site is the smaller angle in a vertical plane from the base of the trajectory to a straight line joining the origin and the target. Vertical interval is the difference in altitude between the target and the origin.
(e) The complementary angle of site is an angle that is algebraically added to the angle of site to compensate for the nonrigidity of the trajectory.
(f) Site is the algebraic sum of the angle of site and the complementary angle of site. Site is computed to compensate for situations in which the target is not at the same altitude as the battery.
(g) Complementary range is the number of meters (range correction) equivalent to the number of mils of complementary angle of site.
(h) The angle of elevation is the vertical angle between the base of the trajectory and the axis of the bore required for a projectile to achieve a prescribed range under standard conditions.
(i) The quadrant elevation is the angle at the origin measured from the base of the trajectory to the line of elevation. It is the algebraic sum of site and the angle of elevation.
(3) Terminal elements. Elements that are characteristic at the point of impact are terminal elements. (See Figure 37.)
(a) The point of impact is the point at which the projectile strikes the target area. (The point of burst is the point at which the projectile bursts in the air.)
(b) The line of fall is the line tangent to the trajectory at the level point.
(c) The angle of fall is the vertical angle at the level point between the line of fall and the base of the trajectory.
(d) The line of impact is a line tangent to the trajectory at the point of impact.
(e) The angle of impact is the acute angle at the point of impact between the line of impact and a plane tangent to the surface at the point of impact. This term should not be confused with angle of fall.
b. Trajectory in a Vacuum.
(1) If a round were fired in a vacuum, gravity would cause the projectile to return to the surface of the earth. The path or trajectory of the projectile would be simple to trace. All projectiles, regardless of size, shape, or weight, would follow paths of the same shape and would achieve the same range for a given muzzle velocity and quadrant elevation.
(2) The factors used to determine the data needed to construct a firing table for firing in a vacuum are the angle of departure, muzzle velocity, and acceleration caused by the force of gravity. The initial velocity imparted to a round has two componentshorizontal velocity and vertical velocity. The relative magnitudes of horizontal and vertical components vary with the angle of elevation. For example, if the elevation were zero, the initial velocity imparted to the round would be horizontal in nature and there would be no vertical component. If, on the other hand, the elevation were 1,600 mils (disregarding the effects of rotation of the earth), the initial velocity would be vertical and there would be no horizontal component.
(3) Gravity causes a projectile in flight to fall to the earth. Because of gravity, the height of the projectile at any instant is less than it would be if no such force were acting on it. In a vacuum, the vertical velocity would decrease from the initial velocity to zero on the ascending branch of the trajectory and increase from zero to the initial velocity on the descending branch, Zero vertical velocity would occur at the summit of the trajectory. For every vertical velocity value on the upward leg of the ascending branch there is an equal vertical velocity value downward on the descending branch. Since there would be no resistance to the forward motion of the projectile in a vacuum, the horizontal velocity component would be a constant. The acceleration caused by the force of gravity (9.81 m/s) affects only the vertical velocity.
c. Trajectory in a Standard Atmosphere.
(1) The resistance of the air to projectile movement depends on the air movement, density, and temperature. As a point of departure for computing firing tables, assumed conditions of air density and air temperature with no wind are used. The air structure is called the standard atmosphere.
(2) The most apparent difference between the trajectory in a vacuum and the trajectory in the standard atmosphere is a net reduction in the range achieved by the projectile. A comparison of the flight of the projectile in a vacuum and in the standard atmosphere is shown in Figure 38.
(3) The difference in range is due to the horizontal velocity component in the standard atmosphere no longer being a constant value. The horizontal velocity component is continually decreased by the retarding effect of the air. The vertical velocity component is also affected by air resistance. The trajectory in the standard atmosphere has the following characteristic differences from the trajectory in a vacuum:
(a) The velocity at the level point is less than the velocity at the origin.
(b) The mean horizontal velocity of the projectile beyond the summit is less than the mean velocity before the projectile reaches the summit; therefore, the projectile travels a shorter horizontal distance. Hence, the descending branch is shorter than the ascending branch. The angle of fall is greater than the angle of elevation.
(c) The spin (rotational motion) initially imparted to the projectile causes it to respond differently in the standard atmosphere because of air resistance. A trajectory in the standard atmosphere, compared to a trajectory in a vacuum, will be shorter and lower at any specific point along the trajectory for the following reasons:
 Horizontal velocity is not a constant value; it decreases with each succeeding time interval.
 Vertical velocity is affected by both gravity and the effects of the atmosphere on the projectile.
 The summit in a vacuum is midway between the origin and the level point; in the standard atmosphere, it is actually nearer the level point.
 The angle of fall in a vacuum is equal to the angle of elevation; in the standard atmosphere, it is greater.
d. Relation of Air Resistance and Projectile Efficiency to Standard Range.
(1) This paragraph concerns only those factors that establish the relationship between the standard range, elevation, and achieved range.
(a) The standard (chart) range is the range opposite a given elevation in the firing tables. It is assumed to have been measured along the surface of a sphere concentric with the earth and passing through the muzzle of a weapon. For all practical purposes, standard range is the horizontal distance from the origin of the trajectory to the level point.
(b) The achieved range is the range attained as a result of firing the cannon at a particular elevation. If actual firing conditions duplicate the ballistic properties and met conditions on which the firing tables are based, then the achieved range and the standard range will be equal.
(c) The corrected range is the range corresponding to the elevation that must be fired to reach the target.
(2) Air resistance affects the flight of the projectile both in range and in direction. The component of air resistance in the direction opposite that of the forward motion of the projectile is called drag. Because of drag, both the horizontal and vertical components of velocity are less at any given time along the trajectory than they would be if drag was zero (as it would be in a vacuum). This decrease in velocity varies directly in magnitude with drag and inversely with the mass of the projectile. Several factors considered in the computation of drag are as follows:
(a) Air density. The drag of a given projectile is proportional to the density of the air through which it passes. For example, an increase in air density by a given percentage increases drag by the same percentage. Since the air density at a specific place, time, and altitude varies widely, the standard trajectories reflected in the firing tables were computed with a fixed relationship between air density and altitude.
(b) Velocity. The faster a projectile moves, the more the air resists its motion. Examination of a set of firing tables reveals that given a constant elevation, the effect of a 1 percent change in air density (and corresponding 1 percent increase in drag) increases with an increase in charge (with the greater muzzle velocity). The drag is approximately proportional to the square of the velocity except when velocity approaches the speed of sound. At the speed of sound, drag increases more rapidly because of the increase in pressure behind the sound wave.
(c) Projectile diameter. Two projectiles of identical shape but of different size will not experience the same drag. For example, a large projectile will offer a larger area for the air to act upon; thus, its drag will be increased by this factor. The drag of projectiles of the same shape is assumed to be proportional to the square of the projectile diameter.
(d) Ballistic coefficient. The ballistic coefficient of a projectile is a measure of its relative efficiency in overcoming air resistance. An increase in the ballistic coefficient reduces the effect of drag and consequently increases range. The reverse is true for a decrease in the ballistic coefficient. The ballistic coefficient can be increased by increasing the ratio of the weight of the projectile to the square of its diameter. It can also be increased by improving the shape of the projectile.
(e) Drag coefficient. The drag coefficient combines several ballistic properties of typical projectiles. These properties include yaw (the angle between the direction of motion and the axis of the projectile) and the ratio of the velocity of the projectile to the speed of sound. Drag coefficients, which have been computed for many projectile types, simplify the work of ballisticians. When a projectile varies slightly in shape from one of the typical projectile types, the drag coefficient can be determined by computing a form factor for the projectile and multiplying the drag coefficient of a typical projectile type by the form factor.
e. Deviations From Standard Conditions. Firing tables are based on actual firings of a piece and its ammunition correlated to a set of standard conditions. Actual firing conditions, however, will never equate to standard conditions. These deviations from standard conditions, if not corrected for when computing firing data will cause the projectile to impact at a point other than the desired location. Corrections for nonstandard conditions are made to improve accuracy.
(1) Range effects. Some of the deviations from standard conditions affecting range are:
 Muzzle velocity.
 Projectile weight.
 Range wind.
 Air temperature.
 Air density.
 Rotation of the earth.
(2) Deflection effects. Some of the deviations from the standard conditions affecting deflection are:
 Drift.
 Crosswind.
 Rotation of the earth.
34. Dispersion and Probability
If a number of rounds of ammunition of the same caliber, lot, and charge are fired from the same position with identical settings used for deflection and quadrant elevation, the rounds will not all impact on a single point but will fall in a scattered pattern. In discussions of artillery fire, this phenomenon is called dispersion, and the array of bursts on the ground is called the dispersion pattern.
35. Causes of Dispersion
a. The points of impact of the projectiles will be scattered both in deflection and in range. Dispersion is caused by inherent (systemic) errors. It should never be confused with roundtoround variations caused by either human or constant errors. Human errors can be minimized through training and supervision. Corrections to compensate for the effects of constant errors can be determined from the TFT. Inherent errors are beyond control or are impractical to measure. Examples of inherent errors are as follows:
(1) Conditions in the bore. The muzzle velocity achieved by a given projectile is affected by the following:
 Minor variations in the weight of the projectile, form of the rotating band, and moisture content and temperature of the propellant grains.
 Differences in the rate of ignition of the propellant.
 Variations in the arrangement of the propellant grains.
 Differences in the rate of ignition of the propellant.
 Variations in the ramming of the projectile.
 Variations in the temperature of the bore from round to round.
For example, variations in the bourrelet and rotating band may cause inaccurate centering of the projectile, which can result in a loss in achieved range because of instability in flight.
(2) Conditions in the carriage. Deflection and elevation are affected by the following:
 Play (looseness) in the mechanisms of the carriage.
 Physical limitations of precision in setting values of deflection and quadrant elevation on the respective scales.
 Nonuniform reactions to firing stress.
(3) Conditions during flight. The flight of the projectile may be affected by the difference in air resistance created by variations in the weight, achieved muzzle velocity, and projectile. Also, the projectile may be affected by minor variations in wind, air density or air pressure, and air temperature from round to round.
b. The distribution of bursts (dispersion pattern) in a given sample of rounds is roughly elliptical (Figure 39) in relation to the line of fire.
c. A rectangle constructed around the dispersion area (excluding any erratic rounds) is called the dispersion rectangle, or 100 percent rectangle. (See Figure 310.)
36. Mean Point of Impact
For any large number of rounds fired, the average (or mean) location of impact can be determined by drawing a diagram of the pattern of bursts as they appear on the ground. A line drawn perpendicular to the line of fire can be used to divide the sample rounds into two equal groups. Therefore, half of the rounds will be over this line when considered in relation to the weapon. The other half of the rounds will be short of this line in relation to the weapon. This dividing line represents the mean range of the sample and is called the mean range line. A second line can be drawn parallel to the line of fire, again dividing the sample into two equal groups. Half of the rounds will be to the right of this line, and half will be to the left. This line represents the mean deflection of the sample and is called the mean deflection line. (See Figure 39.) The intersection of the two lines is the mean point of impact (MPI). (See Figure 310.)
37. Probable Error
Probable error is nothing more than an error that is exceeded as often as it is not exceeded. For example, in Figure 311, consider only those rounds that have impacted over the mean range line (line AB). These rounds all manifest errors in range, since they all impacted over the mean range line. Some of the rounds are more in error than others. At a point beyond the MPI, a second line can be drawn perpendicular to the line of fire to divide the "ovens" into two equal groups (line CD, Figure 311). When the distance from the MPI to line CD is used as a measure of probable error, it is obvious that half of the overs show greater magnitude of error than the other half. This distance is one probable error in range. The range probability curve expresses the following:
a. In a large number of samples, errors in excess and errors in deficiency are equally frequent (probable) as shown by the symmetry of the curve.
b. The errors are not uniformly distributed. Small errors occur more frequently than large errors as shown by the greater number of occurrences near the mean point of impact.
38. Dispersion Zones
If the dispersion rectangle is divided evenly into eight zones in range with the value for 1 probable error in range (PER) used as the unit of measure, the percentage of rounds impacting within each zone is as indicated in Figure 312. The percentage of rounds impacting within each zone has been determined through experimentation. By definition of probable error, 50 percent of all rounds will impact within 1 probable error in range or deflection of the mean point of impact (25 percent over and 25 percent short or 25 percent left and 25 percent right).
39. Range Probable Error
The values for range probable error at various ranges are given in Table G of the tabular firing tables (TFT). These values may be used as an index of the precision of the piece at a particular charge and range. The values for range probable error are listed in meters. Firing Table (FT) values have been determined on the basis of actual firing of ammunition under controlled conditions. For example, FT 155AM2 shows that the value of range probable error for charge 5 green bag (GB) at a range of 6,000 meters is 15 meters. On the basis of the 100 percent rectangle, 50 percent of the rounds will impact within 15 meters (over and short) of the mean range line, 82 percent will impact within 30 meters (over and short), 96 percent will impact within 45meters (over and short), and 100 percent will impact within 60 meters.
310. Fork
The term fork is used to express the change in elevation (in mils) needed to move the mean point of impact 4 probable errors in range. The values of fork are listed in Table F of the firing tables. For example, FT 155AM2 shows that the value of fork for a howitzer firing charge 5GB at a range of 6,000 meters is 4 mils. On the basis of the value for probable error in range (paragraph 39), adding 4 mils to the quadrant elevation would cause the MPI to move 60 meters. Fork is used in the computation of safety data (executive officer's minimum QE).
311. Deflection Probable Error
The values for probable error in deflection (PE_{D}) are listed in Table G of the firing tables. For artillery cannons, the deflection probable error is considerably smaller than the range probable error. Values for PE_{D} are listed in meters. With the same parameters as those used in paragraph 39, the deflection probable error is 4 meters. Therefore, 50 percent of the rounds will impact within 4 meters of the mean deflection line (left and right); 82 percent, within 8 meters (left and right); 96 percent, within 12 meters (left and right); and 100 percent, within 16 meters.
312. TimeToBurst Probable Error
The values of timetoburst probable error (PE_{TB}) (Figure 313) are listed in Table G of the firing tables. Each of these values is the weighted average of the precision of a time fuze timing mechanism in relation to the actual time of flight of the projectile. For example, if a 155mm howitzer fires charge 5GB at a range of 6,000 meters, the value for probable error in time to burst is 0.11 second. As in any other dispersion pattern, 50 percent of the rounds will function within 0.11 second; 82 percent, within 0.22 second; 96 percent, within 0.33 second; and 100 percent within 0.44 second of the mean fuze setting.
313. HeightOfBurst Probable Error
With the projectile fuzed to burst in the air, the heightofburst probable error (PE_{HB}) (Figure 313) is the vertical component of 1 timetoburst probable error. The heightofburst probable error reflects the combined effects of dispersion caused by variations in the functioning of the time fuze and dispersion caused by the conditions described in paragraph 35(a). The values listed (in meters) follow the same pattern of distribution as for those discussed for range dispersion. These values are listed in Table G of the firing tables.
314. RangeToBurst Probable Error
Rangetoburst probable error (PE_{RB}) (Figure 313) is the horizontal component of 1 timetoburst probable error. When this value is added to or subtracted from the expected range to burst, it will produce an interval along the line of fire that should contain 50 percent of the rounds fired. These values are listed in Table G of the firing tables.
NEWSLETTER

Join the GlobalSecurity.org mailing list 

