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Chapter 17 Exterior ballistics
A. Forces Affecting Trajectories
B. Range Tables
C. Practical Application of Range Tables
D. Target-practice Post-firing Analysis
                                           A. Forces Affecting Trajectories

17A1. Introduction

Exterior ballistics is by definition that part of the study of ballistics which deals with projectile motion after the projectile leaves the gun. A study of the entire subject of exterior ballistics includes theoretical aspects of this motion, the forces affecting it, and also the practical problems of laying a gun so that a projectile fired from it will hit the target. This text will deal primarily with the practical aspects of exterior ballistics. In a study of the subject, however, it is desirable to examine the characteristics of trajectories in general, as well as characteristics of trajectories of projectiles fired under specific firing conditions.

17A2. Definitions

The first step in the study of exterior ballistics is an understanding of the characteristics of a trajectory and of the applicable definitions and symbols.

Symbol                  Definition

Horizontal plane.  The imaginary plane tangent to the earth’s surface at a point instantaneously occupied by own ship. Any plane parallel to this one is also horizontal. Angles or distance measured in a horizontal plane are called horizontal angles or distances and are usually identified by the letter h.

Vertical plane.  A plane perpendicular to the horizontal. Angles and distances measured in a vertical plane are called vertical angles or distances and are usually identified by the letter v.

Trajectory.  The path described by a projectile in flight.

R                           Range. The distance from the origin of the trajectory to the target along the LOS.

Rh                         Horizontal Range. The projection of the range in a horizontal plane.

Symbol                  Definition

LOS                      Line of sight. The straight line joining the sight and the target.

E                          Position angle or target elevation. The vertical angle between the LOS and the horizontal plane.

Eg                        Gun elevation (angle of elevation). The vertical angle between the horizontal plane and the axis of the bore.

Vs                        Sight angle. The vertical component of the angle between the line of sight and the axis of the bore.

                            Angle of fall. The vertical angle between the horizontal and the tangent to the trajectory at the point of fall.

                            Maximum ordinate. The vertical distance from the horizontal plane to the highest point of the trajectory.

Tf                        Time of flight. The total time which elapses during the flight of the projectile from the gun to the point of fall (or burst).

Df                        Drift.  The lateral deviation of the trajectory from the vertical plane through the axis of the bore caused by the rotation of the projectile.

Ds                        Sight deflection. The lateral component of the angle between the LOS and the axis of the bore.

I. V.                      Initial velocity of the projectile with respect to the gun muzzle at the instant the projectile leaves the gun (normally given in foot-seconds

17A3. Trajectory analysis

Theoretically, the curved path taken by a projectile fired from a gun, when projected in a vertical plane, is caused by the force of gravity acting on the projectile after it leaves the gun. The path, projected on a horizontal plane, is also a curve, due to gyroscopic properties imparted to the projectile by rotation caused by the rifling in the gun. The effect of this gyroscopic force is called drift, and it is always to the right, because naval guns are rifled with right-hand twist. There are, however, other factors in addition to gravity and gyroscopic precessional force which modify the curvature of a trajectory.

If the projectile traveled in a vacuum, the only external force acting upon it would be that of gravity. Under such circumstances, the result would be vertical deceleration in the ascending branch until the summit was reached, followed by an exactly equal acceleration in the descending branch to the point of fall. This would result in:

1. A trajectory in the form of a parabola with a vertical axis.

2. A maximum ordinate located halfway between the gun and the point of fall.

3. An angle of fall equal to the angle of elevation.

4. A striking velocity equal to the initial velocity.

5. A maximum range obtained from an angle of elevation of 45 degrees.

When the projectile is fired in air, the theoretical trajectory in a vacuum is modified by air resistance, with the following results:

1. The trajectory is not a true parabola.

2. The angle of fall is greater than the angle of elevation.

3. The striking velocity is less than the initial velocity.

4. The maximum range of the projectile is obtained not precisely at, but still close to, an angle of elevation of 45 degrees.

For example, the maximum range of the 55.18-pound AA common projectile for the 5”/38 caliber gun is obtained with an angle of elevation of 44 degrees 35 minutes.

The following factors must be taken into account in ascertaining the difference in the trajectory characteristics of a projectile fired in air from the characteristics of one fired in a vacuum.

1. The density of the atmosphere. The air offers resistance to the projectile which materially alters the characteristics of the trajectory.

Since atmospheric density differs from hour to hour with changes in temperature and barometric pressure, and also according to altitude, the resistance varies not only from time to time, but in the different altitude zones through which the projectile travels.

2. Characteristics of the projectile. The characteristics of a projectile which influence its retardation in passing through air of a given density are:

a. Weight.

b. Cross-sectional area, which is proportional to the square of the projectile’s diameter.

c. Shape. A projectile which has a streamlined front end encounters less resistance than one which has a short, blunt nose. The shape of the base also affects resistance.

3. The initial velocity. With air density and projectile design constant, initial velocity affects the characteristics of the trajectory, because the amount of resistance offered by air varies with velocity.

17A4. Elements of a trajectory

Since the trajectory is curved, because of the factors mentioned, it is obvious that the bore of the gun cannot be pointed directly at the target (except at point-blank ranges), but that it must be directed at some other position in space, as shown in
figure 17A1.
The projectile must describe a definite trajectory in order to hit the target. To obtain this trajectory, the gun must be laid accurately with respect to some reference plane or line. The horizontal plane provides a convenient reference plane from which the vertical angle may be measured. If the gun is mounted on a platform ashore it can be laid from the horizontal plane to the desired elevation by the use of a gunner’s quadrant, an instrument which measures angles of elevation with respect to the true horizontal. At sea such a method is impracticable because of ship’s motion. The most convenient reference is a line joining the positions of gun and target. Since it is normally the line along which the target is sighted from the gun, it is called the line of sight. Laying the gun with respect to this reference line is shown in figure 17A2. It is apparent from the figure that the gun elevation Eg remains constant as the ship rolls, as long as the LOS is held on the target.
The vertical angle through which the gun is elevated with respect to the horizontal is defined as the angle of elevation. If the trunnion axis is horizontal, the vertical angle between the axis of the bore and the line of sight is the same as the sight angle (Vs).

The lateral angular offset of the bore axis from the LOS due to the drift of the projectile is called drift (Df). The total lateral angle due to all factors causing deflection is called sight deflection (Ds).

17A5. Drift

The drift of an elongated, rotating projectile may be considered to result from three causes:

1. Gyroscopic action.

2. The Magnus effect.

3. The cushioning effect.

It is reasonably certain that the last two causes have only a minor effect as compared with the first.

The rifling of the gun barrel causes the projectile to rotate in flight with sufficient rapidity to behave as a gyroscope. This serves to stabilize the flight of the projectile, but it makes the projectile subject to gyroscopic precession. Because of the curvature of the trajectory, air pressure on the underside of the nose of the projectile causes a precession to the right. This shift of the projectile axis to the right increases the air pressure on the left-hand side of the nose, which causes the projectile to precess downward. This train of events continues, causing the axis of the projectile to oscillate about the tangent to the trajectory.

Since the greatest pressure is on the underside of the nose, the over-all precession is to the right.

The initial tendency of a projectile to maintain the original direction of its axis as it falls away from the axis of the bore causes the air stream to strike the projectile’s lower side. With right-hand spin, the air adhering to the right-hand side of the projectile then opposes the air stream created by the projectile’s flight, and the result is an increase of pressure on the right-hand side. At the same time, there is a rarefaction, and the projectile tends to move to the left, to the side of lesser pressure.

This effect is known as the Magnus effect, and is the same phenomenon which causes a golf ball to hook or slice. The Magnus effect can be important on the descending end of the trajectory at extreme elevations.

Since the air tends to pile up on the underside of projectile in motion, it forms a cushion. The projectile tends to roll on this cushion because of the friction imposed by it. This movement is to the right in a projectile with right-hand spin, opposing the Magnus effect but adding to the gyroscopic effect.

17A6. The principle of rigidity of the trajectory

When a gun is elevated with respect to its LOS, it is not necessarily elevated with respect to the horizontal, because the gun and target may not lie in the same horizontal plane. Assuming that the water’s surface is the horizontal, the gun is placed somewhat above this surface, the exact distance depending on its location aboard ship. The LOS is normally directed at the waterline when the target is a ship. In this case, the LOS dips below the horizontal by the amount of the position angle E.

In the surface fire this angle E is small and may be neglected because of the principle of rigidity of the trajectory. This principle states that for small angles of position, a gun which obtains a certain inclined range along a line of position, when elevated the required angle above that line, will have the same range along the horizontal when elevated the same angle above the horizontal.
In surface fire it is unusual to encounter a position angle of more than about 35 minutes, and a value as great as this occurs only at extremely short range, in which case the gun elevation angle is also small. For example, with a 5"/38 caliber gun mounted 50 feet above the waterline and firing at the waterline of a target at a slant range along the LOS of 1,600 yards, the position angle is about -35 minutes and the gun elevation angle is + 43.6 minutes. For these values of E and Eg, the ratio of the inclined range R to the horizontal range Rh is about 1.0001, showing that the difference in range is negligible. Therefore, even though the angles of elevation listed in the range tables are values of Eg (vertical angles measured with respect to the horizontal), these angles, when set on the gun sights as Vs (measured with respect to the line of position), result in the required ranges along the LOS, with insignificant error. This principle is applied only in surface fire, where the angles of position are normally small.

figure 17A3.

The principle of trajectory rigidity does not hold in AA fire, because normally the position angle is considerably larger.