1. Understand the effect of relative target motion on weapon aiming position.

2. Understand the effects of the following upon the trajectory of a weapon gravity, drag wind, drift, and Coriolis Effect.

3. Understand the iterative nature of a prediction algorithm.

4. Know the input/output requirements of a ballistic procedure.

5. Understand the iterative nature of a prediction algorithm.

6. Know the relationships between the ballistic and relative motion procedures of a prediction algorithm.

7. Understand the significance of weapon time-of-flight as it applies to predicting future target position.


In the previous chapter it was shown that a weapon system measured target coordinates and stabilized that measured data to an inertial reference frame suitable for computation. Target information, however, comprises only part of the input data necessary for calculation. An accurate description of the flight path of the weapon is also necessary if the weapon is to be delivered accurately to the target. Weapons, in general, do not follow straight paths to the point of target intercept. Physical phenomena acting on the weapon during its time of flight cause it to miss its target if the weapon were aimed directly at the target. The fire control problem can then be divided into two major categories:

(1) The effect of relative target motion during the time of weapon flight.

(2) The physical phenomena collectively called ballistics, which produce a curved weapon trajectory.

These are illustrated in figure 20-1.


The effect of relative motion results in the requirement to aim a weapon at some point in space that is displaced from the present target position. The amount of displacement or lead depends on the relative velocity and on the time of flight (TOF) of the weapon from the launch point to the target. The point of collision between the weapon and the target is the future target position. A basic assumption that is made when computing the effect of relative motion is that the target will not accelerate during the time of flight of the weapon. Thus, by following a straight-line path, the target simplifies our problem of prediction and thus lowers the target's chances of survival. It might be presumed that a target would avoid this path and use diversionary tactics such as zig-zags, curved paths, etc. Certain factors oppose these choices of target paths. The first is that the shortest distance between two points is a straight line; therefore, a target following a straight-line course will be exposed to danger for a shorter period of time. The target being in danger for a shorter period of time may be more important than the increased danger for that time. Also, most targets tend to follow a straight-line path. It must be noted that an attacking vehicle must press home its attack. If our presence and the fact that we are prepared to launch a counter weapon causes an attacker to make such radical maneuvers that would cause our weapon to miss, then there is a high probability that his weapon will miss also. Therefore, even if evasive maneuvers are taken by a target, these maneuvers will average over time to a straight-line course or one with a slight curve. Secondly, it should be realized that even if the target accelerates and/or turns, the shorter the TOF, the less the error. With the above assumptions in mind, the relative-motion calculation is reduced to a simple linear function of time:

P2 = P1 + VT (20-1)


P2 is the future position of the target

P1 is the present position of the target as measured by the sensor subsystem

V is the total relative target velocity

T is the time of flight of the weapon.

As figure 20-2 illustrates, this relationship in a stabilized reference frame actually must be expressed in each dimension:

X2 = X1 + x T

t (20-2)

Y2 = Y1 + y T

t (20-3)

Z2 = Z1 + z T

t (20-4)

The future target position derived from the above calculations generally comprises the most significant component of the lead angle. The final aiming angles for the weapon are the angles of azimuth and elevation, which include the target motion effect and the corrections as a result of trajectory curvature. The calculation of the relative-motions effect therefore solves only part of the weapon control problem, which answers the following question:

For a known weapon time of flight, what is the position of the target at the end of the time of flight?

Only when a knowledge of target position is gained can the effects of weapon-path curvature by computed.


Exterior ballistics is a generic term used to describe a number of natural phenomena that tend to cause a weapon in flight to change direction or velocity or both. These phenomena are gravity, drag, wind, drift (when considering spin-stabilized weapons) and Coriolis effect.

The computation of relative-motion effect produces the actual point of collision between the weapon and the target. The computation of the effects of ballistics produces aiming corrections as a result of the curvature of the weapon's flight path. These corrections don't change the final collision point, but cause the trajectory of the weapon to intersect the flight path of the target. The ultimate aim of ballistic computation, then, is to produce angular corrections to the azimuth and elevation values derived from the relative-motion results. Inherent in these calculations is the computation of time-of-flight of the weapon, which in turn is used to facilitate the relative-motion computation.

Ideally, it is desired to produce a procedure that totally describes the complex trajectory of a weapon. When provided the coordinates of the future target position as input data, this procedure would produce angles of launch and weapon time-of-flight as output. Figure 20-3 illustrates the conceptual goal of a ballistic solution. It must be emphasized here that the central problem of weapon delivery is the solution of the intersection of two curves, one described by the motion of the target and the other described by the motion of the weapon.

20.3.1 Gravity.

The force of gravity always tends to accelerate an object downward toward the center of the earth. The effects of this force on the trajectory of a weapon are well known and if considered alone result in very simple solution to the trajectory problem. Figure 20-4 is an illustration of the trajectory of a weapon in a vacuum and the effect of gravity upon the weapon. In a vacuum on a "flat" nonrotating earth, the horizontal component of the initial velocity of the weapon (V0 cos ) remain unchanged. However, the vertical component (V0 sin ) is continuously changed as a result of the acceleration due to gravity. This changing vertical velocity results in a parabolic trajectory. A ballistic procedure that will provide an approximation to the desired output can be developed from the equations that describe this trajectory. A two-dimensional example will serve to illustrate this procedure.

The differential equations of motion are:

d2x = 0

dt2 (20-5)

d2z = d = = g

dt2 dt (20-6)

subject to the initial conditions,

x(t = 0) = 0

(t = 0) = v0 cos 0

(t = 0) = v0 sin 0

z(t = 0) = 0

These are solved by separating variables, and integrating

and evaluating constants of integration (through use of initial conditions). Thus, from equation (20-5):

dx = = Constant = vo cos


dx = vocos dt = x = (vo cos ) t + C1

Evaluate constant of integration, C1:

x(0) = 0 + C1 hence, C1 = 0 and x = tvo cos or

t = x (20-7)


Solve equation (20-6) for z:

d = -gdt hence, = -gt + C2

Evaluate constant of integration, C2:

(0) = 0 + C2 = vosin . . C2 = vosin

= dz = -gt + vosin


Solve for z:

dz = (-gt + vosin )dt

z = gt2/2 + t(vosin ) + C3

Evaluate constant of integration, C3:

z(0) = 0 + 0 + C3 = 0 hence, C3 = 0

z = -gt2 + (vosin )t



= arc sin z + .5gT2

voT (20-8)

The procedure for the solution of equations (20-7) and (20-8) is one of successive approximations. The concept is to compute a first approximation to T by using equation (20-7) and = ARCTAN (Z/X). Next compute the effect of gravity on 0 using equation (20-8). Use the results of equation (20-8) to compute a new value for T. Continue this alternate substitution of variables until the values of T and converge to a final value accurate enough for use. This procedure is common to all fire control systems and must be well understood. Sufficient accuracy can be determined by testing the absolute difference between Tn and Tn-1 as it approaches zero, where n is the number of iterations. Figure 20-5 is an annotated flow diagram of this procedure. The above ballistic procedure is valid regardless of the elevation of the target's future position, since equations (20-7) and (20-8) describe the trajectory that passes through the point (X,Z), which defines this position. This validity (in a vacuum) can be shown by comparing the trajectories of a surface target (Z = 0) and one for an air target (Z > 0). (Refer to figure 20-6.)

20.3.2 Drag.

In the vacuum trajectory situation, the solution was easily obtained because of the simplifications and assumptions made in formulating the problem. When the effects of the atmosphere are considered, forces other than gravity act on the weapon, which cause it to deviate from a pure parabolic path. This deviation results specifically from the dissipation of energy as the weapon makes contact with the atmosphere. It has been found that the loss of energy of a projectile in flight is due to:

(1) Creation of air waves--The magnitude of this effect is influenced by the form (shape) of the weapon as well as by the cross-sectional area.

(2) Creation of suction and eddy currents--This phenomenon is chiefly the result of the form of the weapon and in gun projectiles specifically, the form of the after body.

(3) Formation of heat--Energy is dissipated as a result of friction between the weapon and the air mass.

The combined effect of energy loss is a deceleration of the projectile during its time-of-flight. This deceleration acts to decrease the velocity of the weapon as it travels through the atmosphere. For a given projectile form, the retardation in velocity is directly proportional to the atmospheric density and to the velocity itself. This deceleration (called parasitic drag) is given by the following relationship:

D = CAV2/2 (20-9)


C is the drag coefficient derived from empirical data.

is the atmospheric density that is a function of the altitude and the temperature of the atmosphere at any given height.

A is the cross-sectional area of the projectile.

V is the velocity of the projectile at any instant of time during the flight.

Parasitic drag always acts in a direction opposite to that of the motion of the weapon, as shown in figure 20-7. As a result, two fundamental interdependent actions take place while a weapon is in flight. First, as illustrated in equation (20-9), the force of drag at any point in the weapon's trajectory is a direct function of the velocity at that point. Because the drag force is continuously retarding the velocity of the weapon, the force of drag itself is continuously changing over the entire time-of-flight of the weapon, with drag dependent upon velocity and in turn velocity dependent upon drag. Thus, the magnitude of both the velocity and the drag are changing with each dependent upon the other. Second, as the weapon travels the curved path of the trajectory, the angle (figure 20-7) is never constant, which results in the force of drag continuously changing direction throughout the entire flight of the weapon. In a nutshell:

The force of drag constantly changes in magnitude as a function of velocity and in direction as a function of the direction of motion of the weapon.

The end result of drag is to slow the projectile over its entire trajectory, producing a range that is significantly shorter than that which is achievable in a vacuum.

A discussion of the vacuum and atmospheric trajectories is warranted and they are illustrated in figure 20-8. While the vacuum trajectory is symmetric about the mid-range value, with the angle of fall equaling the angle of departure, the atmospheric trajectory is asymmetric. The summit of the trajectory is farther than mid-range, and the angle of fall is greater than the angle of departure. Additionally, the atmospheric trajectory is dependent on projectile mass due to external forces--which also, unlike the vacuum case, causes the atmospheric trajectory's horizontal component of velocity to decrease along the flight path. This results in a computation of the necessary departure angle to effect maximum range, whereas in a vacuum the maximum range's departure angle is 45o.

In consideration of the elevation at which maximum range is achieved, there are two offsetting effects: the change in drag with projectile velocity and the atmospheric density. A projectile's velocity is greatest immediately upon leaving the gun, thus drag is greatest at this time also. For small, lightweight projectiles, deceleration is greatest at the beginning of flight; therefore, to take advantage of improvement in horizontal range while the velocity is greatest, it is necessary to shoot at elevations less than 45o. In large projectiles such as those employed on battleships it is best to employ elevations above 45o to enable the projectiles to quickly reach high altitudes where air density and thus drag are decreased. Range increases result from the approximation of vacuum trajectory conditions in the high-altitude phase of projectile flight.

20.3.3 Wind.

The effects of wind can be divided into two components: range wind and cross wind. Range wind is that component of wind that acts in the plane of the line of fire. It serves to either retard or aid the projectile velocity, thus either increasing or decreasing range.

Cross wind acts in a plane perpendicular to the line of fire and serves to deflect the projectile either to the right or left of the line of fire.

Wind acts on the projectile throughout the time of flight; therefore, the total deviation in range and azimuth is a function of the time-of-flight.

20.3.4 Drift.

Drift of a gun projectile is defined as the lateral displacement of the projectile from the original plane of fire due only to the effect of the rotation of the projectile. The principal cause of the drift lies in the gyroscopic properties of the spinning projectile. According to the laws of the gyroscope, the projectile seeks, first of all, to maintain its axis in the direction of its line of departure from the gun. The center of gravity of the projectile, however, follows the curved path of the trajectory, and the instantaneous direction of its motion, at any point, is that of the tangent to the trajectory at that point. Therefore, the projectile's tendency to maintain the original direction of its axis soon results in leaving this axis pointed slightly above the tangent to the trajectory. The force of the air resistance opposed to the flight of the projectile is then applied against its underside (figure 20-7). This force tends to push the nose of the projectile up, upending it. However, because of the gyroscopic stabilization, this force results in the nose of the projectile going to the right when viewed from above. The movement of the nose of the projectile to the right then produces air resistance forces tending to rotate the projectile clockwise when viewed from above. However, again because of the gyroscopic stabilization, this force results in the nose of the projectile rotating down, which then tends to decrease the first upending force. Thus, the projectile curves or drifts to the right as it goes toward the target. This effect can easily be shown with a gyroscope.

A more detailed analysis of gyroscopic stability would show that a projectile is statically unstable, but because of the gyroscopic effects, becomes dynamically stable. These effects actually cause the axis of the projectile to make oscillatory nutations about its flight path. Also the Magnus Effect, where a rotating body creates lift (figure 20-9) must be considered. The Magnus Effect is what causes a pitched baseball to curve.

20.3.5 Coriolis Effect.

The preceding effects have all considered the earth to be flat and non-rotating. Actually this is not the case, and centripetal acceleration and coriolis acceleration caused by the rotating earth must be considered. The centripetal acceleration of a point on the earth's surface will vary with its distance from the rotation axis. Thus, the difference in the velocities of the launcher and the target because of their different locations on the earth's surface must be considered.

The coriolis acceleration force is created when a body or projectile moves along a radius from the axis of rotation of the earth (or a component of it). This force tends to curve an object to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. An observable example would be that the air moving outward from a high becomes clockwise wind and air moving into a low becomes counterclockwise wind.


The ballistics procedure described earlier is a highly simplified illustration of a concept of trajectory calculations. The calculations involved in a practical trajectory procedure are extremely complex and must take into account factors such as drag, lift, drift, and wind as well as the force of gravity. The two basic equations, (20-7) and (20-8), must be expanded to include all of the forces acting on the weapon over the time-of-flight. Although a practical ballistic procedure is very complex in detail, the logic flow is identical to that described in figure 20-5. The output answers the second major question in the solution of the fire control problem:

For a known point in space, what will be the launch angle and time-of-flight of a weapon to that point?


Now that relative motion and ballistic procedures have been investigated separately, it must be pointed out that in reality each is dependent upon the other. In the statements of the relative-motion and ballistics problems given above, it will be noted that the solution for future target position depends on TOF, and in turn, the solution for the ballistic trajectory and TOF depends upon future target position. To illustrate the problem a bit further: for any given point in space a trajectory and time-of-flight can be computed, but the correct point in space for intercept cannot be given directly since that point depends upon time-of-flight for its calculation. In this context, time-of-flight is a generic term for the time of travel of the weapon. The prediction algorithm described below is general in nature and is directly applicable to all weapons systems. The prediction procedure incorporates both the relative-motion procedure and a ballistic procedure in a closed loop manner. Since at the outset neither the future target position nor consequently the final time-of-flight are known, the solution is an application of the principle of successive approximations. Since the measured present target position is the first-known point in space, this position can be employed to produce a first approximation to the launch angle and time-of-flight. (Trajectory #1 in figure 20-11).

If the weapon were fired based upon this first approximation, it would obviously pass well behind a moving target. Now compute just how far behind the target the projectile would pass. In other words, where is the target at the end of the time-of-flight computed to the present position? Compute the new position and a new point. If the weapon were fired at this new point, it would pass well ahead of the target. Using the time of trajectory #2, compute a new target position, a third trajectory and time (trajectory #3). Continue computing alternate target positions and times until the difference between the two most recently computed times is very small, say 10-5 sec. When this occurs, the point in space used to compute the time-of-flight will be the same point that results when time-of-flight is applied to the relative motion solution and the fire control problem is solved. The above scheme as outlined can be applied to either an analog computer with electro-mechanical feedback loops or to a digital computer that employs numerical techniques and software. Both implementations exist in the Navy today.


This chapter has described general procedures for the solution of a weapon control problem. The overall prediction procedure was divided into sub-procedures of relative motion and weapon ballistics. As input, the relative-motion procedure receives present target position, relative target velocity, and weapon time-of-flight and yields as an output successive approximations to future target position. A ballistics procedure receives as input the approximations to future target position, and its output is successive approximations to the launch angle and time-of-flight. The overall iterative process is terminated when Tn - Tn-1 --> 0 (figure 20-5) which indicates that the refinement of future target position is becoming precise. Figure 20-12 provides the summary of the weapon control problem in flow chart form.


Bliss, Gilbert Ames. Mathematics for Exterior Ballistics. New York: John Wiley and Sons, 1944.

Herrmann, Ernest E. Exterior Ballistics 1935. Annapolis, Md.: U.S. Naval Institute, 1935.

Schmidt, Edward M. Notes on Exterior Ballistics. Ballistic Research Laboratories, 1982.

Wrigley, Walter, and John Hovarka. Fire Control Principles. New York: McGraw-Hill, 1959.