Damaging Effects of Weapons and Ammunition. Igor A. Balagansky

      i.e. the average number of damaged units within a group target is equal to the sum of probabilities of damaging all individual units.

      To use this formula, you must first calculate the probability of each target being damaged from the group by all the shots fired, and then sum up these probabilities.

      Example

      Five independent shots are fired at a group of five ducks. The whole group is fired as a single unit without observing the results or transfer of fire. Each shot aimed at a group can kill no more than one duck. The chances of killing the first, second, third, fourth, and fifth ducks in one shot are, respectively,

      It is necessary to determine the average number of killed ducks as a result of the whole shooting.

      Solution

      Find the probability of killing individual ducks for the whole shooting:

      From here

Formula (I.30) is perfectly universal and valid for any kind of group target and any way of organizing the shooting. However, the probability W_i is simply calculated only if the firing mode of each individual unit is independent of whether other units are damaged or not. This condition is broken, for example when it is possible to transfer fire from one damaged unit to another undamaged unit.

      I.3.7 Evaluation of the Effectiveness of Firing at Area Target

      As an indicator of the effectiveness of firing at an area target, an expected value of the fraction of the damaged area is often used.

      (I.31)

      where U = S_d/S_t is the ratio of the damaged area to the full target area.

      In general, if no assumptions are made about the target, damage area, and firing conditions, the task of evaluating the effectiveness of firing at an area target becomes quite difficult. However, taking into account the overall low accuracy of the information we have about the area target, we can make a number of assumptions and significantly simplify the problem. In particular, it makes no sense to enter into the calculation of the exact configuration of the target and the damage area, and it is possible to replace both areas with rectangles.

      Let's accept the following assumptions:

       the target is a rectangle with dimensions Tx, Ty, with sides parallel to the principal axes of dispersion;

       the damage zone is also a rectangle with the dimensions of Lx, Ly, with the sides parallel to the principal dispersion axes.

      Let's imagine the process of firing at an area target as if each time when the target T is shot, the damage area of L is reset onto the target. The position of zone L with respect to the target T is characterized by one random point O₁, the epicenter of the explosion, which may take one or another position with respect to the origin of coordinates – the aiming point O. If the origin of coordinates O coincides with the center of the target, it is said about aiming at the center of the target; if it does not coincide, it is said about a takeaway aiming point from the center of the target.

Depending on which position the O₁ point will take as a result of the shooting, the damage zone L will cover one or another part of the target area (Figure I.10). For several shots, the damaged target area depends on the concrete location of the centers of all damage zones. The total damaged area S_d does not equal the sum of the areas damaged by the individual shots, as the damage zones may overlap. Where there are overlaps, the increase in the destructive effect is usually neglected in target zones that are covered twice, three times, or more, and the damaged area S_d is taken to be the area covered by at least one damage zone. By dividing the damaged area of S_d by the target area of S_t, we obtain a portion of the damaged area.

      This random value characterizes the success of firing at an area target.

Figure I.10 The mutual position of the target area and its damaged zone.

      Source: From Wentzel [2].

Thus, we are interested in the average fraction (expected value) of the damaged area M = M[U]. The methods of calculating of this value are different depending on whether one or more shots are fired at the target.

      I.3.7.1 Fraction of Damage U with One Shot

Consider the area target T, on which a single shot is fired, with damage area L (Figure I.11). Let us condition all linear dimensions to express not in meters, but in probable deviations on corresponding axes. When it is necessary to express them in meters, we will mark it with a special symbol (m):

      (I.32)

      So, one shot is considered, in which the damage zone L