Читать книгу Damaging Effects of Weapons and Ammunition - Igor A. Balagansky - Страница 31

Most often the task of shooting at a group target is to damage the largest possible number of units in the group. As an indicator of the effectiveness of firing in this case, the average number of damaged units from the group is used.

(I.27)

where the random value of X_d is the number of units damaged.

We will deduce the most general formula for M_d, suitable for all cases without exception when shooting at a group target. For this purpose, let us consider a group target consisting of N units of T₁, T₂, …, T_N (Figure I.9). This target is to be fired with any ammunition (contact or remote), consisting of any number of shots – dependent or independent, under constant or variable conditions. The aim may be to target individual units or a group as a whole.

Figure I.9 Group target.

Source: From Wentzel [2].

Let's deduce the formula for the expected value of the number of damaged units. For this purpose, let us present the total number of damaged units X_d as a sum of N random values:

(I.28)

Each i‐th unit of T_i has its own random value of X_i, which we will define as follows:

 if the unit of Ti is damaged, Xi = 1;

 if unit Ti is not damaged, Xi = 0.

It is not difficult to make sure that the total number of damaged targets X_i simply equals the sum of all X_i values. According to the theorem on the summation of expected values:

(I.29)

Let's denote the probability of damaging the i‐th unit in the whole shooting as W_i . Then by defining the expected value

By substituting (I.29), we get

or finally

(I.30)

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.