Damaging Effects of Weapons and Ammunition. Igor A. Balagansky

Damaging Effects of Weapons and Ammunition

the dependence of shots is either ignored or taken into account approximately. One of the simplest methods of taking into account the dependence of the shots is to calculate the probability of damaging the target using the approximate formula

(I.26) upper W Subscript n Baseline equals upper W 1 plus left-parenthesis upper W Subscript n Superscript left-parenthesis n right-parenthesis Baseline minus upper W 1 right-parenthesis StartRoot 1 minus normal mu squared EndRoot comma

where upper W 1 comma upper W Subscript n Superscript left-parenthesis n right-parenthesis – the probability of target damage at one and n independent shots, respectively; μ – correlation ratio of shots (see I.16).

The relative error of determining the probability of damaging the target with this method at nW₁ ≈ 0.5–5 does not exceed several percents.

Example

One of the hunters from the previous example takes five shots at a duck in a row. The correlation ratio between the shots is μ = 0.5. What's the equal probability of damaging the duck?

Solution

Shots are dependent. The value of nW₁ = 5 ⋅ 0.2 = 1. So, we can use the formula (I.26). Let's put the calculated values from the previous example into the formula

upper W 5 equals 0.2 plus left-parenthesis 0.672 minus 0.2 right-parenthesis StartRoot 1 minus 0.5 squared EndRoot equals 0.608 period

The dependence of the shots always leads to a reduced probability of damaging the target compared to the case of independent shots. This effect is more pronounced with more shots fired and a bigger correlation ratio μ. With a small number of shots (n = 2–4) and relatively small correlation ratio values (μ < 0.5), the correction for the dependence of shots is small and the probability of damaging the target can be calculated as in the case of independent shots. If there are a large number of shots and their correlation is significant, correction for the dependence of shots should be taken into account.

An extremely important issue in efficiency theory is to establish a rational relationship between the group error of firing (E_x0, E_y0) and individual or technical dispersion (B_d, B_s). The optimal characteristics of the individual dispersion must be commensurate with the characteristics of the aiming error (E_x0 ≅ В_d); (E_y0 ≅ В_s). Unfortunately, this is not always the case in existing weapon systems; in particular, for artillery weapon systems and small‐caliber unguided aircraft missiles, the group error of firing often far exceeds the individual dispersion.

The optimal ratio between individual and group firing errors can be achieved either by increasing the aiming accuracy or by increasing the individual dispersion of the ammunition. In the practical case of a firing situation, the effectiveness of the firing was increased by up to five times when the technical dispersion was increased to its optimum value without increasing the aiming accuracy.

Another way to increase the effectiveness of the dependent shots, which is widely used for ground artillery, is to shoot with an artificial dispersion of the shots, which is achieved by shooting one target at several aiming positions.

I.3.6 Evaluation of the Effectiveness of Firing on a Group Target

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) upper M Subscript d Baseline equals upper M left-bracket upper X Subscript d Baseline right-bracket comma

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) upper X Subscript d Baseline equals upper X 1 plus upper X 2 plus midline-horizontal-ellipsis plus upper X Subscript upper N Baseline equals sigma-summation Underscript i equals 1 Overscript upper N Endscripts upper X Subscript i Baseline period

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

upper M left-bracket upper X Subscript i Baseline right-bracket equals upper W Subscript i Baseline dot 1 plus left-parenthesis 1 minus upper W Subscript i Baseline right-parenthesis 0 equals upper W Subscript i Baseline period

By substituting (I.29), we get

upper M left-bracket upper X Subscript d Baseline right-bracket equals upper W 1 plus upper W 2 plus midline-horizontal-ellipsis plus upper W Subscript upper N Baseline equals sigma-summation Underscript i equals 1 Overscript upper N Endscripts upper W Subscript i Baseline comma

or finally

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