Damaging Effects of Weapons and Ammunition. Igor A. Balagansky
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Название: Damaging Effects of Weapons and Ammunition

Автор: Igor A. Balagansky

Издательство: John Wiley & Sons Limited

Жанр: Химия

Серия:

isbn: 9781119779551

isbn:

СКАЧАТЬ [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 Xd 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 Ti has its own random value of Xi, 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 Xi simply equals the sum of all Xi values. According to the theorem on the summation of expected values:

      Let's denote the probability of damaging the i‐th unit in the whole shooting as Wi . 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 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

      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,

p 1 equals 0.1 semicolon p 2 equals 0.2 semicolon p 3 equals 0.2 semicolon p 4 equals 0.1 semicolon p 5 equals 0.1 period

      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:

upper W 1 equals upper W 4 equals upper W 5 equals 1 minus left-parenthesis 1 minus 0.1 right-parenthesis Superscript 5 Baseline equals 0.410 comma upper W 2 equals upper W 3 equals 1 minus left-parenthesis 1 minus 0.2 right-parenthesis Superscript 5 Baseline equals 0.672 period

      From here

upper M Subscript d Baseline equals 3 dot 0.410 plus 2 dot 0.672 equals 2.574 period

      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)upper M equals upper M left-bracket upper U right-bracket comma

      where U = Sd/St 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.