Scientific work in physics
on the topic:
ballistic movement tel

Completed by 10th grade students

Voznesensky Dmitry

Gavrilov Artyom

Theoretical part

The history of the emergence of ballistic motion

- In numerous wars throughout the history of mankind, the warring parties, proving their superiority, first used stones, spears, and arrows, and then cannonballs, bullets, shells, and bombs.

- The success of the battle was largely determined by the accuracy of hitting the target.

- At the same time, an accurate throw of a stone, hitting the enemy with a flying spear or arrow was visually recorded by the warrior. This allowed, with appropriate training, to repeat their success in the next battle.

- Significantly increased with the development of technology, the speed and range of projectiles and bullets made remote battles possible. However, the skill of a warrior, the resolving power of his eye, was not enough to accurately hit the target of an artillery duel first.

- The desire to win stimulated the emergence of ballistics (from the Greek word ballo - I throw).

Ballistics as a science

Ballistics is the science of the movement of projectiles, mines, bullets, unguided rockets during firing (launch). Main sections of ballistics: internal ballistics and external ballistics. The study of real processes occurring during the combustion of gunpowder, the movement of shells, rockets (or their models), etc., is the subject of the ballistics experiment. External ballistics studies the movement of projectiles, mines, bullets, unguided rockets, etc. after the termination of their force interaction with the weapon barrel (launcher), as well as factors affecting this movement. The main sections of external ballistics are: the study of forces and moments acting on a projectile in flight; study of the movement of the center of mass of the projectile to calculate the elements of the trajectory, as well as the movement of the projectile relates. The center of mass in order to determine its stability and dispersion characteristics. Sections of external ballistics are also the theory of corrections, the development of methods for obtaining data for compiling firing tables and external ballistic design. The movement of projectiles in special cases is studied by special sections of external ballistics, aviation ballistics, underwater ballistics, etc.

Basic terms of ballistics

- External ballistics

- Internal ballistics

- Ballistic weapon flexibility

- Ballistic missile

- ballistic track

- Ballistic firing conditions

- Ballistic performance

- ballistic calculator

- ballistic descent

- ballistic similarity

- Ballistic coefficient

- ballistic camera

Law gravity

- Ballistic movement - movement due to gravity, in which the body moves, taking into account the forces of resistance, with acceleration. Isaac Newton studied the laws of motion.

Isaac Newton

The discovery of the law by I. Newton

In his late days, Isaac Newton told how it happened: he was walking in the apple orchard on his parents' estate and suddenly saw the moon in the daytime sky. And right before his eyes, an apple broke off from the branch and fell to the ground. Since Newton was working on the laws of motion at the same time ( cm. Newton's laws of mechanics), he already knew that the apple fell under the influence of the Earth's gravitational field. He also knew that the Moon does not just hang in the sky, but rotates in an orbit around the Earth, and, therefore, some kind of force acts on it, which keeps it from breaking out of orbit and flying away in a straight line, into open space. Then it occurred to him that perhaps it is the same force that makes both the apple fall to the earth and the moon to remain in orbit around the earth.

From the law

The results of Newtonian calculations are now called law of gravity Newton. According to this law, between any pair of bodies in the universe, there is a force of mutual attraction. Like all physical laws, it takes the form of a mathematical equation. If a M and m are the masses of two bodies, and D- the distance between them, then the force F mutual gravitational attraction between them is equal to:

- F =GMm/D2

- where G- gravitational constant determined experimentally. In SI units, its value is approximately 6.67 × 10–11.

Henry Cavendish

G. Cavendish's experience

Establishment Newton law of gravity appeared major event in history physics. Its value is determined primarily by the universality of the gravitational interaction. One of the central sections of astronomy, celestial mechanics, is based on the law of universal gravitation. We feel the force of attraction to the Earth, but the attraction of small bodies to each other is imperceptible. It was required to experimentally prove the validity of the law of universal gravitation for ordinary bodies as well. This is exactly what G. Cavendish did, simultaneously determining the average density of the Earth.

An experience:

Practical part

Application of ballistics in practice

With an increase in the angle of departure of the projectile, at the same initial speed, the flight range decreases, and the height increases.

Another case:

- with an increase in the initial velocity of the projectile, with the same angle of departure, the range and height of the projectile increase

Conclusion:

- With an increase in the angle of departure of the projectile, at the same initial velocity, the flight range decreases, and the height increases, and with an increase in the initial velocity of the projectile, at the same angle of departure, the range and altitude of the projectile increase

Trajectory ballistic missile

guided projectile trajectory

Coordinates that determine the position of the rocket in space

Weightlessness

- Weightlessness- the state observed by us, when the force of interaction of the body with the support ( body weight) arising from gravitational attraction, the action of other mass forces, in particular the force of inertia arising from the accelerated movement of the body, is absent

Overload

- Overload - an increase in body weight caused by the accelerated movement of a support or suspension

- Submarine ballistic missiles(SLBM) - ballistic missiles placed on submarines .

RBPL USSR\Russia

RBPL USA

RS-18, intercontinental ballistic missile

- The RS-18 missile is one of the most advanced intercontinental ballistic missiles in Russia. Its creation began in 1967 at the design bureau of MPO Mashinostroeniya, located in Reutov, Moscow Region.

- Adopted on December 17, 1980. Under this missile, a silo launcher with increased security was created, as well as a new set of means to overcome anti-missile defense. In January 1981, the first regiments with UR-100N UTTKh took up combat duty. In total, 360 RS-18 silo launchers were put on combat duty.

Caliber- bore diameter firearms, as well as the diameter of the projectile (bullet), this is one of the main quantities that determine the power of a firearm.

The caliber is determined for smooth-bore weapons by the inner diameter of the barrel, for rifled weapons - by the distance between opposite fields of rifling, for shells (bullets) - by the largest cross section. guns with conical barrel characterized by input and output calibers.

It is customary to measure the caliber of a hunting rifle not in millimeters, but by the number of spherical bullets that can be cast for a given gun from one English pound of lead, which is equal to 456 grams. Therefore, the smaller the digital designation of the caliber of the gun, the larger its caliber in the millimeter system.

Based on the definition of what is the caliber of a hunting smoothbore gun, i.e. that the nominal caliber is the number of round (ball) bullets cast from one pound (in English units of weight) of pure lead, exactly corresponding to the hole of the receiver tube, then normal weight shot shell in caliber is determined from the formula: C \u003d 454 / K (g), where C is the weight of the projectile in grams, 454 (more precisely - 453.6 g) is the weight equivalent of one English pound of pure lead in grams and K is the caliber of the gun in face value (10, 12, 16, 20, etc.).

From the above formula, the normal weight of the projectile along the diameter of the bore for 24 caliber will be: C \u003d 454/24 \u003d 18.9 (g), or rounded 19 g. Deviations of the weight of the projectile, determined by the formula, by +1.0 g. Considering However, that guns are made significantly lighter than required by the weight of a normal caliber projectile, it is necessary to check the weight of the projectile by the weight of the gun as a whole. It has been established from practice that at average initial projectile velocities from 350 to 375 m / s, the recoil will be tolerable if the weight of the projectile is within: for 12 gauge - from 1/100 to 1/94 of the total weight of the gun, for 16 gauge - 1/100, for 20 gauge - 1/112, for 24 gauge - 1/122, for 28 gauge - 1/136 and for 32 gauge - 1/148 of the total weight of the gun. Thus, with a 2.5 kg gun weighing 2.5 kg, the weight of the projectile will be 20.5 g. From this it can be seen that the weight of this gun corresponds to its caliber. In the production of domestic guns, it most often turns out that the weight of the gun significantly exceeds what should be according to its caliber, and the weight of the projectile, determined by the weight of the gun, will be significantly greater than that which was determined by the caliber of a round bullet. In this case, the normal weight of the projectile, obtained from the caliber of the gun, and not from its weight, should be used. If the weight of the projectile, determined by the weight of the gun, is less than that determined by the caliber, then in this case one should stop at the projectile found from the weight of the gun. In other words, in all cases, take the weight of the projectile, which will be less.

In conclusion, it should be noted that, having made the indicated calculation and verification for a given gun, they stop at the resulting weight of the projectile for the entire time of its existence with a given hunter. All the desired changes in gun action are achieved only by changing the weight of gunpowder and the way the cartridges are loaded.

rifled caliber small arms

The caliber of rifled small arms is indicated in the USA, Great Britain and a number of other countries in fractions of an inch (.308 Winchester; in the USA - in hundredths (0.45 inches), in the UK - in thousandths (0.450 inches). When writing, zero and a comma are replaced by point, and "cal." is used instead of "inch" or omitted altogether (.45 cal.; .450 cal.) In colloquial speech, they say: "forty-five caliber", "four hundred and fifty caliber."

In other countries, it is measured in millimeters - 9 × 18 (the first number is the caliber, the second is the length of the sleeve in millimeters). Here it must be borne in mind that the length of the sleeve is not a characteristic of the caliber, but a characteristic of the cartridge. With the same caliber, cartridges can be of different lengths. It should also be borne in mind that such a "digital" recording is used mainly for army cartridges in the West. For civilian cartridges, the name of the company or model of weapon is usually added to the caliber, for example, the forty-fifth Colt, thirty-eighth Magnum. There are also more complex designations, for example, nine millimeters Browning is short, which is also the three hundred and eightieth car. The above description is due to the fact that almost every arms company has its own patented cartridges of different characteristics. In Russia (formerly in the USSR), the nomenclature of cartridges is unified, therefore it is widely used: 9 mm, 7.62 mm, 5.45 mm, 5.6 mm.

In Russia until 1917 and in a number of other countries, the caliber was measured in lines. One line = 0.1 inch = 2.54 mm. In modern vocabulary, the name "three-line" has taken root, which literally means "a rifle of the Mosin system with a caliber of three lines."

In some countries, the caliber is the distance between the rifling fields (the smallest bore diameter), in others, the distance between the rifling bottoms (the largest diameter). As a result, with the same caliber designations, the diameters of the bullet and the bores are different. Examples are 9x18 Makarov and 9x19 Parabellum.

Makarov has 9 mm - the distance between the fields, the bullet diameter is 9.25 mm.

In Parabellum, the distance between the bottoms is 9 mm, respectively, the diameter of the bullet is 9 mm, and the distance between the fields is 8.8 mm.

Agreed buckshot

Calculation of the diameter of the agreed buckshot is calculated according to the following formula:

Buckshot diameter = n * bore diameter at the muzzle.

n is a constant depending on the number of buckshot in the layer.

If buckshot 3 - n = 0.46;

With 7 buckshots in the layer, the formula takes the form:

Buckshot diameter = diameter of the bore at the muzzle / 3.

N = (21*P) / R3, where:

N - number of pellets

P is the weight of the projectile in grams

R - shot radius in mm

The universal formula for calculating the diameter of the bore:

3–(76500/K), where:

K - caliber expressed in round bullets.

Formulas that may be needed when choosing a gun

1. Balance indicator.

By the balance of a gun, it is customary to mean the location of its center of gravity relative to the breech cut of the barrels, when the gun is assembled and the barrels are closed. A well-balanced gun has a center of gravity located 40-45 mm from the breech, large-scale - 65, 75 mm.

The formula itself: Pb \u003d Vr / Sun, where:

Vp - the total mass of the gun.

Sun is the mass of trunks without forearm.

The balance indicator should be in the limit:

from 2 to 2.3 - for double-barreled smoothbore hunting rifles

from 1.8 to 1.96 - for three-barreled combined hunting rifles

from 1.75 to 1.8 - for double-barreled rifled hunting fittings, rifles and carbines

2. Planting coefficient

The agility of a gun is called its agility, or ease of handling. It depends on the correct distribution of the mass of the gun over the main nodes (barrel with forearm and receiver with a butt), and in the nodes themselves from the distribution of mass closer to the center of gravity of the entire gun, and not to its ends.

Kp = Vk.p. / (Sun+Sun), where:

Vk.p. - mass of the receiver with a butt

Sun - weight of trunks

Vts - the mass of the forearm.

Guns of excellent quality have Kp equal to 1, guns with light barrels have more than 1, and guns with heavy barrels have Kp less than 1.

When buying a gun, it should be borne in mind that its mass should be a certain part of the mass of the shooter:

up to 1/21 from 50-55 kg;

up to 1/22 from 60-65 kg;

up to 1/23 from 70-75 kg;

up to 1/24 from 80-85 kg;

up to 1/25 from 90-95 kg;

up to 1/26 from 100 kg and above

As the mass of the gun increases, the shooter will usually get tired.

Formulas that may be required when sighting a gun

1. Projectile ratio.

A) from the weight of the gun Projectile weight \u003d gun weight / projectile coefficient

The projectile coefficient for 12 gauge is in the range from 94 to 100

For example, for a gun weighing 3.4 kg, the minimum weight of the projectile will be 34 grams (3400/100), the maximum - 36.2 (3400/94) grams.

B) the weight of the projectile by caliber. As you know, the caliber of a smoothbore weapon is the number of round bullets that can be made from 1 pound of lead. Thus, the weight of the projectile will be equal to the result of dividing the mass of the pound by the caliber. At the same time - 1 English pound = 453.592 g, 1 Trinity pound = 373.241 g, 1 French pound = 489.5 g, one Russian pound - 409.512 g. In principle, the standard was the English pound, but I give all types, since the numbers are interesting when calculating. At the same time, the arithmetic average of the projectile weight for all types of pounds for 12 gauge is 35.95 g.

2. Charging ratio.

The weight of the smokeless powder charge is determined by the formula

P \u003d D * B, where:

P is the charge of gunpowder in

D - Shot shell in g

B - Ballistic coefficient component for winter - 0.056; for summer - 0.054

Charge weight = projectile weight / charge factor

The average charge factor for 12 gauge is 16 for smokeless powder; for smoky - 5.5.

A strong primer can give an increase in pressure P up to 100 kgf / cm2 (up to 9810x104 Pa) or more.

An increase in the charge of smokeless powder by 0.05 g leads to an increase in pressure P to 15-17 kgf / cm2 (up to 147.2x104 - 166.8x104 Pa)

With an increase in the mass of the projectile by 1 g, it leads to an increase in pressure P to 5.5-15 kgf/cm2.

Smoke powder burns at a temperature of 2200-2300 degrees Celsius, smokeless - 2400 degrees.

When burning 1 kg of smoke powder, 300 liters of gaseous products are formed, 1 kg of smokeless powder - 900 liters.

Heating a gas for every 273 degrees Celsius increases its volume and elasticity by 100%.

With an increase in the length of the barrel for every 100 mm, the increase in the initial velocity of the projectile is on average 7-8 m / s, the same increase in speed is achieved by adding 0.05 g of smokeless powder.

Powder gases act on the projectile after leaving the barrel at a distance of 25 calibers from the muzzle, and give an increase in muzzle velocity by an average of 2.5%

With an increase in the mass of the projectile by 1 g, the initial velocity decreases by 3.3 m/s.

For shooting rifled weapons: Rifle combat is checked with 3, 4, 5 or 10 rounds. After a predetermined number of shots, the middle point of impact and its deviation from the aiming point vertically and horizontally are determined. Then determine the diameter of the circle containing all the bullet holes or one less if it gave a clear separation to the side. The deviations of the midpoint of the bullets hit vertically and horizontally from the aiming point will show how much you need to move the front sight or rear sight in height or in the lateral direction.

In addition to the magnitude of the deviations of the midpoint of impact from the aiming point, you also need to know the length of the sighting line of a given gun and the firing distance.

The value x of the front sight or rear sight movement is determined by the formula:

X \u003d (Pl * Ov [or Og]) / D, where:

D - firing distance, mm

Pl - aiming line length, mm

Ov (or Og) - deviations of the midpoint of impact from the aiming point, respectively, vertically Ov and horizontally Og

Let us assume that the length of the sighting line Pl is 500 mm, the firing distance is 50,000 mm (50 m) and the deviation of the midpoint of hits in height above the aiming point is 120 mm. Then the value of the front sight correction:

X \u003d 500 * 120 / 50,000 \u003d 1.2 mm.

More about ballistics

When firing in airless space, the maximum horizontal range of the projectile corresponds to an angle of throw of 45 degrees. The angle of throw corresponding to the maximum range of the projectile is commonly called the angle of maximum range in ballistics.

In reality, the angle of greatest range is never 45°, and, depending on the mass and shape of the projectile, varies from 28 to 43 degrees. For modern rifled weapons, the maximum range angle is 35 degrees, for shotguns - 30-32 degrees.

The maximum flight range of a shot is approximately equal to the number of hundreds of meters, which is the number of whole millimeters of the diameter of an individual shot, lined with a maximum initial speed of 375-400 m / s.

With an increase in temperature, the gun “raises”, with a decrease it “lowers”. The normal temperature is 15 degrees C.

As the barometric pressure decreases, the projectile flies farther and hits higher, and vice versa as the barometric pressure increases.

With an increase (or decrease) in temperature for every 10 degrees. The initial speed of the shot projectile increases (or decreases) by 7 m/s.

An imaginary line described in space by the center of gravity of a moving projectile is called trajectory(Fig. 34). It is formed under the action of the following forces: inertia, gravity, air resistance and the force arising from rarefaction of air behind the projectile.

When several forces act simultaneously on the projectile, each of them informs it of a certain movement, and the position of the projectile after a certain period of time is determined by the rule of adding movements that have a different direction. To understand how the trajectory of a projectile in space is formed, it is necessary to consider each of the forces acting on the projectile separately.

In ballistics, it is customary to consider the trajectory above (or below) the horizon of the weapon. By the horizon of arms is an imaginary infinite horizontal plane extending in all directions and passing through the departure point. Departure point called the center of the muzzle of the barrel. The trace from a passing horizontal plane is depicted as a horizontal line.

If we assume that no forces act on the projectile after it leaves the bore, then the projectile, moving by inertia, will fly in space infinitely, rectilinearly in the direction of the bore axis and uniformly. If, after leaving the bore, only one force of gravity acts on it, then in this case it will begin to fall strictly vertically downward towards the center of the Earth, obeying the laws of free fall of bodies.

Ballistics and ballistic movement

Prepared by a student of the 9th "m" class Petr Zaitsev.

Ι Introduction:

1) Goals and objectives of the work:

“I chose this topic because it was recommended to me by the class teacher-teacher of physics in my class, and I also really liked this topic myself. In this work, I want to learn a lot about ballistics and the ballistic motion of bodies.”

ΙΙ Main material:

1) Fundamentals of ballistics and ballistic movement.

a) the history of the emergence of ballistics:

In numerous wars throughout the history of mankind, the warring parties, proving their superiority, first used stones, spears, and arrows, and then cannonballs, bullets, shells, and bombs.

The success of the battle was largely determined by the accuracy of hitting the target.

At the same time, an accurate throw of a stone, hitting the enemy with a flying spear or arrow was recorded by the warrior visually. This allowed, with appropriate training, to repeat their success in the next battle.

The speed and range of projectiles and bullets, which significantly increased with the development of technology, made remote battles possible. However, the skill of a warrior, the resolving power of his eye, was not enough to accurately hit the target of an artillery duel first.

The desire to win stimulated the emergence of ballistics (from the Greek word ballo - I throw).

b) basic terms:

The emergence of ballistics dates back to the 16th century.

Ballistics is the science of the movement of projectiles, mines, bullets, unguided rockets during firing (launch). Main sections of ballistics: internal ballistics and external ballistics. The study of real processes occurring during the combustion of gunpowder, the movement of shells, rockets (or their models), etc., is the subject of the ballistics experiment. External ballistics studies the movement of projectiles, mines, bullets, unguided rockets, etc. after the termination of their force interaction with the weapon barrel (launcher), as well as factors affecting this movement. The main sections of external ballistics are: the study of forces and moments acting on a projectile in flight; study of the movement of the center of mass of the projectile to calculate the elements of the trajectory, as well as the movement of the projectile relates. The center of mass in order to determine its stability and dispersion characteristics. Sections of external ballistics are also the theory of corrections, the development of methods for obtaining data for compiling firing tables and external ballistic design. The movement of projectiles in special cases is studied by special sections of external ballistics, aviation ballistics, underwater ballistics, etc.

Internal ballistics studies the movement of projectiles, mines, bullets, etc. in the bore of a weapon under the action of powder gases, as well as other processes that occur when a shot is fired in the channel or chamber of a powder rocket. The main sections of internal ballistics are: pyrostatics, which studies the patterns of combustion of gunpowder and gas formation in a constant volume; pyrodynamics, which investigates the processes in the bore during firing and establishes a connection between them, the design characteristics of the bore and loading conditions; ballistic design of guns, missiles, small arms. Ballistics (studies the processes of the period of consequences) and internal ballistics of powder rockets (explores the patterns of fuel combustion in the chamber and the outflow of gases through nozzles, as well as the occurrence of forces and actions on unguided rockets).

Ballistic flexibility of a weapon - a property of a firearm that allows it to be expanded combat capabilities increase the effectiveness of the action by changing the ballistic. characteristics. Achieved by changing the ballistic. coefficient (for example, by introducing brake rings) and the initial velocity of the projectile (using variable charges). In combination with a change in the elevation angle, this allows you to get large angles of incidence and less dispersion of projectiles at intermediate ranges.

A ballistic missile is a missile which, except for a relatively small area, follows the trajectory of a freely thrown body. Unlike a cruise missile, a ballistic missile does not have bearing surfaces to create lift when flying in the atmosphere. The aerodynamic stability of the flight of some ballistic missiles is provided by stabilizers. Ballistic missiles include missiles for various purposes, launch vehicles for spacecraft, etc. They are single- and multi-stage, guided and unguided. The first combat ballistic missiles FAU 2- were used by Nazi Germany at the end of the World War. Ballistic missiles with a flight range of over 5500 km (according to foreign classification - over 6500 km) are called intercontinental ballistic missiles. (MBR). Modern ICBMs have a flight range of up to 11,500 km (for example, the American Minuteman is 11,500 km, Titan-2 is about 11,000 km, Trider-1 is about 7,400 km). They are launched from ground (mine) launchers or submarines. (from surface or underwater position). ICBMs are carried out as multi-stage, with liquid or solid propellant propulsion systems, can be equipped with monoblock or multiply charged nuclear warheads.

Ballistic track, spec. equipped on art. polygon area for experiment, study of movement art. shells, mini etc. Appropriate ballistic devices and ballistic equipment are installed on the ballistic track. targets, with the help of which, on the basis of experimental firing, the function (law) of air resistance, aerodynamic characteristics, translational and oscillatory parameters are determined. movement, initial departure conditions and projectile dispersion characteristics.

Ballistic shooting conditions, a set of ballistic. characteristics that have the greatest impact on the flight of the projectile (bullet). Normal, or tabular, ballistic firing conditions are conditions under which the mass and initial velocity of the projectile (bullet) are equal to the calculated (table), the temperature of the charges is 15 ° C, and the shape of the projectile (bullet) corresponds to the established drawing.

Ballistic characteristics, basic data that determine the patterns of development of the firing process and the movement of a projectile (mines, grenades, bullets) in the bore (intra-ballistic) or on a trajectory (external ballistic). The main intra-ballistic characteristics: the caliber of the weapon, the volume of the charging chamber, the density of loading, the length of the path of the projectile in the bore, the relative mass of the charge (its ratio to the mass of the projectile), the strength of gunpowder, max. pressure, forcing pressure, propellant combustion progressiveness characteristics, etc. The main external ballistic characteristics include: initial speed, ballistic coefficient, throw and departure angles, median deviations, etc.

Ballistic computer, an electronic device for firing (usually direct fire) from tanks, infantry fighting vehicles, small-caliber anti-aircraft guns, etc. The ballistic computer takes into account information about the coordinates and speed of the target and its object, wind, temperature and air pressure, initial speed and angles projectile launch, etc.

Ballistic descent, uncontrolled movement of the descent spacecraft (capsule) from the moment of leaving the orbit until reaching the planet specified relative to the surface.

Ballistic similarity, a property of artillery pieces, which consists in the similarity of dependencies characterizing the process of burning a powder charge when fired in the bores of various artillery systems. The conditions of ballistic similarity are studied by the theory of similarity, which is based on the equations of internal ballistics. Based on this theory, ballistic tables are compiled that are used in ballistic. design.

Ballistic coefficient (C), one of the main externally ballistic performance projectile (rocket), reflecting the influence of its shape factor (i), caliber (d), and mass (q) on the ability to overcome air resistance in flight. It is determined by the formula C \u003d (id / q) 1000, where d is in m, and q is in kg. The less ballistic coefficient, the easier the projectile overcomes air resistance.

Ballistic camera, a special device for photographing the phenomenon of a shot and its accompanying processes inside the bore and on the trajectory in order to determine the qualitative and quantitative ballistic characteristics of the weapon. Allows to carry out instant one-time photographing to.-l. phases of the process under study or successive high-speed photography (more than 10 thousand frames) of various phases. According to the method of obtaining exposure B.F. there are spark, with gas-light lamps, with electro-optical shutters and radiographic pulsed ones.

c) speed during ballistic motion.

To calculate the velocity v of the projectile at an arbitrary point of the trajectory, as well as to determine the angle , which forms the velocity vector with the horizontal,

it is enough to know the velocity projections on the X and Y axes (Fig. 1).

If v and v are known, the Pythagorean theorem can be used to find the speed:

The ratio of the leg v opposite the corner to the leg v belonging to

to this corner, determines tg and, accordingly, the angle :

With uniform movement along the X axis, the projection of the speed of movement v remains unchanged and equal to the projection of the initial speed v:

Dependence v(t) is determined by the formula:

into which should be substituted:

Graphs of velocity projections versus time are shown in Fig. 2.

At any point of the trajectory, the projection of the velocity on the X axis remains constant. As the projectile rises, the velocity projection on the Y-axis decreases linearly. At t \u003d 0, it is equal to \u003d sin a. Find the time interval after which the projection of this velocity becomes equal to zero:

0 = vsing- gt , t =

The result obtained coincides with the time of lifting the projectile by maximum height. At the top of the trajectory, the vertical velocity component is equal to zero.

Therefore, the body no longer rises. For t > velocity projection

v becomes negative. This means that this velocity component is directed opposite to the Y axis, i.e. the body begins to fall down (Fig. No. 3).

Since at the top of the trajectory v = 0, the speed of the projectile is:

d) the trajectory of the body in the field of gravity.

Let's consider the main parameters of the trajectory of a projectile flying with an initial speed v from a gun directed at an angle α to the horizon (Fig. 4).

The movement of the projectile occurs in the vertical XY plane containing v.

We choose the origin at the point of departure of the projectile.

In the Euclidean physical space, the movement of the body along the coordinate

the x and y axes can be considered independently.

The gravitational acceleration g is directed vertically downward, so the movement along the X axis will be uniform.

This means that the projection of the velocity v remains constant, equal to its value at the initial time v.

The law of uniform projectile motion along the X axis is: x= x+ vt. (5)

Along the Y axis, the movement is uniform, since the gravitational acceleration vector g is constant.

The law of uniformly variable projectile motion along the Y axis can be represented as follows: y = y+vt + . (6)

The curvilinear ballistic motion of a body can be considered as the result of the addition of two rectilinear motions: uniform motion

along the X axis and equally variable movement along the Y axis.

In the selected coordinate system:

v=vcosα. v=vsinα.

The gravitational acceleration is directed opposite to the Y axis, so

Substituting x, y, v, v, av (5) and (6), we obtain the ballistic law

motion in coordinate form, in the form of a system of two equations:

The projectile trajectory equation, or y(x) dependency, can be obtained by

excluding time from the equations of the system. To do this, from the first equation of the system we find:

Substituting it into the second equation we get:

Reducing v in the first term and taking into account that = tg α, we obtain

projectile trajectory equation: y = x tg α - .(8)

e) Trajectory of ballistic movement.

Let us construct a ballistic trajectory (8).

schedule quadratic function is known to be a parabola. In the case under consideration, the parabola passes through the origin,

since it follows from (8) that y \u003d 0 for x \u003d 0. The branches of the parabola are directed downwards, since the coefficient (-) at x is less than zero. (Fig No. 5).

Let us determine the main parameters of ballistic motion: the time of ascent to the maximum height, the maximum height, the time and range of the flight. Due to the independence of movements along the coordinate axes, the vertical rise of the projectile is determined only by the projection of the initial velocity onto the Y axis.

The maximum lift height can be calculated using the formula

if substituted instead of :

Figure 5 compares vertical and curvilinear motion with the same initial velocity along the Y axis. At any moment in time, a body thrown vertically upwards and a body thrown at an angle to the horizon with the same vertical velocity projection move synchronously along the Y axis.

Since the parabola is symmetrical with respect to the top, the flight time of the projectile is 2 times longer than the time it takes to rise to the maximum height:

Substituting the flight time into the law of motion along the X axis, we obtain maximum range flight:

Since 2 sin cos, a \u003d sin 2, then

e) the application of ballistic movement in practice.

Imagine that several shells were fired from one point, at different angles. For example, the first projectile at an angle of 30°, the second at an angle of 40°, the third at an angle of 60°, and the fourth at an angle of 75° (Fig. 6).

Figure #6 in green shows a graph of a projectile fired at 30°, white at 45°, purple at 60°, and red at 75°. And now let's look at the graphs of the flight of shells and compare them. (The initial speed is the same, and is equal to 20 km / h)

Comparing these graphs, one can deduce a certain pattern: with an increase in the angle of departure of the projectile, at the same initial speed, the flight range decreases, and the height increases.

2)Now consider another case associated with different initial speeds, with the same angle departure. In Figure 7, green color shows a graph of a projectile fired at an initial speed of 18 km/h, white at a speed of 20 km/h, purple at a speed of 22 km/h, and red at a speed of 25 km/h. And now let's look at the graphs of the flight of shells and compare them (the flight angle is the same and equal to 30°). Comparing these graphs, one can deduce a certain pattern: with an increase in the initial velocity of the projectile, at the same angle of departure, the range and height of the projectile increase.

Conclusion: with an increase in the angle of departure of the projectile, at the same initial speed, the flight range decreases, and the height increases, and with an increase in the initial velocity of the departure of the projectile, at the same angle of departure, the range and height of the projectile increase.

2) Application of theoretical calculations to the control of ballistic missiles.

a) the trajectory of a ballistic missile.

The most significant feature that distinguishes ballistic missiles from missiles of other classes is the nature of their trajectory. The trajectory of a ballistic missile consists of two sections - active and passive. On the active site, the rocket moves with acceleration under the action of the thrust force of the engines.

In this case, the rocket stores kinetic energy. At the end of the active part of the trajectory, when the rocket acquires a speed having a given value

and direction, the propulsion system is turned off. After that, the head of the rocket is separated from its body and flies further due to the stored kinetic energy. The second section of the trajectory (after turning off the engine) is called the section of the free flight of the rocket, or the passive section of the trajectory. Below, for brevity, we will usually talk about the free-flight trajectory of a rocket, implying the trajectory of not the entire rocket, but only its head.

Ballistic missiles are launched from launchers vertically upwards. Vertical launch allows you to build the most simple launchers and provides favorable conditions for controlling the rocket immediately after launch. In addition, vertical launch makes it possible to reduce the requirements for the rigidity of the rocket body and, consequently, reduce the weight of its structure.

The missile is controlled in such a way that a few seconds after the launch, while continuing to rise, it begins to gradually tilt towards the target, describing an arc in space. The angle between the longitudinal axis of the rocket and the horizon (pitch angle) changes in this case by 90º to the calculated final value. The required law of change (program) of the pitch angle is set by a software mechanism included in the on-board equipment of the rocket. At the final segment of the active section of the trajectory, the pitch angle is maintained, constant and the rocket flies straight, and when the speed reaches the calculated value, the propulsion system is turned off. In addition to the speed value, on the final segment of the active section of the trajectory, the specified direction of the rocket flight (the direction of its velocity vector) is also set with a high degree of accuracy. The speed of movement at the end of the active part of the trajectory reaches significant values, but the rocket picks up this speed gradually. While the rocket is in the dense layers of the atmosphere, its speed is low, which reduces the energy loss to overcome the resistance of the environment.

The moment of turning off the propulsion system divides the trajectory of the ballistic missile into active and passive sections. Therefore, the point of the trajectory at which the engines are turned off is called the boundary point. At this point, the control of the missile usually ends and it makes the entire further path to the target in free motion. The flight range of ballistic missiles along the Earth's surface, corresponding to the active part of the trajectory, is equal to no more than 4-10% of the total range. The main part of the trajectory of ballistic missiles is the free flight section.

To significantly increase the range, it is necessary to use multi-stage missiles.

Multi-stage rockets consist of separate blocks-stages, each of which has its own engines. The rocket is launched with a working propulsion system of the first stage. When the first stage fuel is used up, the second stage engine is fired and the first stage is reset. After the first stage is dropped, the thrust force of the engine must impart acceleration to a smaller mass, which leads to a significant increase in velocity v at the end of the active part of the trajectory compared to a single-stage rocket having the same initial mass.

Calculations show that already at two steps it is possible to obtain initial speed, sufficient for the flight of the head of the rocket over intercontinental distances.

The idea of ​​using multi-stage rockets to obtain high initial speeds and, consequently, long flight ranges was put forward by K.E. Tsiolkovsky. This idea is used in the creation of intercontinental ballistic missiles and launch vehicles for launching space objects.

b) the trajectory of guided projectiles.

The trajectory of a rocket is a line that its center of gravity describes in space. A guided projectile is an unmanned aerial vehicle that has controls that can be used to influence the movement of the vehicle along the entire trajectory or in one of the flight sections. Projectile control on the trajectory was required in order to hit the target, while remaining at a safe distance from it. There are two main classes of targets: moving and stationary. In turn, a rocket projectile can be launched from a stationary launch device or from a mobile one (for example, from an airplane). With stationary targets and launching devices, the data needed to hit the target is obtained from the known relative location of the launch site and the target. In this case, the trajectory of the projectile can be calculated in advance, and the projectile is equipped with devices that ensure its movement according to a certain calculated program.

In other cases, the relative location of the launch site and the target is constantly changing. To hit the target in these cases, it is necessary to have devices that track the target and continuously determine the relative position of the projectile and the target. The information received from these devices is used to control the movement of the projectile. Control must ensure the movement of the missile to the target along the most advantageous trajectory.

In order to fully characterize the flight of a rocket, it is not enough to know only such elements of its movement as the trajectory, range, altitude, flight speed, and other quantities that characterize the movement of the center of gravity of the rocket. The rocket can occupy various positions in space relative to its center of gravity.

A rocket is a body of significant size, consisting of many components and parts, made with a certain degree of accuracy. In the process of movement, it experiences various perturbations associated with the restless state of the atmosphere, the inaccuracy of work power plant, various kinds of interference, etc. The combination of these errors, not provided for by the calculation, leads to the fact that the actual movement is very different from the ideal one. Therefore, in order to effectively control a rocket, it is necessary to eliminate the undesirable influence of random disturbing influences, or, as they say, to ensure the stability of the rocket's movement.

c) coordinates that determine the position of the rocket in space.

The study of the various and complex movements made by a rocket can be greatly simplified if the rocket's movement is represented as the sum of the translational movement of its center of gravity and rotational movement about the center of gravity. The examples given above clearly show that in order to ensure the stability of the rocket's movement, it is extremely important to have its stability relative to the center of gravity, i.e., the angular stabilization of the rocket. Rocket rotation relative to the center of gravity can be represented as the sum of rotational movements about three perpendicular axes that have a certain orientation in space. Fig. No. 7 shows an ideal feathered rocket flying along a calculated trajectory. The origin of the coordinate systems, relative to which we will stabilize the rocket, will be placed at the center of gravity of the rocket. Let's direct the X-axis tangentially to the trajectory in the direction of the rocket's movement. The Y axis will be drawn in the plane of the trajectory perpendicular to the X axis, and the axis

Z - perpendicular to the first two axes, as shown in Fig. No. 8.

Associate with the rocket a rectangular coordinate system XYZ, similar to the first one, and the X axis must coincide with the axis of symmetry of the rocket. In a perfectly stabilized rocket, the X, Y, Z axes coincide with the X, Y, Z axes, as shown in Fig. 8

Under the action of perturbations, the rocket can rotate around each of the oriented axes X, Y, Z. The rotation of the rocket around the X axis is called the roll of the rocket. The bank angle lies in the YOZ plane. It can be determined by measuring in this plane the angle between the Z and Z or Y and Y axes. Rotation around the axis

Y is the yaw of the rocket. The yaw angle is in the XOZ plane as the angle between the X and X or Z and Z axes. The angle of rotation around the Z-axis is called the pitch angle. It is determined by the angle between the X and X or Y and Y axes that lie in the path plane.

Automatic rocket stabilization devices should give it such a position when = 0 or . To do this, the rocket must have sensitive devices capable of changing its angular position.

The trajectory of the rocket in space is determined by the current coordinates

X, Y, Z of its center of gravity. The starting point of the rocket is taken as the starting point. For missiles long range the X-axis is taken as a straight line tangent to the arc of the great circle connecting the start with the target. In this case, the Y axis is directed upward, and the Z axis is perpendicular to the first two axes. This coordinate system is called terrestrial (Fig. 9).

The calculated trajectory of ballistic missiles lies in the XOY plane, called the firing plane, and is determined by two coordinates X and Y.

Conclusion:

“In this work, I learned a lot about ballistics, the ballistic movement of bodies, about the flight of missiles, finding their coordinates in space.”

Bibliography

Kasyanov V.A. - Physics grade 10; Petrov V.P. - Missile control; Zhakov A.M. -

Control of ballistic missiles and space objects; Umansky S.P. - Cosmonautics today and tomorrow; Ogarkov N.V. - Military Encyclopedic Dictionary.

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BALLISTICS, the science of motion under the action of some forces of a heavy body thrown into space. Ballistics attached Ch. arr. to the study of the movement of an artillery projectile or bullet fired with the help of one or another kind of throwing weapon. Ballistics is also applied to the study of the movement of a bomb dropped from an aircraft. The methods of higher mathematics and experiment are used to establish the laws of scientific ballistics. Ballistics is divided into external and internal.

External ballistics considers the laws of motion of a projectile in air and other media, as well as the laws of the action of projectiles on various subjects. The main task of external ballistics is to establish the dependence of the projectile flight curve (trajectory) on the initial speed v 0, throwing angle ϕ, caliber 2R, weight P and the shape of the projectile, as well as on all kinds of circumstances accompanying firing (for example, meteorological). The first studies in the field of external ballistics belong to Tartaglia (1546). Galileo established that the trajectory of a body thrown in airless space is a parabola (Fig. 1).

The equation for this parabola is:

The trajectory is symmetrical about the vertex A, so that Aa is the axis of the parabola; the angle of incidence ϴ c is equal to the angle of throw ϕ; the speed v c at the point of incidence C is equal to the initial speed v 0 ; the projectile has the lowest speed at vertex A; the flight times for the ascending and descending branches are equal.

Flight range X in airless space is determined from the expression

which indicates that the greatest range is obtained at an angle of throw ϕ = 45°. The total flight time T in airless space is found from the expression

Newton in 1687 showed that the trajectory of a body thrown in the air is not a parabola, and on the basis of a series of experiments he came to the conclusion that the force of air resistance is proportional to the square of the speed of the body. Euler, Legendre and others also assumed it to be proportional to the square of the speed. The analytical expression of the air resistance force was derived both theoretically and on the basis of experimental data. The first systematic work on this issue belongs to Robins (1742), who studied the resistance of air to the movement of spherical bullets. In 1839-1840. Piober, Morin, and Didion at Metz made experiments of the same kind on spherical projectiles. The introduction of rifled weapons and oblong projectiles gave a strong impetus to the study of the laws of air resistance to the flight of a projectile. As a result of Bashfort's experiments in England (1865-1880) on oblong and spherical projectiles, based on the work of Maievsky in Russia (1868-1869), the Krupp factory in Germany (1881-1890) and Hozhel in Holland ( 1884) it turned out to be possible to express the force of air resistance ϱ by such a monomial:

where λ is a coefficient depending on the shape of the projectile, A is a numerical coefficient, π is the ratio of the circumference to the diameter, R is the radius of the cylindrical part of the projectile, P is the air density during firing and P 0 \u003d 1.206 kg is the air density at 15 °, pressure atmosphere at 750 mm and humidity 50%. Coefficient A and exponent n are determined from experience and are different for different speeds, namely:

The general properties of the trajectory of a non-rotating projectile in the air are established on the basis of the differential equations of motion of its center of gravity in the vertical plane of fire. These equations look like:

In them: ϱ is the force of air resistance, P is the weight of the projectile, ϴ is the angle of inclination of the tangent at a given point of the trajectories to the horizon, v is the speed of the projectile at a given point, v 1 \u003d v∙cos ϴ is the horizontal projection of the speed, s is the length of the arc trajectories, t - time, g - acceleration of gravity. Based on these equations, S.-Rober indicated the following main properties of the trajectory: it is curved above the horizon, its top is closer to the point of incidence, the angle of incidence is greater than the angle of incidence, the horizontal velocity projection gradually decreases, the lowest speed and the greatest curvature of the trajectory are behind the top, descending the branch of the trajectory has an asymptote. Professor N. Zabudsky, in addition, added that the flight time in the descending branch is longer than in the ascending one. The trajectory of the projectile in the air is shown in Fig. 2.

When the projectile moves in the air, the angle of greatest range is generally less than 45 °, but m. b. cases where this angle is greater than 45°. The differential equations of motion of the center of gravity of the projectile are not integrated, and therefore the main problem of external ballistics in the general case does not have an exact solution. A rather convenient method for an approximate solution was given for the first time by Didion. In 1880, Siacci proposed a method convenient for practice for solving the problem of aimed shooting (i.e., when ϕ ≤ 15°), which is still used today. For the convenience of Siacci's calculations, appropriate tables have been compiled. To solve the problems of mounted shooting (i.e., at ϕ > 15°), when the initial speed is less than 240 m/sec, a method was given and the necessary Otto tables were compiled, subsequently modified by Siacci and Lordillon. Bashfort also gives a method and tables for solving the problems of mounted shooting at speeds over 240 m/sec. Professor N. Zabudsky for solving the problems of mounted shooting at initial speeds from 240 to 650 m/s takes the force of air resistance proportional to the 4th degree of speed and gives a solution method under this assumption. At initial speeds exceeding 650 m/s, to solve the problems of mounted shooting, it is necessary to divide the trajectory into three parts, with the extreme parts calculated using the Siacci method, and the middle part using the Zabudsky method. Per last years a method for solving the main problem of external ballistics, based on the Shtormer method - the numerical integration of differential equations, has become widespread and generally recognized. The application of this method to solving problems of ballistics was first made by Academician A. N. Krylov. The numerical integration method is universal, since it is suitable for any speeds and throwing angles. With this method, it is easy and with great accuracy m. the change in air density with height is taken into account. This last one has great importance when firing at large throwing angles, up to 90 °, with significant initial speeds, of the order of 800-1000 m / s (shooting at air targets), and especially when firing at so-called ultra-long range, i.e., at a distance of 100 or more km.

The basis for resolving the issue of shooting at such distances is the following idea. A projectile fired at a very high initial velocity, for example, 1500 m/s, at an angle of throw of 50-55°, quickly flies in the ascending branch of its trajectory to such layers of the atmosphere in which the air density is extremely low. It is believed that at an altitude of 20 km, the air density is 15 times, and at an altitude of 40 km, 350 times less than the density of air on the surface of the earth; as a result, the force of air resistance decreases in the same corresponding number of times at these heights. That. we can consider the part of the trajectory passing in the layers of the atmosphere lying above 20 km as a parabola. If the tangent to the trajectory at an altitude of 20 km has an inclination of 45° to the horizon, then the range in airless space will be the greatest. To ensure an angle of 45° at an altitude of 20 km, a projectile must be thrown from the ground at an angle greater than 45°, i.e. at an angle of 50-55°, depending on the initial velocity, caliber and weight of the projectile. For example, (Fig. 3): a projectile, thrown, at an angle to the horizon of 55 ° with an initial speed of 1500 m / s; at the point a of the ascending branch, its speed became equal to 1000 m / s, and the tangent to the trajectory at this point makes an angle of 45 ° with the horizon.

Under these conditions, the flight range ab in airless space will be:

and the horizontal range of the point of standing of the OS gun will be more than 102 km for the sum of the OA and AF sections, the calculation of the values ​​​​of which is more convenient and most accurate can be done by numerical integration. When accurately calculating an ultra-long trajectory, one has to take into account the influence of the rotation of the earth, and for trajectories with a range of several hundred kilometers (a theoretically possible case), also the spherical shape of the earth and the change in the acceleration of gravity both in magnitude and in direction.

The first significant theoretical studies of the motion of an elongated projectile rotating about its axis were made in 1859 by S. Robert, whose memoirs served as the basis for Maievsky's work on this issue in Russia. Analytical studies led Maievsky to the conclusion that the axis of the projectile figure, when the forward speed is not too small, has an oscillatory movement around the tangent to the trajectory, and made it possible to study this movement for the case of aimed shooting. De-Sparre succeeded in reducing this problem to quadratures, and Professor N. Zabudsky extended de-Sparre's conclusion to the case of mounted shooting. The differential equations for the rotational motion of the projectile, with the adoption of some practically possible assumptions, have the form:

here: δ is the angle between the tangent to the trajectory and the axis of the projectile figure; v is the angle between the vertical plane passing through the axis of the gun channel and the plane passing through the tangent to the trajectory and the axis of the projectile figure; k is the moment of air resistance force relative to the center of gravity of the projectile; A is the moment of inertia of the projectile about the axis; p 0 - projection of the angular velocity of the projectile on its axis; ϴ - angle of inclination of the tangent at a given point of the trajectory to the horizon; t - time.

These equations do not exactly integrate. The study of the rotational motion of an elongated projectile leads to the following main conclusion: in aimed shooting, the axis of the projectile is always deviated to one side from the firing plane, namely, in the direction of projectile rotation, if you look at it from behind; with mounted shooting, this deviation can be in the opposite direction. If we imagine a plane that always remains perpendicular to the tangent to the trajectory and is always at the same distance from its center of gravity during the flight of the projectile, then the axis of the figure of the projectile will draw on this plane a complex curve of the type shown in Fig. four.

Large loops of this curve are the result of the oscillatory movement of the axis of the projectile figure around the tangent to the trajectory, this is the so-called. precession; small loops and waviness of the curve are the result of a mismatch between the instantaneous axis of rotation of the projectile and the axis of its figure, this is the so-called. nutation. To obtain greater accuracy of the projectile, it is necessary to achieve a decrease in nutation. The deviation of the projectile from the plane of fire due to the deviation of its axis is called derivation. Maievsky derived a simple formula for the amount of derivation in aimed shooting; the same formula can be. applied in mounted shooting. Due to the derivation, the projection of the trajectory onto the horizon, the plane, acquires the form shown in Fig. 5.

That. the trajectory of a rotating projectile is a curve of double curvature. For the correct flight of an elongated projectile, it must be given an appropriate speed of rotation around the axis. Professor N. Zabudsky gives an expression for the minimum rotation speed necessary for the stability of the projectile in flight, depending on its design data. The questions of the projectile's rotational motion and the influence of this motion on its flight are extremely complex and little studied. Only in recent years has a number of serious studies of this question been undertaken. arr. in France as well as in America.

The study of the action of shells on various subjects is carried out by external ballistics Ch. arr. through experiments. Based on the experiments of the Metsk Commission, formulas are given for calculating the depths of projectiles in solid media. The experiments of the Le Havre Commission provided material for the derivation of armor penetration formulas. The Spanish artilleryman de la Love, on the basis of experience, gave formulas for calculating the volume of a funnel formed when a projectile breaks in the ground; this volume is proportional to the weight of the bursting charge and depends on the speed of the projectile, its shape, the quality of the soil and the properties explosive. Methods for solving problems of external ballistics serve as the basis for compiling firing tables. Calculation of tabular data is carried out after determining by shooting at 2-3 distances some coefficients characterizing the projectile and gun.

Internal ballistics considers the laws of projectile motion in the gun channel under the action of powder gases. Only knowing these laws, it is possible to design a tool of the required power. That. The main task of internal ballistics is to establish the functional dependence of the pressure of powder gases and the speed of the projectile in the channel on the path it passes. To establish this dependence, internal ballistics uses the laws of thermodynamics, thermochemistry and the kinetic theory of gases. S.-Robert was the first to use the principles of thermodynamics in the study of internal ballistics; then the French engineer Sarro gave a number of fundamental works (1873-1883) on internal ballistics, which served as the basis for further work by various scientists, and this laid the foundation for the modern rational study of the issue. The phenomena that occur in the channel of a given gun depend to a large extent on the composition of the gunpowder, the shape and size of its grains. The burning time of a powder grain depends mainly on its smallest size - thickness - and the burning rate of the powder, i.e., the speed of the flame penetrating into the thickness of the grain. The rate of combustion primarily depends on the pressure under which it occurs, as well as on the nature of the gunpowder. The impossibility of an accurate study of the combustion of gunpowder forces one to resort to experiments, hypotheses and assumptions that simplify the solution of the general problem. Sarro expressed the rate of burning and gunpowder as a function of pressure

where A is the burning rate at a pressure of 1 kg / cm 2, a v is an indicator depending on the type of gunpowder; v, generally speaking, is less than unity, but very close to it, so Seber and Hugognot simplified the Sarro formula, taking v = 1. When burning under variable pressure, which takes place in the gun channel, the burning rate of gunpowder is also a variable value. According to the works of Viel, it can be considered that smokeless powders burn in concentric layers, while combustion black powder such a law does not obey and is very wrong. The law of development of pressure of powder gases in closed vessels was established by Noble in the following form:

P 0 - atmospheric pressure; w 0 - the volume of decomposition products of 1 kg of gunpowder at 0 ° and a pressure of 760 mm, considering the water as gaseous; T 1 - absolute temperature decomposition of gunpowder; W is the volume of the vessel in which combustion occurs; w is the weight of the charge; α - covolum, i.e., the volume of decomposition products of 1 kg of gunpowder at an infinitely high pressure (generally, α \u003d 0.001w 0 is taken); Δ - loading density, equal to w/W in metric measures; f = RT 1 - powder force, measured in units of work per unit charge weight. To simplify the solution of the general problem of the movement of a projectile in the gun channel, it is assumed: 1) that the ignition of the entire charge occurs simultaneously, 2) that the burning rate of gunpowder during the entire process is proportional to pressure, 3) that the combustion of grains occurs in concentric layers, 4) that the amount of heat, separated by each equal share of the charge, the volumes and composition of the gases, as well as the strength of the powder, are constant during the entire time the charge is burning, 5) that there is no transfer of heat to the walls of the gun and the projectile, 6) that there are no losses of gases and 7) that there is no wave-like movement of the explosion products. Taking these basic assumptions and some more, various authors give a solution to the main problem of internal ballistics in the form of one or another system of differential equations of projectile motion. Integrate into general view these equations are not possible, and therefore resort to approximate methods of solution. All these methods are based on the classical solution of the problem of internal ballistics, proposed by Sarro, which consists in integrating the differential equations of projectile motion using a change of variables. After the classical formulas of Sarro, the most famous are the formulas proposed by Charbonnier and Sugo.

Ballisticians Bianchi (Italy), Kranz (Germany) and Drozdov (Russia) also give their own methods for solving the main problem. All of the above methods present significant difficulties for practical application due to their complexity and the need for tables to calculate various kinds of auxiliary functions. By the method of numerical integration of differential equations, the problem of internal ballistics can also be resolved. For practical purposes, some authors give empirical dependencies, using which one can quite accurately solve the problems of internal ballistics. The most satisfactory of these dependencies are the formulas of Heidenreich, le-Duc, Oekkinghaus and the differential formulas of Kisnemsky. The law of pressure development and the law of projectile velocities in the gun channel are graphically represented in Fig. 6.

A detailed consideration of the question of the influence of the shape and size of the powder grain on the development of pressure in the gun channel leads to the conclusion that such a grain is possible in which the pressure, having reached a certain value, will not decrease as the projectile moves in the channel, but will remain so until complete combustion charge. Such gunpowder will have, as they say, complete progressiveness. With the help of such gunpowder, the projectile will receive the highest initial velocity at a pressure not exceeding a predetermined one.

The study of the rotational motion of the projectile in the channel under the action of rifling has the ultimate goal of determining the forces acting on the leading parts, which is necessary for calculating their strength. The pressure at the moment on the combat edge of the rifling or ledge of the leading belt

where λ is a coefficient depending on the projectile, is in the range of 0.55-0.60 for the accepted designs of projectiles; n is the number of grooves; P - gas pressure; s is the cross-sectional area of ​​the channel; α - the angle of inclination of the rifling to the generating channel; m is the mass of the projectile; v - projectile speed; y \u003d f (x) - the equation of the cutting curve, deployed on a plane (for cutting of constant steepness)

The most common type of slicing is a constant, which is a straight line when unrolled onto a plane. The steepness of the cut is determined by the speed of rotation of the projectile around the axis necessary for its stability in flight. The manpower of the projectile's rotational motion is about 1% of the manpower of its translational motion. In addition to communicating translational and rotational movements, the energy of powder gases is spent on overcoming the resistance of the leading belt of the projectile to cutting into rifling, friction on the combat edges, friction of the combustion products of gunpowder, atmospheric pressure, air resistance, the weight of the projectile and the work of stretching the walls of the barrel. All these circumstances m. taken into account to some extent either by theoretical considerations or on the basis of experimental material. The loss of heat by gases for heating the walls of the barrel depends on the conditions of firing, caliber, temperature, thermal conductivity, etc. Theoretical considerations on this issue are very difficult, but direct experiments regarding this loss have not been made; so arr. this question remains open. Developing in the bore when fired is extremely high pressures(up to 3000-4000 kg / cm 2) and temperatures have a devastating effect on the channel walls - the so-called. burning it out. There are several hypotheses explaining the phenomenon of burnout, the most important of which belong to Professor D. Chernov, Viel and Charbonnier.