Time in ballistics. Start in science

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 (e.g. by inserting brake rings) and initial speed 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 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 provide greatest influence on the flight of a 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, electronic device for firing (usually direct fire) from tanks, infantry fighting vehicles, small-caliber anti-aircraft guns etc. The ballistic calculator takes into account information about the coordinates and speed of the target and its object, wind, temperature and air pressure, initial velocity and angles of the projectile, 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 sequential high-speed photography (more than 10 thousand frames / s) 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.

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 according to 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. Differential Equations rotary motion projectile when making 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; when mounted shooting, this deviation can be in reverse side. 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 presented 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. Pressure in this moment on the combat edge of the groove 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 motions to the projectile, the energy of the 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.

In the section on the question Physics. ballistic movement. Help find initial speed. given by the author Eldar Nezametdinov the best answer is If alpha is the angle with the horizon line, i.e. the direction OX, then Uo must be decomposed into a vertical (along the OY axis and horizontal components, i.e. Uoy \u003d Uo Sin (alfa) and Uox \u003d UoCos (alfa)
The change in speed along the OY axis in a scalar expression when moving up (that is, we have already taken into account the direction of the velocity and acceleration vector)
Uy=Uoy -gt=Uo Sin alfa - gt/2 =0, where t is the time of the entire flight
i.e. Uo=(gt)/(2 Sin(alfa))=(10x2)/(2x0.5)=20 (m/s)
Eldar Nezametdinov
Thinker
(5046)
where did the two come from?
The thing is
Uy = Uosina - gT*T/2
you have written
Uy = Uosina - gT/2
I don’t understand) how did you get rid of T * T so that you did T .... and equal to 2ke)

Answer from 22 answers[guru]

Hello! Here is a selection of topics with answers to your question: Physics. ballistic movement. Help find initial speed.

Answer from Leonid Fursov[guru]
solution. x(t)=v0*(cos(a))*t; y(t)=v0*(sin(a))*t-0.5*g*t^2; vy=v0*(sin(a))-g*t;
1. vy=0 (condition for finding the maximum height of the ascent. First find the time of ascent, then substitute in the formula y(t)=v0*(sin(a))*t-0.5*g*t^2 and find maximum height lift).
2. y(t)=0 - a condition for finding the flight duration, and according to it, the flight range.

Karpov Yaroslav Aleksandrovich, Bakkasov Damir Rafailevich

Relevance of the topic: Ballistics is an important and ancient science, it is used in military affairs and in criminalistics.

Field of study - Mechanics.

Subject of study- bodies passing part of the way as a freely thrown body.

Goals: to study the patterns characteristic of ballistic motion and check their implementation with the help of laboratory work.

Tasks of this work:

1. The study of additional material on mechanics.

2. Introduction to the history and types of ballistics.

3. Carry out laboratory work to study the patterns of ballistic motion.

Research methods: collection of information, analysis, generalization, study of theoretical material, laboratory work.

In the theoretical part The work deals with the basic theoretical information on ballistic motion.

In the research part the results of laboratory work are presented.

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Karpov Yaroslav Aleksandrovich, Bakkasov Damir RafailevichGrade 9 "A" GBOU secondary school № 351

VOUO DO Moscow

Scientific adviser: Kucherbaeva O.G.

"The study of ballistic motion using the digital laboratory "Archimedes"

Annotation.

Relevance of the topic: Ballistics is an important and ancient science, it is used in military affairs and in forensic science.

Field of study - Mechanics.

Subject of study- bodies passing part of the way as a freely thrown body.

Goals: to study the patterns characteristic of ballistic motion and check their implementation with the help of laboratory work.

Tasks of this work:

The study of additional material on mechanics.

Introduction to the history and types of ballistics.

Carry out laboratory work to study the patterns of ballistic motion.

Research methods:collection of information, analysis, generalization, study of theoretical material, laboratory work.

In the theoretical part work the basic theoretical information on ballistic motion is considered.

In the research partthe results of laboratory work are presented.

The purpose of the experiments:

1) Use a ballistic pistol to determine at what angle of departure the range of the projectile is the greatest.

2) Find out at what angles of departure the flight range is approximately the same

3) Shoot a video with the movement of the body at an angle to the horizon and use the digital laboratory "Archimedes" to analyze the resulting movement trajectories.

When firing on a horizontal surface at different angles to the horizon, the range of the projectile is expressed by the formula

ℓ = (2V²cosα sinα)/g

or

ℓ = (V²sin(2α))/g

It follows from this formula that when the projectile departure angle changes from 90 to 0°, the range of its fall first increases from zero to a certain maximum value, and then again decreases to zero, the fall range is maximum when the products of cosα and sinα are greatest. In this work, we decided to test this dependence experimentally using a ballistic pistol.

We set up the gun at various angles: 20, 30, 40, 45, 60 and 70° and fired 3 shots at each angle. See the table for the results.

From the table, we see that the range of the projectile at a departure angle of 45 ° is maximum. This is confirmed by the formula. When the products of the cosine of an angle and the sine of an angle are greatest. It can also be seen from the table that the flight range at angles of 20° and 70°, as well as 30° and 60° are equal. This is confirmed by the same formula. When the product of the cosines of the angles and the sines of the angles are equal.

o Filming a short film showing planar motion (movement of a body thrown at an angle to the horizon).

o Convert digital video footage to QuickTime format on an Apple computer using iMovie or on a PC using QuickTime Pro. A feature of these programs is that they allow you to control the parameters of the output file.

o Processing the received video file in the Multilab program, in fact, digitizing the trajectory, and then mathematical processing of the graphs.

3.Conclusion

Ballistics is an important and ancient science, it is used in military affairs and in forensic science. With the help of our experiment, we have confirmed a certain relationship between the angle of departure and the range of the projectile. I would also like to note that studying ballistics, we see a close connection between the two sciences: physics and mathematics

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District scientific and industrial complex "Children-creators of the XXI century" Physics "Research of ballistic movement" Authors: Karpov Yaroslav Alexandrovich Bakkasov Damir Rafailevich GBOU secondary school No. 351, 9 "A" class Supervisor: teacher of physics Kucherbaeva Olga Gennadievna Moscow, 2011

Introduction Ballistics is an important and ancient science, it is used in military affairs and in criminalistics. At the same time, it is interesting from the point of view of the connection of subjects: mathematics and physics.

Objectives to study the patterns characteristic of ballistic movement to verify their implementation using laboratory work.

The objectives of this work The study of additional material on mechanics. Introduction to the history and types of ballistics. Conduct laboratory work on the study of the patterns of ballistic movement using a ballistic pistol and using the digital laboratory "Archimedes"

The history of the emergence of ballistics The emergence of ballistics as a science dates back to the 16th century. The first works on ballistics are the books of the Italian N. Tartaglia " new science"(1537) and "Questions and discoveries relating to artillery shooting" (1546). In the 17th century the fundamental principles of external ballistics were established by G. Galileo, who developed the parabolic theory of projectile motion, the Italian E. Torricelli and the Frenchman M. Mersenne, who proposed calling the science of projectile motion ballistics (1644). I. Newton conducted the first studies on the movement of a projectile, taking into account air resistance - "The Mathematical Principles of Natural Philosophy" (1687). In the 17-18 centuries. The movement of projectiles was studied by the Dutchman H. Huygens, the Frenchman P. Varignon, the Swiss D. Bernoulli, the Englishman Robins, and the Russian scientist L. Euler, and others. The experimental and theoretical foundations of internal ballistics were laid in the 18th century. in the works of Robins, C. Hetton, Bernoulli, and others. In the 19th century. the laws of air resistance were established (the laws of N. V. Maievsky, N. A. Zabudsky, the Le Havre law, the law of A. F. Siacci). At the beginning of the 20th century an exact solution was given to the main problem of internal ballistics - the work of N.F. Drozdov (1903, 1910), the issues of combustion of gunpowder in a constant volume - the work of I.P. Grave (1904) and the pressure of powder gases in the bore - the work of N.A. Zabudsky (1904, 1914), as well as the Frenchman P. Charbonnier and the Italian D. Bianchi .. As an independent, specific field of science, ballistics has been widely developed since the middle of the XlX century.

Ballistics in the USSR In the USSR, a great contribution to the further development of ballistics was made by scientists from the Commission for Special Artillery Experiments (KOSLRTOP) in 1918-26. During this period, V. M. Trofimov, A. N. Krylov, D. A. Venttsel, V. V. Mechnikov, G. V. Oppokov, N. Okunev and others carried out a number of works to improve methods for calculating the trajectory, developing the theory corrections and for the study of the rotational motion of the projectile. The studies of N. E. Zhukovsky and S. A. Chaplygin on the aerodynamics of artillery shells formed the basis for the work of E. A. Berkalov and others on improving the shape of shells and increasing their flight range. V. S. Pugachev was the first to solve the general problem of the motion of an artillery shell.

The main sections of ballistics "BALLISTICS - the science of the laws of flight of bodies (shells, mines, bombs, bullets) passing part of the way as a freely thrown body" - they write in Ozhegov's dictionary. Ballistics is divided into: internal and external, as well as "terminal" (final) ballistics. 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. 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. "Terminal" (final) ballistics, is related to the interaction of the projectile and the body into which it hits, and the movement of the projectile after the hit, that is, it considers the physics of the destructive effect of the weapon on the targets it hits, including the explosion phenomenon. Terminal ballistics is handled by gunsmiths-specialists in shells and bullets, strength and other specialists in armor and protection, as well as forensic specialists. To imitate the action of fragments and bullets that hit a person, shots are fired at massive targets made of gelatin. Similar experiments belong to the so-called. wound ballistics. Their results make it possible to judge the nature of the wounds that a person can receive. The information provided by research on wound ballistics makes it possible to optimize the effectiveness different types weapons designed to destroy enemy manpower.

The concept of forensic ballistics Forensic ballistics is a branch of forensic technology that studies the patterns of occurrence of traces of a crime, the event of which is associated with the use of firearms. The objects of ballistic research are: 1. Traces that appear on the parts of weapons, shells and bullets, formed as a result of a shot. 2. Traces that appear on an obstacle when a projectile hits it. 3. Firearms and its parts. 4. Ammunition and parts thereof. 5. Explosive devices. 6. Edged weapons.

Velocity during ballistic motion To calculate the velocity v of a projectile at an arbitrary point of the trajectory, as well as to determine the angle α that forms the velocity vector with the horizontal, it is enough to know the velocity projections on the X and Y axes. If vX and v Y are known, using the Pythagorean theorem, you can find the velocity : v \u003d √ vX ² + v Y ². With uniform movement along the X axis, the projection of the velocity of movement vX remains unchanged and equal to the projection of the initial velocity v: v = v cos α. Dependence v (t) is determined by the formula: v = v + a t. into which should be substituted: v = v sinα, a = -g.

Then v = v sin - gt . 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. Let's find the time interval after which the projection of this speed becomes equal to zero: 0 = v sin - gt , t = The result obtained coincides with the time the projectile rises to the maximum height. At the top of the trajectory, the vertical velocity component is equal to zero. Therefore, the body no longer rises. At t> the projection of velocity v becomes negative. This means that this velocity component is directed opposite to the Y axis, i.e., the body begins to fall down. Since at the top of the trajectory v = 0, the velocity of the projectile is: v = v = v cosα

Journal of research The purpose of the experiments: 1) To establish at what angle of departure the flight range of the projectile is the greatest. 2) Find out at what angles of departure the flight range is approximately the same 3) Check the data using the digital laboratory "Archimedes"

When firing on a horizontal surface at different angles to the horizon, the range of the projectile is expressed by the formula ℓ = (2V²cosα sinα)/g Or ℓ = (V²sin(2α))/g the flight range of its fall first increases from zero to some maximum value, and then again decreases to zero; the fall distance is maximum when the products of cosα and sinα are greatest. In this work, we decided to test this dependence experimentally using a ballistic pistol

We set up the gun at various angles: 20, 30, 40, 45, 60 and 70° and fired 3 shots at each angle. Flight angle 20º 30º 40º 45º 60º 70º Flight range of the "projectile" ℓ, m 1.62 1.90 2.00 2.10 1.61 1.25 1.54 1.90 2.00 2.05 1.55 1 ,20 1.54 1.86 1.95 2.12 1.55 1.30 that the range of the projectile at a departure angle of 45 ° is maximum. This is confirmed by the formula. When the products of the cosine of an angle and the sine of an angle are greatest. It can also be seen from the table that the flight range at angles of 20° and 70°, as well as 30° and 60° are equal. This is confirmed by the same formula. When the product of the cosines of the angles and the sines of the angles are equal

Ballistic Missile Trajectory The most significant feature that distinguishes ballistic missiles from other classes of missiles 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. Thereafter head part 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. 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 trajectory is set with a high degree accuracy as well as the given direction of the rocket's flight (the direction of its velocity vector). 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. 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. In the process of movement, the rocket 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.

Conclusion Ballistics is an important and ancient science, it is used in military affairs and in forensic science. With the help of our experiment, we have confirmed a certain relationship between the angle of departure and the range of the projectile. I would also like to note that studying ballistics, we see a close connection between the two sciences: physics and mathematics.

List of used literature E.I. Butikov, A.S. Kondratiev, Physics for in-depth study, volume 1. Mechanics. G.I. Kopylov, Only kinematics, Library "Quantum", issue 11. M .: Nauka, 1981 Physics. Textbook for grade 10. Myakishev G.Ya., Bukhovtsev B.B. (1982.)

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MOUSOSH No. 8 Ballistic movement Completed by: Muzalevskaya Veronika 10 "I" 2007 Purpose To study ballistic movement . Explain why and how it came about. Consider all sorts of examples and basic parameters based on ballistic motion. Learn to make charts. To reveal the meaning of the speed of ballistic motion and speed in the atmosphere. Understand why and for what purposes it is used. And most importantly, learn to solve problems using the knowledge of ballistic motion. Ballistic motion 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, the exact throw of a stone, the defeat of the enemy by a flying spear or arrow was recorded by the warrior visually. This allowed (with appropriate training) to repeat their success in the next battle. Ballistics is a branch of mechanics that studies the motion of bodies in the Earth's gravitational field. Bullets, projectiles and bombs, as well as tennis and soccer balls, and the core of an athlete, move along a ballistic trajectory during flight. To describe the ballistic motion, as a first approximation, it is convenient to introduce an idealized model, considering the body as a material point moving with a constant gravitational acceleration g. At the same time, the change in the height of the body, air resistance, the curvature of the Earth's surface and its rotation around its own axis are neglected. This approximation greatly facilitates the calculation of the trajectory of bodies. However, such consideration has certain limits of applicability. For example, when flying an intercontinental ballistic missile, one cannot neglect the curvature of the Earth's surface. In free-falling bodies, air resistance cannot be ignored. The trajectory of a body in a gravitational field. Let us consider the main parameters of the trajectory of a projectile flying with an initial velocity U0 from a gun directed at an angle ą to the horizon. X U0 U0y = U0 sin ą ą 0 Y U0x = U0 cos ą The projectile moves in the vertical XY plane containing U0. We choose the origin at the projectile departure point. In the Euclidean physical space, the movement of a body along the X and Y coordinate axes can be considered independently. The gravitational acceleration g is directed downward, so the movement along the X axis will be uniform. This means that the velocity projection Ux remains constant, equal to its value at the initial time U0x. The law of uniform projectile motion along the X axis has the form X = X0 + U0xt. Along the Y axis, the movement is uniformly variable, since the gravitational acceleration vector g is constant. The law of uniform motion along the Y axis can be represented as Y = Y0 + U0yt + ayt²/2 0, Y0 = 0; U0x = U0 cos ą, U0y = U0 sin ą. Gravity is opposite to the Y axis, so ay = -g. Substituting X0, Y0, U0x, U0y, ay, we obtain the law of ballistic motion in coordinate form: X = (U0 cos ą) t, Y = (U0 sin ą) t - gt²/2. Ballistic movement chart. Let's build a ballistic trajectory Y = X tg ą - gx²/2U²0 cos² ą Graph quadratic function is known to be a parabola. In the case under consideration, the parabola passes through the origin, since it follows from the formula that Y = 0 for X = 0. The branches of the parabola are directed downwards, since the coefficient (g / 2U²0 cos² ą) at X² is less than zero. 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 U0y onto the Y axis. In accordance with the formula tmax = U0/g obtained for a body thrown upwards with an initial velocity U0, the time for the projectile to rise to the maximum height is tmax = U0y /g = U0 siną/g. 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 along the Y axis in the same way. Y tmax = U²0/2g U0 sin ą/g Ymax tp = 2U0 ą/g U0 U0 U²0y/2g = U²0 sin² ą/2g U0y ą U0x = Ux U²0 /g sin 2ą X tp of the projectile is 2 times more than the time of its rise to the maximum height: Tp = 2tmax = 2U0 sin ą/g. Representing the flight time in the law of motion along the X axis, we obtain the maximum flight range: Xmax = U0 cos ą 2U0 sin ą/g. Since 2 sin ą cos ą = sin 2ą, then Xmax = U²0/g sin 2ą. Consequently, the flight range of a body at the same initial speed depends on the angle at which the body is thrown to the horizon. Flight range is maximum when sin 2ą is maximum. The maximum value of the sine is equal to one at an angle of 90º, i.e. Sin 2ą = 1, 2ą = 90º, ą = 45º. Y 75º 60º 45º 30º 15º 0 X Ballistic speed. To calculate the velocity U of the projectile at an arbitrary point of the trajectory, as well as to determine the angle β that forms the velocity vector with the horizontal, it is sufficient to know the projections of the velocity on the X and Y axes. If Ux and Uy are known, then by the Pythagorean theorem one can find the velocity U = √ U²x + U²y 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 = 0, it is equal to Uy = U0 sin ą. Let us find the time interval after which the projection of this velocity becomes equal to zero: 0 = U0 sin ą – gt, t = U0 sin ą/g. Y u uy = 0 u Uy β Ux U0y Uy U0 β U ą Ux ą U0x = Ux Uy Uy = - Uoy U The result obtained coincides with the time the projectile rises to the maximum height. At the top of the trajectory, the vertical velocity component is equal to zero. Ballistic motion in the atmosphere. The results obtained are valid for the idealized case, when air resistance can be neglected. The real movement of bodies in earth's atmosphere occurs along a ballistic trajectory, which differs significantly from a parabolic one due to air resistance. As the speed of the body increases, the force of air resistance increases. The greater the speed of the body, the greater the difference between the ballistic trajectory and the parabola. Y, m in vacuum in air 0 200 400 600 800 1000 X, m We only note that the calculation of the ballistic trajectory of the launch and insertion into the required orbit of Earth satellites and their landing in a given area is carried out with great accuracy by powerful computer stations. A ball thrown at an angle of 45º to the horizontal, rebounding elastically from a vertical wall, located at a distance L from the point of throw, hits the Earth at a distance ℓ from the wall. With what initial speed was the ball thrown? Problem Y 45º 0 ℓ L X Problem solution Given: ą = 45º L; ℓ U0 - ? Solution: X(T) = U0t cos ą, Y(t) = U0t sin ą - gt²/2 gT²/2. We express T from the first equation and substitute it into the second, we get: T = L + ℓ/U0 cos ą; 0 = U0 sin ą – g(L + ℓ)/2U0 cos ą; U²0 sin 2ą = g(L + ℓ); U0 = √g (L + ℓ)/sin 2ą = = √g (L + ℓ) . Answer: U0 = √g (L + ℓ) . √g (L + ℓ)/sin 2 · 45º = Test 1. A branch of mechanics that studies the motion of bodies in the Earth's gravity field. a) kinematics b) electrodynamics c) ballistics d) dynamics 2. A coin is thrown horizontally from a window of a house from a height of 19.6 m at a speed of 5 m/s. Neglecting air resistance, find the time interval after which the coin will fall to the Earth? How far horizontally from the house is the point of impact? a) 2 s; 10 m b) 5 s; 25 m c) 3 s; 15 mg d) 1 s; 5 m 3. Using the condition of problem 2, find the speed of the fall of the coin and the angle that the velocity vector forms with the horizon at the point of fall. a) 12.6 m/s; 58º b) 20.2 m/s; 78.7º c) 18 m/s; 89.9º d) 32.5 m/s; 12.7º 4. The length of a flea's jump on a table, jumping at an angle of 45º to the horizon, is 20 cm. How many times does the height of its rise above the table exceed its own length, which is 0.4 mm? a) 55.8 b) 16 c) 125 d) 159 5. At what angle to the horizon must the hunter point the barrel of the gun to hit a bird sitting at a height H on a tree located at a distance ℓ from the hunter? At the moment of the shot, the bird falls freely down to the ground. a) ą = cos (H/ℓ) b) ą = sin (H/ℓ) c) ą = ctg (H/ℓ) d) ą = arctg (H/ℓ)