To find the magnitude of the reactive force R there is no need to consider in detail the distribution of pressure along the inner and outer walls of the jet apparatus. The reactive force can be determined in its final form using the equation of momentum. While flying, the body produces disturbances in the environment. It is always possible to single out some rather large, for example, cylindrical, region, the boundaries of which go beyond the perturbed part of the flow (Fig.). At the lateral boundaries of this area, the pressure and flow velocity (we assume the engine is stationary and the air moving at the flight speed) are equal to their values at infinity in front of the engine. Let the axis NS coincides with the direction of flight and is the axis of symmetry of the engine; project on the axis NS forces acting on the engine and on the surface of the selected contour. Since the pressure forces in the fluid are normal to the surface, the projections onto the axis NS forces acting on the lateral surfaces of the contour vanish. Therefore, the Euler equation will be written as follows:

Here, the areas over which the integrals extend and the region of integration of the first term on the right-hand side are infinite. Power R is taken with the + sign because here the jet engine imparts work to the gas; - second air mass flowing into the circuit through the section F; - additional second mass of fuel supplied to the engine.

If we take the left end surface far in front of the engine, then the pressure on it is constant and equal to atmospheric (), and the flow velocity is equal to the flight speed (). In addition, it can be assumed that in the transverse direction, already at a certain finite distance from the surface of the engine, the flow is undisturbed and the area F, to which the integrals of the left-hand side extend, should be considered finite; in the same way, the region of integration in the first term of the right-hand side will also be finite. Then you should write:

In a large number of cases, the perturbation caused by a flying body is so insignificant that in the nozzle exit plane but(outside the exhaust gas jet) the pressure of the flowing stream differs little from the pressure at infinity (). Then the pressure forces on the front and rear end surfaces of the contour are balanced everywhere, except for the area corresponding to the cross section of the exhaust jet (). The flow rates in all elementary streams, except for those passing through the engine, are the same (we neglect the influence of friction, vortex and wave losses on the outer surface of the engine). Therefore, the change in momentum is obtained only in the jet flowing through the engine. Then the Euler equation takes the following form:

whence the basic formula for the reactive force is obtained

In these expressions, the average expiration rate.

It should be emphasized that the obtained relation is valid only if the velocity and pressure in the plane a (except for the section of the working jet) are exactly equal to their values at infinity in front of the engine. In addition, here we neglect the external drag of the engine, which can always be taken into account separately.

At the design operating mode of the jet engine, the pressure in the exhaust jet is equal to the ambient air pressure (); in this case, the thrust is equal to the change in the amount of motion of the gas that has passed through the engine:

In jet engines, the second term on the right-hand side is small and often neglected, i.e. taken for air-breathing engines in the calculated case

The thrust of a liquid-propellant jet engine, which does not use atmospheric air, is determined for the design mode by the formula

or off-design

Here GO is the second mass flow rate of the oxidizer.

About the place of application of the reactive force.

Let us find out where the reactive force is applied in the engine. Let's consider the simplest case - an ideal ramjet engine (fig.). Let the speed in the inlet be equal to the flight speed (); then the pressure in the inlet is equal to atmospheric (), in addition, we assume that the engine is operating at the design mode, i.e., the pressure in the outlet is also equal to atmospheric (). At a low speed of gas movement in the combustion chamber, the pressure in the latter can be considered constant (). In the described ideal engine, the pressure differences in the diffuser and nozzle are the same:

However, due to the fact that the air in the nozzle has a higher temperature than in the diffuser, the area of the motor outlet must be larger than the area of the inlet. Indeed, in an ideal engine, the velocity head at the outlet is equal to the velocity head of the incoming flow, i.e. in the case under consideration, the velocity head in the inlet

Taking this equality into account, from the continuity equation we obtain:

Therefore, when heat is supplied in the combustion chamber (), we have:

So, the average pressure acting on the walls of the diffuser and the nozzle is the same, and the projection of the diffuser wall onto the plane perpendicular to the engine axis is greater than the corresponding projection of the nozzle wall. As a result of the above, the force of pressure from the inside on the diffuser () is greater than on the nozzle (); the directions of these forces, as is evident from Fig., are opposite.

If the outer contours of the engine are very smooth, then the air pressure on the outer surface of the engine is very close to atmospheric, that is, the force of pressure on the outer surface can be neglected. In the considered ideal case, the reactive force acting on the engine is reduced to the difference in the forces applied to the diffuser and the nozzle, respectively:

The magnitudes of the forces acting on the diffuser and the nozzle are, respectively, equal

According to the above conditions

Consider an engine with low speeds in the combustion chamber, i.e., with a chamber area that is significantly larger than the area of the inlet and outlet openings:

In this case, we arrive at the following simple formula for the reactive force, determined by subtracting the force applied to the nozzle from the force applied to the diffuser:

The same result can be obtained directly from the formula for the reactive force

or, given the above condition,

So, the thrust is obtained due to the fact that the pressure force on the diffuser is greater than on the nozzle. This is a consequence of the heating of the gas, in connection with which the area of the outlet opening has to be made larger than the cross-sectional area of the incoming jet.

In a ramjet engine, the reactive force is the result of pressure forces applied to the walls of the inner and outer bypasses of the engine.

The useful part of the reactive force, equal to the difference between the reactive force and the total external resistance of the propulsion system, is called effective thrust:

The reactive force of the engine, determined by formula (105), can be considered as the difference between the output pulse of the gas jet, calculated from the excess pressure at the nozzle exit:

and the input impulse of the jet of the oncoming undisturbed air flow sucked into the engine:

The internal thrust of the engine (without taking into account external resistance) is estimated using the relative impulse (116)

The quantity

is called the lost relative momentum of the nozzle.

Completed by the student:

MOU "SOSH S. Zubovka"

Masaeva Alisat (grade 9),

Head: V.G. Melshina

2011

On the cusp of the space age

Jet propulsion

Jet engine

Jet engines and the environment

Conclusion

On the cusp of the space age

The principle of jet propulsion has been known for a long time. The ball Gerona. Solid propellant rocket motors- powder rockets appeared in China in the 10th century. n. NS. For hundreds of years, such missiles were used first in the East, and then in Europe as fireworks, signal, and combat missiles.

Segner's wheel is an engine based on the reactive action of flowing water. It was invented by the Hungarian scientist J.A. Segner in 1750. The first hydraulic turbine... Horizontal wheel without a rim, in which the spokes are replaced by tubes with bent ends so that the resulting water drives the Segner wheel into rotation.

The idea of rocket flying, which seems so bold and new to many today, actually has a half-century history behind it, a good three-quarters of which has taken place entirely in our fatherland.

The first thought about a rocket plane was born in the bright head of the young revolutionary, First March, Nikolai Ivanovich Kibalchich.

In 1903 K.E. Tsiolkovsky in his work "Exploration of world spaces by reactive devices" for the first time in the world put forward the main provisions of the theory liquid propellant rocket engines and proposed the main elements of a liquid-fueled taxiway device.

Jet propulsion

The law of conservation of momentum in many cases makes it possible to find the velocities of interacting bodies even when the values of the acting forces are unknown. An example is jet propulsion. When firing from a gun, recoil occurs - the projectile moves forward, and the gun rolls back. The projectile and the weapon are two interacting bodies. The speed that the weapon gains upon recoil depends only on the speed of the projectile and the mass ratio

Reactive motion is understood as the movement of a body that occurs when a certain part of it is separated at a certain speed relative to the body,

for example, when the combustion products flow out from the nozzle of a jet aircraft. In this case, the so-called reactive force appears, imparting acceleration to the body.

Observing jet propulsion is very simple. Inflate the baby's rubber ball and release it. The ball will skyrocket upward. The movement, however, will be short-lived. The reactive force acts only as long as the flow of air continues.

Meshchersky and Tsiolkovsky equations

If there are no external forces, then the rocket, together with the ejected material, is closed system. Pulse such a system cannot change over time.

, where

- weight rockets

Her acceleration

- speed gas outflow

Mass consumption fuel per unit time

Since the flow rate of combustion products (working fluid) is determined by the physicochemical properties of the fuel components and the design features of the engine, being a constant value for not very large changes in the operating mode of the jet engine, the magnitude of the reactive force is determined mainly mass second fuel consumption.

Proof

Before starting work engines pulse rockets and fuel was equal to zero, therefore, after switching on, the sum of changes in vectors impulse rockets and impulse outflowing gases is equal to zero:, where

Change speed rockets

We divide both sides of the equality into the interval time t, during which rocket engines:

Work masses rockets m to acceleration its motion a is by definition equal to strength causing this acceleration:

Meshchersky's equation

If on rocket, except reactive force, the external power, then the equation of the dynamics of motion will take the form:

Formula Meshchersky is a generalization Newton's second law for motion of bodies of variable mass. Acceleration body of variable mass is determined not only by external forces acting on the body, but also by the reactive force due to the change in the mass of the moving body:

Tsiolkovsky's formula

Applying Meshchersky equation to the movement rockets, on which external forces do not act, and, integrating the equation, we obtain Tsiolkovsky's formula

Relativistic a generalization of this formula is:

, where - speed of light.

Conclusions from the laws:

Let's analyze the resulting expression. We see that the speed of the rocket is the greater, the greater the speed of the ejected gases and the greater the ratio of the mass of the working body (ie, the mass of the fuel) to the final ("dry") mass of the rocket.

Meshchersky's formula is approximate. It does not take into account that as the fuel burns, the mass of the flying rocket becomes less and less. The exact formula for the rocket speed was first obtained in 1897 by K.E. Tsiolkovsky and therefore bears his name.

To impart a speed to the rocket that exceeds the speed of the outflow of gases by 4 times (V p = 16 km / s), it is necessary that the initial mass of the rocket (together with the fuel) exceeds the final ("dry") mass of the rocket 55 times (m 0 / m = 55). This means that the lion's share of the entire mass of the rocket at the start should be precisely the mass of the fuel. The payload, in comparison with it, should have a very low mass.

A significant reduction in the launch mass of a rocket can be achieved by using multistage rockets, where the rocket stages separate as the fuel burns out. The process of the subsequent acceleration of the rocket excludes the masses of containers in which there was fuel, spent engines, control systems, etc. It is along the path of creating economical multistage missiles that modern rocketry is developing.

Jet engine... Jet engine classes

Jet engine- engine-mover, which creates the traction force necessary for movement by means of transformation potential energy fuel in kinetic energy jet stream racombat body.

Components of a jet engine:

The combustion chamber("Chemical reactor") - chemical energy is released in it fuel and its transformation into thermal energy gases.

Jet nozzle("Gas tunnel") - in which the thermal energy of gases passes into their kinetic energy, when gases flow out of the nozzle at a high speed, thereby creating jet thrust.

Jet engines are divided into two classes:

Rocket

Air-jet

In rocket engines, the fuel and the oxidizer necessary for its combustion are located directly inside the engine or in its fuel tanks.

Solid fuel rocket engines

The figure shows a diagram of a solid propellant rocket engine. Gunpowder or some other solid fuel capable of burning in the absence of air is placed inside the combustion chamber of the engine.

Reactive force

When the fuel burns, gases are formed that have a very high temperature and exert pressure on the walls of the chamber. The force of pressure on the front wall of the chamber is greater than on the back, where the nozzle is located. The gases flowing out through the nozzle do not encounter a wall on their way on which they could exert pressure. The result is a force that propels the rocket forward.

The narrowed part of the chamber - the nozzle serves to increase the speed of the outflow of combustion products, which in turn increases the reactive force. The narrowing of the gas jet causes an increase in its velocity, since in this case the same gas mass must pass through the smaller cross section per unit time as with the larger cross section.

Diagram of a turbocharged type air jet engine.

The hot gases (combustion products) exiting through the nozzle rotate the gas turbine that drives the compressor. Turbocharger engines are installed in our Tu-134, Il-62, Il-86 and others. Jet engines are used not only for rockets, but also for most modern aircraft.

The first Soviet liquid-propellant rocket engines - ORM, ORM-1, ORM-2 were designed by V.P. Glushko and under his leadership were created in 1930-31 in Gas dynamic laboratory... For the first time, an electrothermal engine was created and tested by Glushko at the GDL in 1929-1933. In 1939, the USSR tested rockets with ramjet engines designed by I.A.Merkulov.

Nuclear rocket motors

Nuclear rocket engines make it possible to achieve a significantly higher (in comparison with chemical rocket engines) specific impulse value due to the high velocity of the working fluid outflow (from 8,000 m / s to 50 km / s and more). At the same time, the total thrust of the NRE can be comparable to the thrust of chemical rocket engines, which creates the preconditions for replacing chemical rocket engines with nuclear ones in the future. The main problem when using NRE is radioactive contamination of the environment by the engine exhaust plume, which makes it difficult to use NRE (except, possibly, gas-phase) on the stages of launch vehicles operating within the Earth's atmosphere. However, a constructively perfect GFNRD, based on its calculated thrust characteristics, can easily solve the problem of creating a fully reusable single-stage launch vehicle.

Application of jet engines

Most military and civil aircraft all over the world are equipped with turbojet engines and bypass turbojet engines, they are used in helicopters. These rocket launchers are suitable for flights at both subsonic and supersonic speeds; they are also installed on projectile aircraft; supersonic turbojet engines can be used in the first stages of aerospace aircraft. The ramjet engines are installed on anti-aircraft guided missiles, cruise missiles, and supersonic interceptor fighters. Subsonic ramjet engines are used in helicopters (installed at the ends of the rotor blades). Pulsating jet engines have low thrust and are intended only for subsonic aircraft. During World War II, 1939-45, these engines were equipped with FAU-1 projectile aircraft.

Liquid propellant rocket engines are used on spacecraft launch vehicles and spacecraft in

as sustainer, brake and control engines, as well as on guided ballistic missiles. Solid-propellant rocket engines are used in ballistic, anti-aircraft, anti-tank, and other military missiles, as well as in launch vehicles and spacecraft. Small solid-fuel engines are used as accelerators for aircraft takeoff. Electric rocket engines and nuclear rocket engines can be used in spacecraft.

Squid jet engine

The squid is the largest invertebrate inhabitant of the ocean depths. Of greatest interest is the squid jet engine. When moving slowly, the squid uses a large diamond-shaped fin that bends periodically. He uses a jet engine for a quick throw. The animal sucks water into the mantle cavity, and then abruptly throws out a stream of water through a narrow nozzle. This nozzle is equipped with a special valve, and muscles can turn it, changing the direction of movement. In this case, all ten tentacles of the squid gather in a knot above the head, and it takes on a streamlined shape.

The squid engine is very economical, it is capable of speeds up to 60 - 70 km / h. (Some researchers believe that even up to 150 km / h!) No wonder the squid is called a “live torpedo”. Engineers have already created an engine similar to that of a squid. It is called a water cannon. In it, water is sucked into the chamber. And then it is thrown out of it through the nozzle; the ship is moving in the direction opposite to the direction of the jet ejection. The water is sucked in using a conventional gasoline or diesel engine.

Salpa is a sea animal with a transparent body, when it moves, it receives water through the front opening, and the water enters a wide cavity, inside which the gills are stretched diagonally. As soon as the animal takes a long sip of water, the hole closes. Then the longitudinal and transverse muscles of the salpa contract, the whole body contracts, and water is pushed out through the posterior opening. The reaction of the flowing jet pushes the salpa forward.

Dragonfly larva

The hind gut of the dragonfly larva, in addition to its main function, also plays the role of an organ of movement. Water fills the hind gut, then it is thrown out with force, and the larva moves according to the principle of jet propulsion by 6-8 cm. The hind gut also serves the nymphs to breathe, which, like a pump, constantly pumps oxygen-rich water through the anus.

Bilimovich B.F. "Physics quizzes"

Squirting cucumber

Examples of jet propulsion can also be found in the plant world.

In southern countries (and on our Black Sea coast too) a plant called "mad cucumber" grows. One has only to lightly touch the ripe fruit, similar to a cucumber, as it rebounds from the stalk, and through the hole formed from the fruit, a liquid with seeds flies out with a fountain at a speed of up to 10 m / s.

The cucumbers themselves fly off in the opposite direction. A mad cucumber (otherwise it is called a "lady's pistol") shoots more than 12 m.

HOME EXPERIENCE

"Reactive bank"

Take an empty tin can with no top lid. At equal distances along the top rim of the can, make three small holes and insert strong threads into them, with which you can hang the can from the water tap. At the bottom on the side wall of the jar, make a couple of holes opposite each other with a diameter of about 5 cm. Hang the jar on the water tap and turn on the tap with water to fill the jar.

Environment

Heat engines (including jet engines) are a necessary attribute of modern civilization. With their help, ≈ 80% of electricity is generated. It is impossible to imagine modern transport without heat engines. At the same time, the widespread use of heat engines is associated with a negative impact on the environment.

Fuel combustion is accompanied by the release of carbon dioxide into the atmosphere, which is capable of absorbing thermal infrared (IR) radiation from the Earth's surface. An increase in the concentration of carbon dioxide in the atmosphere, increasing the absorption of infrared radiation, leads to an increase in its temperature (greenhouse effect). Every year the temperature of the Earth's atmosphere rises by 0.05 єС. This effect can create a threat of melting glaciers and a catastrophic rise in the level of the World Ocean.

Hydrocarbons reacting with ozone in the atmosphere form chemical compounds that adversely affect the life of plants, animals and humans.

Oxygen consumption during fuel combustion reduces its content in the atmosphere.

For environmental protection, it widely uses treatment facilities that prevent the release of harmful substances into the atmosphere, sharply limit the use of heavy metal compounds added to the fuel.

Conclusion:

The basis of jet propulsion is the law of conservation of momentum of a body, which is fulfilled only for a closed system of bodies.

The speed of movement of the jet device is the greater, the greater the mass of the substance is separated from the body in 1 s.

The simplest models of jet engines and devices can be made by ourselves.

A manifestation of jet propulsion is recoil, which must be taken into account in practice (when shooting, jumping off a boat, skateboard, etc.).

The result of recoil depends on the mass and speed of the separating body or substance.

Reactive motion has found wide application in technology

Literature

http :// class - fizika . narod . ru /9_19. htm

A.A. Kosmodemyanskiy Tsiolkovsky K.E. (M., "Science", 1976)

Arlazorov A. Tsiolkovsky K.E. (M., "Young Guard", 1963)

Myakishev G.Ya. Physics: [Text]: textbook for the 10th grade of educational institutions / G.Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky. - 11th ed. - M .: Education, 2003 .-- 306 p.

G.S.Lansberg Elementary textbook of physics [Text]: G.S.Lansberg, - M .: Nauka, 1985 - 460 p.

Kirik L.A. Physics-9: [Text]: Multilevel independent and control works. - Kharkov: Gymnasium, 2001 .-- 160 p.

Complete course of physics of the XXI century [Electronic resource]: Computer program for the study of physics. - Access mode:

Newton's second law \ (~ m \ vec a = \ vec F \) can be written in a different form, which is given by Newton himself in his main work "Mathematical Principles of Natural Philosophy".

If a constant force acts on a body (material point), then acceleration is also constant

\ (~ \ vec a = \ frac (\ vec \ upsilon_2 - \ vec \ upsilon_1) (\ Delta t) \),

where \ (~ \ vec \ upsilon_1 \) and \ (~ \ vec \ upsilon_2 \) are the initial and final values of the body's velocity.

Substituting this acceleration value into Newton's second law, we get:

\ (~ \ frac (m \ cdot (\ vec \ upsilon_2 - \ vec \ upsilon_1)) (\ Delta t) = \ vec F \) or \ (~ m \ vec \ upsilon_2 - m \ vec \ upsilon_1 = \ vec F \ Delta t \). (one)

A new physical quantity appears in this equation - the momentum of a material point.

The impulse of the material points are called a value equal to the product of the mass of a point by its speed.

Let us denote momentum (sometimes also called momentum) by the letter \ (~ \ vec p \). Then

\ (~ \ vec p = m \ vec \ upsilon \). (2)

It can be seen from formula (2) that momentum is a vector quantity. As m> 0, then the impulse has the same direction as the velocity.

The unit of momentum has no specific name. Its name is derived from the definition of this quantity:

[p] = [m] · [ υ ] = 1 kg · 1 m / s = 1 kg · m / s.

Another form of writing Newton's second law

We denote by \ (~ \ vec p_1 = m \ vec \ upsilon_1 \) the momentum of a material point at the initial moment of the interval Δ t, and after \ (~ \ vec p_2 = m \ vec \ upsilon_2 \) - the impulse at the end of this interval. Then \ (~ \ vec p_2 - \ vec p_1 = \ Delta \ vec p \) is change in momentum in time Δ t... Now equation (1) can be written as follows:

\ (~ \ Delta \ vec p = \ vec F \ Delta t \). (3)

Since Δ t> 0, then the directions of the vectors \ (~ \ Delta \ vec p \) and \ (~ \ vec F \) coincide.

According to formula (3)

the change in the momentum of a material point is proportional to the force applied to it and has the same direction as the force.

This is how it was first formulated Newton's second law.

The product of force by the time of its action is called impulse of power... Do not confuse momentum \ (~ m \ vec \ upsilon \) of a material point and impulse of force \ (\ vec F \ Delta t \). These are completely different concepts.

Equation (3) shows that the same changes in the momentum of a material point can be obtained as a result of the action of a large force during a short time interval or a small force over a long time interval. When you jump from a certain height, then the stop of your body occurs due to the action of force from the side of the ground or floor. The shorter the duration of the collision, the greater the braking force. To reduce this force, it is necessary that braking occurs gradually. This is why when jumping high, athletes land on soft mats. Sagging, they gradually slow down the athlete. Formula (3) can be generalized to the case when the force changes over time. For this, the entire time interval Δ t the action of the force must be divided into such small intervals Δ t i, so that on each of them the value of the force can be considered constant without a large error. For each small time interval, formula (3) is valid. Summing up the changes in impulses over small time intervals, we get:

\ (~ \ Delta \ vec p = \ sum ^ (N) _ (i = 1) (\ vec F_i \ Delta t_i) \). (4)

The symbol Σ (Greek sigma) stands for sum. Indexes i= 1 (bottom) and N(top) means that it is summed N terms.

To find the impulse of the body, they do the following: mentally break the body into separate elements (material points), find the impulses of the received elements, and then sum them up as vectors.

The momentum of a body is equal to the sum of the impulses of its individual elements.

Changing the impulse of the system of bodies. Momentum conservation law

When considering any mechanical problem, we are interested in the motion of a certain number of bodies. The set of bodies, the motion of which we study, is called mechanical system or just a system.

Changing the momentum of a system of bodies

Consider a three-body system. These can be three stars that are affected by neighboring cosmic bodies. External forces act on the bodies of the system \ (~ \ vec F_i \) ( i- body number; for example, \ (~ \ vec F_2 \) is the sum of external forces acting on body number two). Forces \ (~ \ vec F_ (ik) \), called internal forces, act between the bodies (Fig. 1). Here is the first letter i in the index means the number of the body on which the force \ (~ \ vec F_ (ik) \) acts, and the second letter k means the number of the body from which the given force acts. Based on Newton's third law

\ (~ \ vec F_ (ik) = - \ vec F_ (ki) \). (five)

Due to the action of forces on the bodies of the system, their impulses change. If for a short period of time the force does not change noticeably, then for each body of the system it is possible to write down the change in momentum in the form of equation (3):

\ (~ \ Delta (m_1 \ vec \ upsilon_1) = (\ vec F_ (12) + \ vec F_ (13) + \ vec F_1) \ Delta t \), \ (~ \ Delta (m_2 \ vec \ upsilon_2) = (\ vec F_ (21) + \ vec F_ (23) + \ vec F_2) \ Delta t \), (6) \ (~ \ Delta (m_3 \ vec \ upsilon_3) = (\ vec F_ (31) + \ vec F_ (32) + \ vec F_3) \ Delta t \).

Here, on the left side of each equation, there is a change in the momentum of the body \ (~ \ vec p_i = m_i \ vec \ upsilon_i \) in a short time Δ t... More details \ [~ \ Delta (m_i \ vec \ upsilon_i) = m_i \ vec \ upsilon_ (ik) - m_i \ vec \ upsilon_ (in) \] where \ (~ \ vec \ upsilon_ (in) \) - speed in beginning, and \ (~ \ vec \ upsilon_ (ik) \) - at the end of the time interval Δ t.

Let us add the left and right sides of equations (6) and show that the sum of the changes in the momenta of individual bodies is equal to the change in the total momentum of all bodies in the system, which is equal to

\ (~ \ vec p_c = m_1 \ vec \ upsilon_1 + m_2 \ vec \ upsilon_2 + m_3 \ vec \ upsilon_3 \). (7)

Really,

\ (~ \ Delta (m_1 \ vec \ upsilon_1) + \ Delta (m_2 \ vec \ upsilon_2) + \ Delta (m_3 \ vec \ upsilon_3) = m_1 \ vec \ upsilon_ (1k) - m_1 \ vec \ upsilon_ (1n) + m_2 \ vec \ upsilon_ (2k) - m_2 \ vec \ upsilon_ (2n) + m_3 \ vec \ upsilon_ (3k) - m_3 \ vec \ upsilon_ (3n) = \) \ (~ = (m_1 \ vec \ upsilon_ ( 1k) + m_2 \ vec \ upsilon_ (2k) + m_3 \ vec \ upsilon_ (3k)) - (m_1 \ vec \ upsilon_ (1n) + m_2 \ vec \ upsilon_ (2n) + m_3 \ vec \ upsilon_ (3n)) = \ vec p_ (ck) - \ vec p_ (cn) = \ Delta \ vec p_c \).

Thus,

\ (~ \ Delta \ vec p_c = (\ vec F_ (12) + \ vec F_ (13) + \ vec F_ (21) + \ vec F_ (23) + \ vec F_ (31) + \ vec F_ (32 ) + \ vec F_1 + \ vec F_2 + \ vec F_3) \ Delta t \). (eight)

But the forces of interaction of any pair of bodies add up to zero, since according to formula (5)

\ (~ \ vec F_ (12) = - \ vec F_ (21); \ vec F_ (13) = - \ vec F_ (31); \ vec F_ (23) = - \ vec F_ (32) \).

Therefore, the change in the momentum of the system of bodies is equal to the momentum of external forces:

\ (~ \ Delta \ vec p_c = (\ vec F_1 + \ vec F_2 + \ vec F_3) \ Delta t \). (nine)

We came to an important conclusion:

the momentum of a system of bodies can only be changed by external forces, and the change in the momentum of the system is proportional to the sum of external forces and coincides with it in direction. Internal forces, changing the impulses of individual bodies of the system, do not change the total impulse of the system.

Equation (9) is valid for any time interval if the sum of external forces remains constant.

Momentum conservation law

An extremely important consequence follows from equation (9). If the sum of the external forces acting on the system is equal to zero, then the change in the momentum of the system is also equal to zero \ [~ \ Delta \ vec p_c = 0 \]. This means that no matter what time interval we take, the total impulse at the beginning of this interval \ (~ \ vec p_ (cn) \) and at its end \ (~ \ vec p_ (ck) \) is the same \ [~ \ vec p_ (cn) = \ vec p_ (ck) \]. The momentum of the system remains unchanged, or, as they say, persists:

\ (~ \ vec p_c = m_1 \ vec \ upsilon_1 + m_2 \ vec \ upsilon_2 + m_3 \ vec \ upsilon_3 = \ operatorname (const) \). (10)

Momentum conservation law is formulated as follows:

if the sum of external forces acting on the bodies of the system is equal to zero, then the momentum of the system is conserved.

The bodies can only exchange impulses, the total value of the impulse does not change. It is only necessary to remember that the vector sum of the impulses is saved, and not the sum of their modules.

As can be seen from our conclusion, the law of conservation of momentum is a consequence of Newton's second and third laws. A system of bodies that is not acted upon by external forces is called closed or isolated. In a closed system of bodies, momentum is conserved. But the field of application of the law of conservation of momentum is wider: even if external forces act on the bodies of the system, but their sum is equal to zero, the momentum of the system is still conserved.

The result obtained can be easily generalized to the case of a system containing an arbitrary number N of bodies:

\ (~ m_1 \ vec \ upsilon_ (1n) + m_2 \ vec \ upsilon_ (2n) + m_3 \ vec \ upsilon_ (3n) + \ ldots + m_N \ vec \ upsilon_ (Nn) = m_1 \ vec \ upsilon_ (1k) + m_2 \ vec \ upsilon_ (2k) + m_3 \ vec \ upsilon_ (3k) + \ ldots + m_N \ vec \ upsilon_ (Nk) \). (eleven)

Here \ (~ \ vec \ upsilon_ (in) \) are the velocities of bodies at the initial moment of time, and \ (~ \ vec \ upsilon_ (ik) \) - at the final one. Since the momentum is a vector quantity, equation (11) is a compact record of three equations for the projections of the momentum of the system on the coordinate axes.

When is the momentum conservation law satisfied?

All real systems, of course, are not closed, the sum of external forces can rarely be equal to zero. Nevertheless, in very many cases the law of conservation of momentum can be applied.

If the sum of the external forces is not zero, but the sum of the projections of the forces on some direction is equal to zero, then the projection of the momentum of the system on this direction is preserved. For example, a system of bodies on the Earth or near its surface cannot be closed, since gravity acts on all bodies, which changes the vertical momentum according to equation (9). However, along the horizontal direction, the force of gravity cannot change the momentum, and the sum of the projections of the impulses of the bodies on the horizontally directed axis will remain unchanged if the action of the resistance forces can be neglected.

In addition, during fast interactions (explosion of a projectile, a shot from a weapon, collisions of atoms, etc.), the change in the momenta of individual bodies will actually be caused only by internal forces. In this case, the momentum of the system is preserved with great accuracy, because such external forces as the force of gravity and the force of friction, which depends on the speed, do not noticeably change the momentum of the system. They are small compared to the internal forces. So, the speed of projectile fragments during an explosion, depending on the caliber, can vary within 600 - 1000 m / s. The time interval for which the force of gravity could impart such a velocity to bodies is

\ (~ \ Delta t = \ frac (m \ Delta \ upsilon) (mg) \ approx 100 c \)

The internal forces of gas pressure impart such velocities in 0.01 s, i.e. 10,000 times faster.

Jet propulsion. Meshchersky's equation. Reactive force

Under jet propulsion understand the movement of a body that occurs when some part of it is separated at a certain speed relative to the body,

for example, when the combustion products flow out from the nozzle of a jet aircraft. In this case, the so-called reactive force appears, imparting acceleration to the body.

Observing jet propulsion is very simple. Inflate the baby's rubber ball and release it. The ball will rapidly rise upward (Fig. 2). The movement, however, will be short-lived. The reactive force acts only as long as the flow of air continues.

The main feature of the reactive force is that it arises without any interaction with external bodies. There is only interaction between the rocket and the stream of matter flowing out of it.

The force that imparts acceleration to a car or a pedestrian on the ground, a steamer on water or a propeller plane in the air, arises only due to the interaction of these bodies with the earth, water or air.

When the products of fuel combustion flow out, due to the pressure in the combustion chamber, they acquire a certain speed relative to the rocket and, therefore, a certain momentum. Therefore, in accordance with the law of conservation of momentum, the rocket itself receives the same pulse in modulus, but directed in the opposite direction.

The mass of the rocket decreases over time. A rocket in flight is a body of variable mass. To calculate its motion, it is convenient to apply the law of conservation of momentum.

Meshchersky's equation

Let's derive the equation of motion of the rocket and find an expression for the reactive force. We will assume that the velocity of gases flowing out of the rocket relative to the rocket is constant and equal to \ (~ \ vec u \). External forces do not act on the rocket: it is in outer space far from stars and planets.

Let at some point in time the rocket speed relative to the inertial system associated with the stars is \ (~ \ vec \ upsilon \) (Fig. 3), and the rocket mass is M... After a short time interval Δ t the mass of the rocket will be equal

\ (~ M_1 = M - \ mu \ Delta t \),

where μ - fuel consumption ( fuel consumption is called the ratio of the mass of the burned fuel to the time of its combustion).

During the same period of time, the rocket speed will change to \ (~ \ Delta \ vec \ upsilon \) and become equal to \ (~ \ vec \ upsilon_1 = \ vec \ upsilon + \ Delta \ vec \ upsilon \). The gas outflow velocity relative to the selected inertial reference system is \ (~ \ vec \ upsilon + \ vec u \) (Fig. 4), since before combustion the fuel had the same velocity as the rocket.

Let us write the law of conservation of momentum for the rocket - gas system:

\ (~ M \ vec \ upsilon = (M - \ mu \ Delta t) (\ vec \ upsilon + \ Delta \ vec \ upsilon) + \ mu \ Delta t (\ vec \ upsilon + \ vec u) \).

Expanding the brackets, we get:

\ (~ M \ vec \ upsilon = M \ vec \ upsilon - \ mu \ Delta t \ vec \ upsilon + M \ Delta \ vec \ upsilon - \ mu \ Delta t \ Delta \ vec \ upsilon + \ mu \ Delta t \ vec \ upsilon + \ mu \ Delta t \ vec u \).

The term \ (~ \ mu \ Delta t \ vec \ upsilon \) can be neglected in comparison with the others, since it contains the product of two small quantities (this is a quantity, as they say, of the second order of smallness). After bringing similar terms, we will have:

\ (~ M \ Delta \ vec \ upsilon = - \ mu \ Delta t \ vec u \) or \ (~ M \ frac (\ Delta \ vec \ upsilon) (\ Delta t) = - \ mu \ vec u \ ). (12)

This is one of Meshchersky's equations for the motion of a body of variable mass, obtained by him in 1897.

If we enter the notation \ (~ \ vec F_r = - \ mu \ vec u \), then equation (12) coincides in the form of notation with Newton's second law. However, body weight M here it is not constant, but decreases with time due to the loss of matter.

The value \ (~ \ vec F_r = - \ mu \ vec u \) is called reactive force... It appears due to the outflow of gases from the rocket, is applied to the rocket and is directed opposite to the speed of the gases relative to the rocket. The reactive force is determined only by the rate of outflow of gases relative to the rocket and the fuel consumption. It is essential that it does not depend on the details of the engine device. It is only important that the engine provides the outflow of gases from the rocket at a speed \ (~ \ vec u \) with a fuel consumption μ ... The reactive force of space rockets reaches 1000 kN.

If external forces act on the rocket, then its movement is determined by the reactive force and the sum of the external forces. In this case, equation (12) will be written as follows:

\ (~ M \ frac (\ Delta \ vec \ upsilon) (\ Delta t) = \ vec F_r + \ vec F \). (13)

Jet engines

Jet engines are now widely used in connection with the exploration of outer space. They are also used for meteorological and military rockets of various ranges. In addition, all modern high-speed aircraft are powered by jet engines.

In outer space, it is impossible to use any other engines besides jet ones: there is no support (solid, liquid or gaseous), pushing off from which the spacecraft could get acceleration. The use of jet engines for airplanes and rockets that do not leave the atmosphere is due to the fact that it is jet engines that are capable of providing the maximum flight speed.

Jet engines are divided into two classes: missile and air-jet.

In rocket engines, the fuel and the oxidizer necessary for its combustion are located directly inside the engine or in its fuel tanks.

Figure 5 shows a schematic of a solid propellant rocket engine. Gunpowder or some other solid fuel capable of burning in the absence of air is placed inside the combustion chamber of the engine.

When the fuel burns, gases are formed that have a very high temperature and exert pressure on the walls of the chamber. The force of pressure on the front wall of the chamber is greater than on the back, where the nozzle is located. The gases flowing out through the nozzle do not encounter a wall on their way, on which they could exert pressure. The result is a force that propels the rocket forward.

Liquid propellant rocket engines are also used.

In liquid jet engines (LRE), kerosene, gasoline, alcohol, aniline, liquid hydrogen, etc. can be used as fuel, and liquid oxygen, nitric acid, liquid fluorine, hydrogen peroxide, etc. can be used as an oxidizing agent required for combustion. The fuel and the oxidizer are stored separately in special tanks and are pumped into the chamber, where, during fuel combustion, a temperature of up to 3000 ° C and a pressure of up to 50 atm develop (Fig. 6). Otherwise, the engine operates in the same way as a solid fuel engine.

The hot gases (combustion products) exiting through the nozzle rotate the gas turbine that drives the compressor. Turbocharger engines are installed in our Tu-134, Il-62, Il-86, etc.

Jet engines are equipped not only with rockets, but also with most modern aircraft.

Advances in space exploration

The foundations of the theory of a jet engine and scientific proof of the possibility of flights in interplanetary space were first expressed and developed by the Russian scientist K.E. Tsiolkovsky in his work "Exploration of world spaces by jet devices".

K.E. Tsiolkovsky also owns the idea of using multistage rockets. The individual stages that make up the rocket are supplied with their own engines and fuel reserves. As the fuel burns out, each successive stage is separated from the rocket. Therefore, in the future, no fuel is consumed to accelerate its body and engine.

Tsiolkovsky's idea of building a large satellite station in orbit around the Earth, from which rockets will be launched to other planets of the solar system, has not yet been implemented, but there is no doubt that sooner or later such a station will be created.

At present, the prophecy of Tsiolkovsky is becoming a reality: "Humanity will not remain forever on Earth, but in the pursuit of light and space, it will first timidly penetrate beyond the atmosphere, and then conquer the entire solar space."

Our country has the great honor of launching the first artificial Earth satellite on October 4, 1957. Also for the first time in our country on April 12, 1961, a spacecraft flight with cosmonaut Yu.A. Gagarin on board.

These flights were carried out on rockets designed by Russian scientists and engineers under the leadership of S.P. Queen. American scientists, engineers and astronauts are of great service in space exploration. Two American astronauts from the crew of the Apollo 11 spacecraft - Neil Armstrong and Edwin Aldrin - landed on the moon for the first time on July 20, 1969. The first steps were taken by man on the cosmic body of the solar system.

With the advent of man into space, not only the possibilities of exploring other planets were opened up, but truly fantastic opportunities for studying the natural phenomena and resources of the Earth were presented, which could only be dreamed of. Space science arose. Previously, the general map of the Earth was compiled bit by bit, like a mosaic panel. Now images from orbit, covering millions of square kilometers, allow you to select the most interesting areas of the earth's surface for research, thereby saving forces and resources. From space, it was possible to discover a new type of geological formations, ring structures similar to the craters of the Moon and Mars,

Now on orbital complexes have developed technologies for obtaining materials that cannot be produced on Earth, but only in a state of prolonged weightlessness in space. The cost of these materials (ultrapure single crystals, etc.) is close to the cost of launching spacecraft.

Literature

Physics: Mechanics. 10th grade: Textbook. for in-depth study of physics / M.M. Balashov, A.I. Gomonova, A.B. Dolitsky and others; Ed. G.Ya. Myakisheva. - M .: Bustard, 2002 .-- 496 p.

Any problem in mechanics can be solved using Newton's laws. However, the application of the law of conservation of momentum in many cases greatly simplifies the solution. The law of conservation of momentum is of great importance for the study of jet propulsion.

What kind of movement is called reactive?

Reactive motion is understood as the movement of a body that occurs when some part of it is separated at a certain speed relative to the body, for example, when combustion products flow out of the nozzle of a jet aircraft. In this case, the so-called reactive force appears, imparting acceleration to the body.

Observing jet propulsion is very simple. Inflate the baby's rubber ball and release it. The ball will skyrocket upward (Fig. 5.4). The movement, however, will be short-lived. The reactive force acts only as long as the flow of air continues.

Rice. 5.4

The main feature of the reactive force is that it arises without any interaction with external bodies. There is only interaction between the rocket and the stream of matter flowing out of it.

The mass of the rocket decreases over time. A rocket in flight is a body of variable mass. To calculate its motion, it is convenient to apply the law of conservation of momentum.

Meshchersky's equation

Let's derive the equation of motion of the rocket and find an expression for the reactive force. We will assume that the velocity of gases flowing out of the rocket relative to the rocket is constant and equal. External forces do not act on the rocket: it is in outer space far from stars and planets.

Let at some point in time the velocity of the rocket relative to the inertial system associated with the stars is (Fig.5.5, a), and the mass of the rocket is M. After a short time interval Δt, the mass of the rocket will become

where μ is the fuel consumption (1).

Rice. 5.5

During this time interval, the rocket speed will change by Δ and become equal to 1 = + Δ. The speed of the outflow of gases relative to the selected inertial reference system is + (Fig. 5.5, b), since before the start of combustion, the fuel had the same speed as the rocket.

Let us write the law of conservation of momentum for the rocket - gas system:

Expanding the brackets, we get:

The term μΔtΔ can be neglected in comparison with the others, since it contains the product of two small quantities (this is a quantity, as they say, of the second order of smallness). After bringing similar terms, we will have:

This is one of Meshchersky's equations (2) for the motion of a body of variable mass, obtained by him in 1897.

If we introduce the designation p = -μ, then the equation (5.4.1) coincides in the form of notation with Newton's second law. However, the body mass M is not constant here, but decreases with time due to the loss of matter.

The value p = -μ is called reactive force. It appears due to the outflow of gases from the rocket, is attached to the rocket and is directed opposite to the speed of the gases relative to the rocket. The reactive force is determined only by the rate of outflow of gases relative to the rocket and the fuel consumption. It is essential that it does not depend on the details of the engine device. It is only important that the engine provides the outflow of gases from the rocket at a speed with a fuel consumption μ. The reactive force of space rockets reaches 1000 kN.

If external forces act on the rocket, then its movement is determined by the reactive force and the sum of the external forces. In this case, equation (5.4.1) will be written as follows:

The principle of jet propulsion is based on the fact that gases flowing out of a jet engine receive an impulse. The rocket acquires the same modulus of impulse.

Self-test questions

(1) Fuel consumption is the ratio of the mass of the burned fuel to the time of its combustion.

(2) Meshchersky I.V. (1859-1935) - professor at the St. Petersburg Polytechnic Institute. His works on the mechanics of bodies of variable mass became the theoretical basis for rocketry.

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Traction force can be determined through the net power, and the vehicle speed (v):

For a car going up a hill, which has a slope, the mass of the car m, the traction force (F T) will enter the equation:

where a is the acceleration with which the car is moving.

Traction units

The basic unit of measure for force in the SI system is: = N

In the SGS: = din

Traction force formula

In the event that the body has an acceleration when moving, then, among all others, a certain force necessarily acts on it, which is the traction force at the moment in time. In fact, if the body is moving in a straight line and at a constant speed, then the traction force also acts, since the body must overcome resistance forces. Usually, the thrust force is found by considering the forces acting on the body, finding the resultant and applying Newton's second law. There is no hard-coded formula for the thrust force.

It should not be assumed that the traction force, for example, of a vehicle acts from the side of the engine, since internal forces cannot change the speed of the system as a whole, which would contradict the law of conservation of momentum. However, it should be noted that in order to obtain the required direction from the static friction force, the motor rotates the wheels, the wheels "cling to the road" and a traction force is generated. Theoretically, it would be possible not to use the concept of "pulling force", but to talk about the force of friction at rest or the force of reaction of the air. But it is more convenient to divide the external forces that act on the transport into two parts, while some forces are called traction forces, and others - resistance forces. This is done so that the equations of motion do not lose their universal form and the useful mechanical power (P) has a simple expression:

Examples of problem solving

Example

The task. A car with a mass of 1 ton when it moves on a horizontal surface is affected by a friction force that is equal to = 0.1 of the force of gravity. What will be the traction force if the car is moving with an acceleration of 2 m / s?

Solution. Let's make a drawing.