Law of conservation of momentum. Jet propulsion. Study of the law of conservation of momentum

His movements, i.e. size .

Pulse is a vector quantity coinciding in direction with the velocity vector.

SI unit of impulse: kg m/s .

The momentum of a system of bodies is equal to the vector sum of the momentum of all bodies included in the system:

Law of conservation of momentum

If a system of interacting bodies is additionally acted upon by external forces, for example, then in this case the relation is valid, which is sometimes called the law of momentum change:

For a closed system (in the absence of external forces), the law of conservation of momentum is valid:

The action of the law of conservation of momentum can explain the phenomenon of recoil when shooting from a rifle or during artillery shooting. Also, the law of conservation of momentum underlies the operating principle of all jet engines.

When solving physical problems, the law of conservation of momentum is used when knowledge of all the details of the movement is not required, but the result of the interaction of bodies is important. Such problems, for example, are problems about the impact or collision of bodies. The law of conservation of momentum is used when considering the motion of bodies of variable mass such as launch vehicles. Most of the mass of such a rocket is fuel. During the active phase of the flight, this fuel burns out, and the mass of the rocket in this part of the trajectory quickly decreases. Also, the law of conservation of momentum is necessary in cases where the concept is not applicable. It is difficult to imagine a situation where a stationary body acquires a certain speed instantly. In normal practice, bodies always accelerate and gain speed gradually. However, with the movement of electrons and other subatomic particles the change in their state occurs abruptly without staying in intermediate states. In such cases, the classical concept of “acceleration” cannot be applied.

Examples of problem solving

EXAMPLE 1

Exercise	A projectile weighing 100 kg, flying horizontally along a railway track at a speed of 500 m/s, hits a carriage with sand weighing 10 tons and gets stuck in it. What speed will the car get if it moved at a speed of 36 km/h in the direction opposite to the movement of the projectile?
Solution	The wagon + projectile system is closed, therefore in in this case the law of conservation of momentum can be applied. Let's make a drawing, indicating the state of the bodies before and after the interaction. When the projectile and the car interact, an inelastic impact occurs. The law of conservation of momentum in this case will be written as: Choosing the direction of the axis to coincide with the direction of movement of the car, we write the projection of this equation onto the coordinate axis: where does the speed of the car come from after a projectile hits it: We convert the units to the SI system: t kg. Let's calculate:
Answer	After the shell hits, the car will move at a speed of 5 m/s.

EXAMPLE 2

Exercise	A projectile weighing m=10 kg had a speed v=200 m/s at the top point. At this point it broke into two parts. The smaller part with a mass m 1 =3 kg received a speed v 1 =400 m/s in the same direction at an angle to the horizontal. At what speed and in what direction will most of the projectile fly?
Solution	The trajectory of the projectile is a parabola. The speed of the body is always directed tangentially to the trajectory. At the top point of the trajectory, the projectile speed is parallel to the axis. Let's write down the law of conservation of momentum: Let's move from vectors to scalar quantities. To do this, let’s square both sides of the vector equality and use the formulas for: Taking into account that , and also that , we find the speed of the second fragment: Substituting into the resulting formula numerical values physical quantities, we calculate: We determine the flight direction of most of the projectile using: Substituting numerical values ​​into the formula, we get:
Answer	Most of the projectile will fly down at a speed of 249 m/s at an angle to the horizontal direction.

EXAMPLE 3

Exercise

The mass of the train is 3000 tons. The friction coefficient is 0.02. What type of locomotive must be in order for the train to reach a speed of 60 km/h 2 minutes after the start of movement?

Solution

Since the train is acted upon by (an external force), the system cannot be considered closed, and the law of conservation of momentum is not satisfied in this case. Let's use the law of momentum change: Since the friction force is always directed in the direction opposite to the movement of the body, the friction force impulse will enter the projection of the equation onto the coordinate axis (the direction of the axis coincides with the direction of motion of the train) with a “minus” sign:

Simple observations and experiments prove that rest and motion are relative, the speed of a body depends on the choice of the reference system; according to Newton's second law, regardless of whether the body was at rest or moving, a change in the speed of its movement can only occur under the influence of force, i.e., as a result of interaction with other bodies. However, there are quantities that can be conserved during the interaction of bodies. These quantities are energy And pulse.

Body impulse is called a vector physical quantity, which is a quantitative characteristic of the translational motion of bodies. The impulse is indicated by . The momentum of a body is equal to the product of the mass of the body and its speed: . The direction of the momentum vector p coincides with the direction of the body's velocity vector. The unit of impulse is .

For the momentum of a system of bodies, the conservation law is satisfied, which is valid only for closed physical systems. In general, a closed system is a system that does not exchange energy and mass with bodies and fields that are not part of it. In mechanics closed called a system that is not affected by external forces or the action of these forces is compensated. In this case, where is the initial impulse of the system, and is the final one. In the case of two bodies included in the system, this expression has the form , where are the masses of the bodies, and are the velocities before interaction, and are the velocities after interaction (Fig. 4). This formula is the mathematical expression of the law of conservation of momentum: the momentum of a closed physical system is conserved during any interactions occurring within this system. In other words: in a closed physical system geometric sum momenta of bodies before interaction is equal to the geometric sum of the momenta of these bodies after interaction. In the case of an open system, the momentum of the bodies of the system is not conserved. However, if the system also has a direction in which external forces do not act or their action is compensated, then the projection of the impulse in this direction is preserved. In addition, if the interaction time is short (shot, explosion, impact), then during this time, even in the case of an open system, external forces slightly change the impulses of the interacting bodies. Therefore, for practical calculations in this case, the law of conservation of momentum can also be applied.

Experimental studies of the interactions of various bodies - from planets and stars to atoms and elementary particles- showed that in any system of interacting bodies, in the absence of action on the part of other bodies not included in the system, or the sum is equal to zero active forces the geometric sum of the momenta of the bodies really remains unchanged.

In mechanics, the law of conservation of momentum and Newton's laws are interconnected. If a force acts on a body of mass over time and the speed of its movement changes from to , then the acceleration of motion a of the body is equal to . Based on Newton's second law for force, we can write , this implies

. - a vector physical quantity that characterizes the action of a force on a body over a certain period of time and is equal to the product of the force and the time of its action is called impulse of force. The unit of force impulse is .

The law of conservation of momentum underlies jet propulsion. Jet propulsion- this is the movement of the body that occurs after the separation of its part from the body.

Let the body mass be at rest. Some part of it has been separated from the body by mass with speed. Then the remaining part will move in the opposite direction with speed, the mass of the remaining part. Indeed, the sum of the impulses of both parts of the body before separation was equal to zero and after separation will be equal to zero:

From here.

Much credit for the development of the theory of jet propulsion belongs to K. E. Tsiolkovsky.

He developed the theory of flight of a body of variable mass (a rocket) in a uniform gravitational field and calculated the fuel reserves necessary to overcome the force of gravity; basics of fluid theory jet engine, as well as elements of its design; the theory of multistage rockets, and proposed two options: parallel (several jet engines operate simultaneously) and sequential (jet engines operate one after another). K. E. Tsiolkovsky strictly scientifically proved the possibility of flying into space using rockets with a liquid jet engine, and proposed special landing trajectories spacecraft to Earth, put forward the idea of ​​​​creating interplanetary orbital stations and examined in detail the living conditions and life support on them. Tsiolkovsky's technical ideas are used in the creation of modern rocket and space technology. Movement using a jet stream according to the law of conservation of momentum is the basis of a hydrojet engine. The movement of many marine mollusks (octopus, jellyfish, squid, cuttlefish) is also based on the reactive principle.

Common Mistakes

1. There were applicants who made a serious mistake when explaining the principle of operation of a jet engine. They argued that the movement jet plane is caused by the interaction of emitted gases and air: the plane acts on the air, and the air, according to Newton’s third law, acts on the plane, as a result of which it moves. This is, of course, not true. The real reason for the movement of a jet aircraft is the interaction of gases escaping from the nozzle, which are formed during the combustion of fuel. Due to the high pressure in the combustion chamber, these gases acquire some momentum, therefore, according to the law of conservation of momentum, the aircraft receives an impulse of the same magnitude, but opposite in direction. So the plane doesn't push away from the air. Against, atmospheric air is only an obstacle to the movement of the aircraft.

2. Some students cannot give a complete and correct answer to the question: in what cases can the law of conservation of momentum be applied? Good to remember following criteria its applicability:

1. the system of bodies is closed, i.e. the bodies of this system are not acted upon by external forces;
2. external forces act on the bodies of the system, but their vector sum is zero
3. the system is not closed, but the sum of the projections of all external forces onto any coordinate axis is equal to zero; then the sum of the projections of the impulses of all bodies of the system onto this axis remains constant.
4. the time of interaction between bodies is short (for example, the time of impact, shot, explosion); in this case, the impulse of external forces can be neglected and the system can be considered as closed.

I'll start with a couple of definitions, without knowing which further consideration the question will be meaningless.

The resistance that a body exerts when trying to set it in motion or change its speed is called inertia.

Measure of inertia – weight.

Thus, the following conclusions can be drawn:

1. The greater the mass of a body, the greater its resistance to the forces that try to bring it out of rest.
2. The greater the mass of a body, the more it resists the forces that try to change its speed if the body moves uniformly.

To summarize, we can say that the inertia of the body counteracts attempts to give the body acceleration. And mass serves as an indicator of the level of inertia. The greater the mass, the great strength must be applied to influence the body to give it acceleration.

Closed system (isolated)- a system of bodies that is not influenced by other bodies not included in this system. Bodies in such a system interact only with each other.

If at least one of the two conditions above is not met, then the system cannot be called closed. Let there be a system consisting of two material points, having speeds and respectively. Let's imagine that an interaction occurred between the points, as a result of which the velocities of the points changed. Let us denote by and the increments of these speeds during the interaction between the points. We will assume that the increments have opposite directions and are related by the relation . We know that the coefficients do not depend on the nature of the interaction of material points - this has been confirmed by many experiments. The coefficients are characteristics of the points themselves. These coefficients are called masses ( inert masses). The given relationship for the increment of velocities and masses can be described as follows.

The ratio of the masses of two material points is equal to the ratio of the increments in the velocities of these material points as a result of the interaction between them.

The above relationship can be presented in another form. Let us denote the velocities of the bodies before the interaction as and , respectively, and after the interaction as and . In this case, the speed increments can be presented in the following form - and . Therefore, the relationship can be written as follows - .

Momentum (amount of energy of a material point)– a vector equal to the product of the mass of a material point and its velocity vector –

Momentum of the system (amount of motion of the system of material points)– vector sum of the momenta of the material points of which this system consists - .

We can conclude that in the case of a closed system, the momentum before and after the interaction of material points should remain the same - , where and . We can formulate the law of conservation of momentum.

The momentum of an isolated system remains constant over time, regardless of the interaction between them.

Required definition:

Conservative forces – forces whose work does not depend on the trajectory, but is determined only by the initial and final coordinates of the point.

Formulation of the law of conservation of energy:

In a system in which only conservative forces act, the total energy of the system remains unchanged. Only the transformation of potential energy into kinetic energy and vice versa is possible.

The potential energy of a material point is a function only of the coordinates of this point. Those. potential energy depends on the position of the point in the system. Thus, the forces acting on a point can be defined as follows: can be defined as follows: . – potential energy of a material point. Multiply both sides by and get . Let's transform and get an expression proving law of energy conservation .

Elastic and inelastic collisions

Absolutely inelastic impact - a collision of two bodies, as a result of which they connect and then move as one.

Two balls, with and experience a completely inelastic gift with each other. According to the law of conservation of momentum. From here we can express the speed of two balls moving after a collision as a single whole - . Kinetic energies before and after impact: And . Let's find the difference

,

Where - reduced mass of balls . This shows that in a completely inelastic collision of two balls there is a loss kinetic energy macroscopic movement. This loss is equal to half the product of the reduced mass and the square of the relative velocity.

>>Physics: Law of conservation of momentum

For momentum, a fundamental law of nature is true, called the law of conservation of momentum (or momentum). Descartes, who discovered this law, wrote in one of his letters: “I accept that in the Universe, in all created matter, there is a certain amount of motion that never increases or decreases, and thus, if one body sets another in motion, then loses as much of its movement as it communicates.”

In the simplest case law of conservation of momentum can be formulated as follows:

When two bodies interact, their total momentum remains unchanged (i.e., conserved).

Let's do an experiment. Let's hang two identical steel balls on thin threads (Fig. 18). Let's move the left ball to the side and release it. We will see that after the collision of the balls, the left ball will stop, and the right one will begin to move. The height to which the right ball will rise will coincide with the one to which the left ball was previously deflected. This suggests that during the collision, the left ball transfers all its momentum to the right ball. By how much the momentum of the first ball decreases, the momentum of the second ball increases by the same amount. The total (total) momentum of the balls remains unchanged, i.e., is conserved.

Most often, the law of conservation of momentum is used in the analysis of collisions of bodies. Let's look at a simple example. Suppose a boy with a mass of 50 kg jumps at a speed of 3 m/s onto a skateboard with a mass of 2 kg standing motionless in front of him. At what speed V will it begin to move after this? To answer this question, let's first calculate the total momentum that the boy and the skateboard had before the collision. We find: 50 kg 3 m/s = 150 kgm/s. According to the law of conservation, the same impulse should remain after the boy is on the skateboard. But now the boy and the skateboard form a system of mass 52 kg, moving with speed V, which we have to find. Let's make an equation:
52 kg V= 150 kgm/s. Solving it, we find: V= 2.9 m/s.

??? 1. Who discovered the law of conservation impulse? 2. How does the law of conservation of momentum manifest itself when bodies collide? 3. What will happen in the system of identical elastic balls shown in Figure 19 after the leftmost ball is pulled aside and released?

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In classical mechanics, there are two conservation laws: the law of conservation of momentum and the law of conservation of energy.

Body impulse

The concept of momentum was first introduced by a French mathematician, physicist, and mechanic. and the philosopher Descartes, who called impulse amount of movement .

From Latin, “impulse” is translated as “push, move.”

Any body that moves has momentum.

Let's imagine a cart standing still. Its momentum is zero. But as soon as the cart starts moving, its momentum will no longer be zero. It will begin to change as the speed changes.

Momentum of a material point, or amount of movement – a vector quantity equal to the product of the mass of a point and its speed. The direction of the point's momentum vector coincides with the direction of the velocity vector.

If we are talking about a solid physical body, then the momentum of such a body is called the product of the mass of this body and the speed of the center of mass.

How to calculate the momentum of a body? One can imagine that a body consists of many material points, or a system of material points.

If - the impulse of one material point, then the impulse of a system of material points

That is, momentum of a system of material points is the vector sum of the momenta of all material points included in the system. It is equal to the product of the masses of these points and their speed.

The unit of impulse in the international system of units SI is kilogram-meter per second (kg m/sec).

Impulse force

In mechanics, there is a close connection between the momentum of a body and force. These two quantities are connected by a quantity called impulse of force .

If a constant force acts on a bodyF over a period of time t , then according to Newton's second law

This formula shows the relationship between the force that acts on a body, the time of action of this force and the change in the speed of the body.

The quantity equal to the product of the force acting on a body and the time during which it acts is called impulse of force .

As we see from the equation, the impulse of force is equal to the difference between the impulses of the body at the initial and final moments of time, or the change in impulse over some time.

Newton's second law in momentum form is formulated as follows: the change in the momentum of a body is equal to the momentum of the force acting on it. It must be said that Newton himself originally formulated his law in exactly this way.

Force impulse is also a vector quantity.

The law of conservation of momentum follows from Newton's third law.

It must be remembered that this law operates only in a closed, or isolated, physical system. A closed system is a system in which bodies interact only with each other and do not interact with external bodies.

Let us imagine a closed system of two physical bodies. The forces of interaction of bodies with each other are called internal forces.

The force impulse for the first body is equal to

According to Newton's third law, the forces that act on bodies during their interaction are equal in magnitude and opposite in direction.

Therefore, for the second body the momentum of the force is equal to

By simple calculations we obtain a mathematical expression for the law of conservation of momentum:

Where m 1 And m 2 – body masses,

v 1 And v 2 – velocities of the first and second bodies before interaction,

v 1" And v 2" – velocities of the first and second bodies after interaction .

p 1 = m 1 · v 1 - momentum of the first body before interaction;

p 2 = m 2 · v 2 - momentum of the second body before interaction;

p 1 "= m 1 · v 1" - momentum of the first body after interaction;

p 2 "= m 2 · v 2" - momentum of the second body after interaction;

That is

p 1 + p 2 = p 1" + p 2"

In a closed system, bodies only exchange impulses. And the vector sum of the momenta of these bodies before their interaction is equal to the vector sum of their momenta after the interaction.

So, as a result of firing a gun, the momentum of the gun itself and the momentum of the bullet will change. But the sum of the impulses of the gun and the bullet in it before the shot will remain equal to the sum of the impulses of the gun and the flying bullet after the shot.

When firing a cannon, there is recoil. The projectile flies forward, and the gun itself rolls back. The projectile and the gun are a closed system in which the law of conservation of momentum operates.

The momentum of each body in a closed system can change as a result of their interaction with each other. But the vector sum of the impulses of bodies included in a closed system does not change when these bodies interact over time, that is, it remains constant. That's what it is law of conservation of momentum.

More precisely, the law of conservation of momentum is formulated as follows: the vector sum of the impulses of all bodies of a closed system is a constant value if there are no external forces acting on it, or their vector sum is equal to zero.

The momentum of a system of bodies can change only as a result of the action of external forces on the system. And then the law of conservation of momentum will not apply.

It must be said that closed systems do not exist in nature. But, if the time of action of external forces is very short, for example, during an explosion, shot, etc., then in this case the influence of external forces on the system is neglected, and the system itself is considered as closed.

In addition, if external forces act on the system, but the sum of their projections onto one of coordinate axes is equal to zero (that is, the forces are balanced in the direction of this axis), then in this direction the law of conservation of momentum is satisfied.

The law of conservation of momentum is also called law of conservation of momentum .

Most shining example application of the law of conservation of momentum - reactive motion.

Jet propulsion

Reactive motion is the movement of a body that occurs when some part of it is separated from it at a certain speed. The body itself receives an oppositely directed impulse.

The simplest example of jet propulsion is flight. balloon from which air comes out. If we inflate a balloon and release it, it will begin to fly in the direction opposite to the movement of the air coming out of it.

An example of jet propulsion in nature is the release of liquid from a fetus mad cucumber when it bursts. At the same time, the cucumber itself flies in the opposite direction.

Jellyfish, cuttlefish and other inhabitants of the deep sea move by taking in water and then throwing it out.

Jet thrust is based on the law of conservation of momentum. We know that when a rocket with a jet engine moves, as a result of fuel combustion, a jet of liquid or gas is ejected from the nozzle ( jet stream ). As a result of the interaction of the engine with the escaping substance, Reactive force . Since the rocket, together with the ejected substance, is closed system, then the momentum of such a system does not change with time.

Reactive force arises from the interaction of only parts of the system. External forces have no effect on its appearance.

Before the rocket began to move, the sum of the impulses of the rocket and the fuel was zero. Consequently, according to the law of conservation of momentum, after the engines are turned on, the sum of these impulses is also zero.

where is the mass of the rocket

Gas flow rate

Changing rocket speed

∆mf - fuel consumption

Suppose the rocket operated for a period of time t .

Dividing both sides of the equation by ∆ t, we get the expression

According to Newton's second law, the reactive force is equal to

Reaction force, or jet thrust, ensures the movement of the jet engine and the object associated with it in the direction opposite to the direction of the jet stream.

Jet engines are used in modern aircraft and various missiles, military, space, etc.