The momentum of the body is equal to the change in the momentum of the body. body momentum

Impulse(momentum) of a body is called a physical vector quantity, which is a quantitative characteristic of the translational motion of bodies. The momentum is denoted R. The momentum of a body is equal to the product of the mass of the body and its speed, i.e. it is calculated by the formula:

The direction of the momentum vector coincides with the direction of the body's velocity vector (directed tangentially to the trajectory). The unit of impulse measurement is kg∙m/s.

The total momentum of the system of bodies equals vector sum of impulses of all bodies of the system:

Change in momentum of one body is found by the formula (note that the difference between the final and initial impulses is vector):

where: p n is the momentum of the body at the initial moment of time, p to - to the end. The main thing is not to confuse the last two concepts.

Absolutely elastic impact– an abstract model of impact, which does not take into account energy losses due to friction, deformation, etc. No interactions other than direct contact are taken into account. With an absolutely elastic impact on a fixed surface, the speed of the object after the impact is equal in absolute value to the speed of the object before the impact, that is, the magnitude of the momentum does not change. Only its direction can change. At the same time, the angle of incidence equal to the angle reflections.

Absolutely inelastic impact- a blow, as a result of which the bodies are connected and continue their further movement as a single body. For example, a plasticine ball, when it falls on any surface, completely stops its movement, when two cars collide, an automatic coupler is activated and they also continue to move on together.

Law of conservation of momentum

When bodies interact, the momentum of one body can be partially or completely transferred to another body. If external forces from other bodies do not act on a system of bodies, such a system is called closed.

In a closed system, the vector sum of the impulses of all bodies included in the system remains constant for any interactions of the bodies of this system with each other. This fundamental law of nature is called the law of conservation of momentum (FSI). Its consequences are Newton's laws. Newton's second law in impulsive form can be written as follows:

As follows from this formula, if the system of bodies is not affected by external forces, or the action of external forces is compensated (the resultant force is zero), then the change in momentum is zero, which means that the total momentum of the system is preserved:

Similarly, one can reason for the equality to zero of the projection of the force on the chosen axis. If external forces do not act only along one of the axes, then the projection of the momentum on this axis is preserved, for example:

Similar records can be made for other coordinate axes. One way or another, you need to understand that in this case the impulses themselves can change, but it is their sum that remains constant. The law of conservation of momentum in many cases makes it possible to find the velocities of interacting bodies even when the values active forces unknown.

Saving the momentum projection

There are situations when the law of conservation of momentum is only partially satisfied, that is, only when designing on one axis. If a force acts on a body, then its momentum is not conserved. But you can always choose an axis so that the projection of the force on this axis is zero. Then the projection of the momentum on this axis will be preserved. As a rule, this axis is chosen along the surface along which the body moves.

Multidimensional case of FSI. vector method

In cases where the bodies do not move along one straight line, then in the general case, in order to apply the law of conservation of momentum, it is necessary to describe it along all the coordinate axes involved in the problem. But the solution of such a problem can be greatly simplified by using the vector method. It is applied if one of the bodies is at rest before or after the impact. Then the momentum conservation law is written in one of the following ways:

From the rules of vector addition it follows that the three vectors in these formulas must form a triangle. For triangles, the law of cosines applies.

A 22-caliber bullet has a mass of only 2 g. If someone throws such a bullet, he can easily catch it even without gloves. If you try to catch such a bullet that has flown out of the muzzle at a speed of 300 m / s, then even gloves will not help here.

If a toy cart is rolling towards you, you can stop it with your toe. If a truck is rolling towards you, you should keep your feet out of the way.

Let's consider a problem that demonstrates the connection between the momentum of a force and a change in the momentum of a body.

Example. The mass of the ball is 400 g, the speed acquired by the ball after the impact is 30 m/s. The force with which the foot acted on the ball was 1500 N, and the impact time was 8 ms. Find the momentum of the force and the change in the momentum of the body for the ball.

Change in body momentum

Example. Estimate the average force from the side of the floor acting on the ball during impact.

1) During the impact, two forces act on the ball: support reaction force, gravity.

The reaction force changes during the impact time, so it is possible to find the average floor reaction force.

2) Change in momentum body shown in the picture

3) From Newton's second law

The main thing to remember

1) Formulas for body impulse, force impulse;
2) The direction of the momentum vector;
3) Find the change in body momentum

General derivation of Newton's second law

F(t) chart. variable force

The force impulse is numerically equal to the area of ​​the figure under the graph F(t).

If the force is not constant in time, for example, it increases linearly F=kt, then the momentum of this force is equal to the area of ​​the triangle. You can replace this force with such a constant force that will change the momentum of the body by the same amount in the same period of time.

Average resultant force

LAW OF CONSERVATION OF MOMENTUM

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Closed system of bodies

This is a system of bodies that interact only with each other. There are no external forces of interaction.

In the real world, such a system cannot exist, there is no way to remove any external interaction. A closed system of bodies is a physical model, just like a material point is a model. This is a model of a system of bodies that allegedly interact only with each other, external forces are not taken into account, they are neglected.

Law of conservation of momentum

In a closed system of bodies vector the sum of the momenta of the bodies does not change when the bodies interact. If the momentum of one body has increased, then this means that at that moment the momentum of some other body (or several bodies) has decreased by exactly the same amount.

Let's consider such an example. Girl and boy are skating. A closed system of bodies - a girl and a boy (we neglect friction and other external forces). The girl stands still, her momentum zero, since the speed is zero (see the body momentum formula). After the boy, moving at some speed, collides with the girl, she will also begin to move. Now her body has momentum. The numerical value of the momentum of the girl is exactly the same as the momentum of the boy decreased after the collision.

One body of mass 20kg moves with a speed of , the second body of mass of 4kg moves in the same direction with a speed of . What is the momentum of each body. What is the momentum of the system?

Impulse of the body system is the vector sum of the impulses of all bodies in the system. In our example, this is the sum of two vectors (since two bodies are considered) that are directed in the same direction, therefore

Now let's calculate the momentum of the system of bodies from the previous example if the second body moves in the opposite direction.

Since the bodies move in opposite directions, we get the vector sum of the multidirectional impulses. More on the sum of vectors.

The main thing to remember

1) What is a closed system of bodies;
2) Law of conservation of momentum and its application

Pulse (Quantity of movement) - vector physical quantity, which is a measure mechanical movement body. In classical mechanics, the momentum of a body is equal to the product of the mass m this body at its speed v, the direction of the momentum coincides with the direction of the velocity vector:

System momentum particles is the vector sum of the momenta of its individual particles: p=(sums) pi, where pi – i-th impulse particles.

Theorem on the change in the momentum of the system: the total momentum of the system can only be changed by the action of external forces: Fext=dp/dt(1), i.e. the time derivative of the momentum of the system is equal to the vector sum of all external forces acting on the particles of the system. As in the case of a single particle, it follows from expression (1) that the increment of the momentum of the system is equal to the momentum of the resultant of all external forces for the corresponding period of time:

p2-p1= t & 0 F ext dt.

In classical mechanics, complete momentum system of material points is called a vector quantity equal to the sum of the products of the masses of material points at their speed:

accordingly, the quantity is called the momentum of one material point. It is a vector quantity directed in the same direction as the particle's velocity. The unit of momentum in international system units (SI) is kilogram meter per second(kg m/s).

If we are dealing with a body of finite size, which does not consist of discrete material points, to determine its momentum, it is necessary to break the body into small parts, which can be considered as material points and sum over them, as a result we get:

The momentum of a system that is not affected by any external forces (or they are compensated), preserved in time:

The conservation of momentum in this case follows from Newton's second and third laws: having written Newton's second law for each of the material points that make up the system and summing it over all the material points that make up the system, by virtue of Newton's third law we obtain equality (*).

In relativistic mechanics, the three-dimensional momentum of a system of non-interacting material points is the quantity

,

where m i- weight i-th material point.

For a closed system of non-interacting material points, this value is preserved. However, the three-dimensional momentum is not a relativistically invariant quantity, since it depends on the frame of reference. A more meaningful value will be a four-dimensional momentum, which for one material point is defined as

In practice, the following relationships between the mass, momentum, and energy of a particle are often used:

In principle, for a system of non-interacting material points, their 4-momenta are summed. However, for interacting particles in relativistic mechanics, one should take into account the momenta not only of the particles that make up the system, but also the momentum of the field of interaction between them. Therefore, a much more meaningful quantity in relativistic mechanics is the energy-momentum tensor, which fully satisfies the conservation laws.

Pulse Properties

· Additivity. This property means that the momentum of a mechanical system consisting of material points, is equal to the sum impulses of all material points included in the system.

· Invariance with respect to the rotation of the frame of reference.

· Preservation. The momentum does not change during interactions that change only the mechanical characteristics of the system. This property is invariant with respect to Galilean transformations. The properties of conservation of kinetic energy, conservation of momentum and Newton's second law are sufficient to derive the mathematical formula for momentum.

Law of conservation of momentum (Law of conservation of momentum)- the vector sum of the impulses of all bodies of the system is a constant value, if the vector sum of the external forces acting on the system is equal to zero.

In classical mechanics, the law of conservation of momentum is usually derived as a consequence of Newton's laws. From Newton's laws, it can be shown that when moving in empty space, momentum is conserved in time, and in the presence of interaction, the rate of its change is determined by the sum of the applied forces.

Like any of the fundamental conservation laws, the law of conservation of momentum is associated, according to Noether's theorem, with one of the fundamental symmetries - the homogeneity of space

The change in momentum of a body is equal to the momentum of the resultant of all forces acting on the body. This is another formulation of Newton's second law.

The definition looks like:

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Subtitles

The history of the term

Formal definition of momentum

Impulse called a conserved physical quantity associated with the homogeneity of space (invariant under translations).

Electromagnetic field impulse

The electromagnetic field is like any other material object, has momentum, which can be easily found by integrating the Poynting vector over volume :

The existence of momentum electromagnetic field explains, for example, such a phenomenon as pressure electromagnetic radiation.

Momentum in quantum mechanics

Formal definition

The momentum modulus is inversely proportional to the wavelength λ (\displaystyle \lambda ):), momentum modulus is equal to p = m v (\displaystyle p=mv)(where m (\displaystyle m) is the mass of the particle), and

Consequently, the de Broglie wavelength is the smaller, the greater the momentum modulus.

In vector form, this is written as:

Topics USE codifier: momentum of a body, momentum of a system of bodies, law of conservation of momentum.

Pulse body is a vector quantity equal to the product of the mass of the body and its speed:

There are no special units for measuring momentum. The momentum dimension is simply the product of the mass dimension and the velocity dimension:

Why is the concept of momentum interesting? It turns out that it can be used to give Newton's second law a slightly different, also extremely useful form.

Newton's second law in impulsive form

Let be the resultant of the forces applied to the body of mass . We start with the usual notation of Newton's second law:

Given that the acceleration of the body is equal to the derivative of the velocity vector, Newton's second law is rewritten as follows:

We introduce a constant under the sign of the derivative:

As you can see, the derivative of the momentum is obtained on the left side:

. ( 1 )

Relation ( 1 ) is new form writing Newton's second law.

Newton's second law in impulsive form. The derivative of the momentum of a body is the resultant of the forces applied to the body.

We can also say this: the resulting force acting on the body is equal to the rate of change of the momentum of the body.

The derivative in the formula ( 1 ) can be replaced by the ratio of final increments:

. ( 2 )

In this case, there is an average force acting on the body during the time interval . The smaller the value , the closer the relation to the derivative , and the closer the average force to its instantaneous value in this moment time.

In tasks, as a rule, the time interval is quite small. For example, it can be the time of impact of the ball with the wall, and then - the average force acting on the ball from the side of the wall during the impact.

The vector on the left side of relation ( 2 ) is called momentum change during . The momentum change is the difference between the final and initial momentum vectors. Namely, if is the momentum of the body at some initial moment of time, is the momentum of the body after a period of time , then the change in momentum is the difference:

We emphasize once again that the change in momentum is the difference of vectors (Fig. 1):

Let, for example, the ball flies perpendicular to the wall (the momentum before the impact is ) and bounces back without loss of speed (the momentum after the impact is ). Despite the fact that the modulo momentum has not changed (), there is a change in momentum:

Geometrically, this situation is shown in Fig. 2:

The modulus of change in momentum, as we see, is equal to twice the modulus of the initial momentum of the ball: .

Let's rewrite the formula ( 2 ) as follows:

, ( 3 )

or, writing the momentum change as above:

The value is called force impulse. There is no special unit of measurement for the impulse of force; the dimension of the force impulse is simply the product of the dimensions of force and time:

(Note that turns out to be another possible unit of measure for body momentum.)

The verbal formulation of equality ( 3 ) is as follows: the change in the momentum of the body is equal to the momentum of the force acting on the body for a given period of time. This, of course, is again Newton's second law in impulsive form.

Force Calculation Example

As an example of applying Newton's second law in impulsive form, let's consider the following problem.

A task. A ball of mass r, flying horizontally with a speed of m/s, hits a smooth vertical wall and bounces off it without loss of speed. The angle of incidence of the ball (that is, the angle between the direction of the ball and the perpendicular to the wall) is . The hit lasts s. Find the average strength
acting on the ball during impact.

Solution. First of all, we will show that the angle of reflection is equal to the angle of incidence, that is, the ball will bounce off the wall at the same angle (Fig. 3).

According to (3) we have: . It follows that the momentum change vector co-directed with vector , i.e. directed perpendicular to the wall towards the rebound of the ball (Fig. 5).

Rice. 5. To the task

Vectors and
equal in modulo
(because the speed of the ball has not changed). Therefore, the triangle made up of the vectors , and , is isosceles. This means that the angle between the vectors and is equal to , that is, the angle of reflection is indeed equal to the angle of incidence.

Now note in addition that our isosceles triangle has an angle (this is the angle of incidence); that is, given triangle- equilateral. From here:

And then the desired average force acting on the ball:

Impulse of the body system

Let's start with a simple situation of a two-body system. Namely, let there be body 1 and body 2 with momenta and respectively. The impulse of the body data system is the vector sum of the impulses of each body:

It turns out that for the momentum of a system of bodies there is a formula similar to Newton's second law in the form ( 1 ). Let's derive this formula.

All other objects with which bodies 1 and 2 under consideration interact, we will call external bodies. The forces with which external bodies act on bodies 1 and 2 are called external forces. Let - the resulting external force acting on body 1. Similarly - the resulting external force acting on body 2 (Fig. 6).

In addition, bodies 1 and 2 can interact with each other. Let body 2 act on body 1 with force . Then body 1 acts on body 2 with force . According to Newton's third law, the forces and are equal in absolute value and opposite in direction: . Forces and is internal forces, operating in the system.

Let's write for each body 1 and 2 Newton's second law in the form ( 1 ):

, ( 4 )

. ( 5 )

Let's add equalities ( 4 ) and ( 5 ):

On the left side of the resulting equality is the sum of the derivatives, which is equal to the derivative of the sum of the vectors and . On the right side we have, by virtue of Newton's third law:

But - this is the impulse of the system of bodies 1 and 2. We also denote - this is the resultant of external forces acting on the system. We get:

. ( 6 )

In this way, the rate of change of momentum of a system of bodies is the resultant of external forces applied to the system. Equality ( 6 ), which plays the role of Newton's second law for the system of bodies, is what we wanted to obtain.

Formula (6) was derived for the case of two bodies. Let us now generalize our reasoning to the case of an arbitrary number of bodies in the system.

The impulse of the system of bodies bodies is called the vector sum of the impulses of all bodies included in the system. If the system consists of bodies, then the momentum of this system is equal to:

Then everything is done in exactly the same way as above (only technically it looks a little more complicated). If for each body we write equalities similar to ( 4 ) and ( 5 ), and then add all these equalities, then on the left side we again get the derivative of the momentum of the system, and on the right side only the sum of external forces remains (internal forces, adding up in pairs, will give zero due to Newton's third law). Therefore, equality (6) will remain valid in the general case.

Law of conservation of momentum

The body system is called closed if the actions of external bodies on the bodies of a given system are either negligible or compensate each other. Thus, in the case of a closed system of bodies, only the interaction of these bodies with each other is essential, but not with any other bodies.

The resultant of external forces applied to a closed system is equal to zero: . In this case, from ( 6 ) we get:

But if the derivative of the vector vanishes (the rate of change of the vector is zero), then the vector itself does not change with time:

Law of conservation of momentum. The momentum of a closed system of bodies remains constant over time for any interactions of bodies within this system.

The simplest problems on the law of conservation of momentum are solved according to the standard scheme, which we will now show.

A task. A body of mass r moves at a speed m/s on a smooth horizontal surface. A body of mass r is moving towards it with a speed of m/s. An absolutely inelastic impact occurs (the bodies stick together). Find the speed of the bodies after the impact.

Solution. The situation is shown in Fig. 7. Let's direct the axis in the direction of motion of the first body.

Rice. 7. To the task

Because the surface is smooth, there is no friction. Since the surface is horizontal, and the movement occurs along it, the force of gravity and the reaction of the support balance each other:

Thus, the vector sum of the forces applied to the system of these bodies is equal to zero. This means that the system of bodies is closed. Therefore, it satisfies the law of conservation of momentum:

. ( 7 )

The impulse of the system before the impact is the sum of the impulses of the bodies:

After an inelastic impact, one body of mass was obtained, which moves with the desired speed:

From the momentum conservation law ( 7 ) we have:

From here we find the speed of the body formed after the impact:

Let's move on to the projections on the axis:

By condition, we have: m/s, m/s, so that

The minus sign indicates that the sticky bodies move in the direction opposite to the axis. Target speed: m/s.

Momentum projection conservation law

The following situation often occurs in tasks. The system of bodies is not closed (the vector sum of external forces acting on the system is not equal to zero), but there is such an axis, the sum of the projections of external forces on the axis is zero at any point in time. Then we can say that along this axis, our system of bodies behaves as a closed one, and the projection of the momentum of the system onto the axis is preserved.

Let's show this more strictly. Project equality ( 6 ) onto the axis :

If the projection of the resultant external forces vanishes, then

Therefore, the projection is a constant:

Law of conservation of momentum projection. If the projection onto the axis of the sum of external forces acting on the system is equal to zero, then the projection of the momentum of the system does not change with time.

Let's look at an example of a specific problem, how the law of conservation of momentum projection works.

A task. A mass boy, skating on smooth ice, throws a mass stone with speed at an angle to the horizon. Find the speed with which the boy rolls back after the throw.

Solution. The situation is schematically shown in Fig. eight . The boy is depicted as a rectangle.

Rice. 8. To the task

The momentum of the "boy + stone" system is not conserved. This can be seen at least from the fact that after the throw, a vertical component of the system's momentum appears (namely, the vertical component of the stone's momentum), which was not there before the throw.

Therefore, the system that the boy and the stone form is not closed. Why? The fact is that the vector sum of external forces is not equal to zero during the throw. The value is greater than the sum, and due to this excess, it is precisely the vertical component of the system's momentum that appears.

However, external forces act only vertically (no friction). Therefore, the projection of momentum on the horizontal axis is preserved. Before the throw, this projection was equal to zero. Directing the axis in the direction of the throw (so that the boy went in the direction of the negative semi-axis), we get.