What is the momentum of the body formula. Saving the momentum projection

Pulse (Quantity of movement) is a vector physical quantity, which is a measure of the mechanical movement of the 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 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 momentum 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.

Let the body mass m for some small time interval Δ t force acted Under the influence of this force, the speed of the body changed by Therefore, during the time Δ t the body moves with acceleration

From the basic law of dynamics ( Newton's second law) follows:

The physical quantity equal to the product of the mass of the body and the speed of its movement is called body momentum(or amount of movement). The momentum of the body is a vector quantity. The SI unit of momentum is kilogram-meter per second (kg m/s).

The physical quantity equal to the product of the force and the time of its action is called momentum of force . The momentum of a force is also a vector quantity.

In new terms Newton's second law can be formulated as follows:

ANDthe change in the momentum of the body (momentum) is equal to the momentum of the force.

Denoting the momentum of the body by the letter Newton's second law can be written as

It is in such general view Newton himself formulated the second law. The force in this expression is the resultant of all forces applied to the body. This vector equality can be written in projections onto the coordinate axes:

Thus, the change in the projection of the momentum of the body on any of the three mutually perpendicular axes is equal to the projection of the momentum of the force on the same axis. Consider as an example one-dimensional movement, i.e., the movement of the body along one of the coordinate axes (for example, the axis OY). Let the body fall freely from initial speedυ 0 under the action of gravity; the fall time is t. Let's direct the axis OY vertically down. The momentum of gravity F t = mg during t equals mgt. This momentum is equal to the change in momentum of the body

This simple result coincides with the kinematicformulafor the speed of uniformly accelerated motion. In this example, the force remained unchanged in absolute value over the entire time interval t. If the force changes in magnitude, then the average value of the force must be substituted into the expression for the impulse of the force F cf on the time interval of its action. Rice. 1.16.1 illustrates a method for determining the impulse of a time-dependent force.

Let us choose a small interval Δ on the time axis t, during which the force F (t) remains virtually unchanged. Impulse of force F (t) Δ t in time Δ t will be equal to the area of ​​the shaded bar. If the entire time axis on the interval from 0 to t split into small intervals Δ ti, and then sum the force impulses on all intervals Δ ti, then the total impulse of the force will be equal to the area formed by the step curve with the time axis. In the limit (Δ ti→ 0) this area is equal to the area bounded by the graph F (t) and axis t. This method for determining the momentum of a force from a graph F (t) is general and applicable to any laws of force change with time. Mathematically, the problem is reduced to integration functions F (t) on the interval .

The impulse of force, the graph of which is shown in fig. 1.16.1, on the interval from t 1 = 0 s to t 2 = 10 s is equal to:

In this simple example

In some cases, the average force F cp can be determined if the time of its action and the impulse imparted to the body are known. For example, swipe a football player on a ball weighing 0.415 kg can tell him the speed υ = 30 m/s. The impact time is approximately equal to 8·10 -3 s.

Pulse p acquired by the ball as a result of a stroke is:

Therefore, the average force F cf, with which the football player's foot acted on the ball during the kick, is:

This is very great power. It is approximately equal to the weight of a body weighing 160 kg.

If the movement of the body during the action of the force occurred along a certain curvilinear trajectory, then the initial and final momenta of the body may differ not only in absolute value, but also in direction. In this case, to determine the change in momentum, it is convenient to use pulse diagram , which depicts the vectors and , as well as the vector constructed according to the parallelogram rule. As an example, in fig. 1.16.2 shows an impulse diagram for a ball bouncing off a rough wall. ball mass m hit the wall with a speed at an angle α to the normal (axis OX) and rebounded from it with a speed at an angle β. During contact with the wall, a certain force acted on the ball, the direction of which coincides with the direction of the vector

With a normal fall of a ball with a mass m on an elastic wall with a speed , after the rebound the ball will have a speed . Therefore, the change in the momentum of the ball during the rebound is

In projections on the axis OX this result can be written in the scalar form Δ px = –2mυ x. Axis OX directed away from the wall (as in Fig. 1.16.2), so υ x < 0 и Δpx> 0. Therefore, the module Δ p momentum change is related to the modulus υ of the ball speed by the relation Δ p = 2mυ.

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. The angle of incidence is equal to the angle of reflection.

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 (equal to acting force is zero), then the change in momentum is zero, which means that the total momentum of the system is conserved:

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 ​​of the acting forces are 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.

Problems with moving bodies in physics, when the speed is much less than the speed of light, are solved using the laws of Newtonian, or classical mechanics. In it, one of the important concepts is momentum. The basics in physics are given in this article.

Momentum or momentum?

Before giving the formulas for the momentum of a body in physics, let's get acquainted with this concept. For the first time, a quantity called impeto (impulse) was used by Galileo in the description of his works at the beginning of the 17th century. Subsequently, Isaac Newton used another name for it - motus (motion). Since the figure of Newton had a greater influence on the development of classical physics than the personality of Galileo, it was initially customary to talk not about the momentum of the body, but about the amount of motion.

The amount of motion is understood as the product of the speed of movement of the body by the inertial coefficient, that is, by the mass. The corresponding formula looks like:

Here p¯ is a vector whose direction is the same as v¯, but the modulus is m times greater than the modulus of v¯.

Change in p¯

The concept of momentum is currently used less frequently than momentum. And this fact is connected directly with the laws of Newtonian mechanics. Let's write it in the form that is given in school textbooks on physics:

We replace the acceleration a¯ with the corresponding expression for the derivative of the speed, we get:

Transferring dt from the denominator of the right side of the equality to the numerator of the left side, we get:

We have obtained an interesting result: in addition to the fact that the acting force F¯ leads to the acceleration of the body (see the first formula of this paragraph), it also changes the momentum of the body. The product of force and time, which is on the left side, is called the impulse of the force. It turns out to be equal to the change in p¯. Therefore, the last expression is also called the momentum formula in physics.

Note that dp¯ is also, but unlike p¯, it is directed not as the speed v¯, but as the force F¯.

A striking example of a change in the vector of momentum (momentum) is the situation when a football player hits the ball. Before the impact, the ball moved towards the player, after the impact - away from him.

Law of conservation of momentum

Formulas in physics that describe the conservation of p¯ can be given in several ways. Before writing them down, let's answer the question of when momentum is conserved.

Let's look at the expression from the previous paragraph:

It says that if the sum of external forces acting on the system is zero (closed system, F¯= 0), then dp¯= 0, that is, no change in momentum will occur:

This expression is common for the momentum of a body and the law of conservation of momentum in physics. Note two important moments, which you should know about in order to successfully apply this expression in practice:

• The momentum is conserved along each coordinate, that is, if before some event the value of p x of the system was 2 kg * m / s, then after this event it will be the same.
• The momentum is conserved regardless of the nature of the collisions of rigid bodies in the system. Two ideal cases of such collisions are known: absolutely elastic and absolutely plastic collisions. In the first case, kinetic energy is also conserved, in the second, part of it is spent on the plastic deformation of bodies, but the momentum is still preserved.

Elastic and inelastic interaction of two bodies

A special case of using the momentum formula in physics and its conservation is the motion of two bodies that collide with each other. Consider two fundamental different cases mentioned in the paragraph above.

If the impact is absolutely elastic, that is, the transfer of momentum from one body to another is carried out through elastic deformation, then the conservation formula p will be written as follows:

m 1 * v 1 + m 2 * v 2 = m 1 * u 1 + m 2 * u 2

Here it is important to remember that the sign of the speed must be substituted taking into account its direction along the considered axis (opposite speeds have different signs). This formula shows that under the condition of a known initial state of the system (values ​​m 1 , v 1 , m 2 , v 2) in the final state (after a collision) there are two unknowns (u 1 , u 2). You can find them if you use the corresponding law of conservation of kinetic energy:

m 1 *v 1 2 + m 2 *v 2 2 = m 1 *u 1 2 + m 2 *u 2 2

If the impact is absolutely inelastic or plastic, then after the collision the two bodies begin to move as a whole. In this case, the expression takes place:

m 1 * v 1 + m 2 * v 2 \u003d (m 1 + m 2) * u

As seen, we are talking about only about one unknown (u), so this one equality is enough to determine it.

The momentum of a body while moving in a circle

Everything that was said above about momentum refers to the linear displacements of bodies. How to be in case of rotation of objects around an axis? For this, another concept has been introduced in physics, which is similar to a linear momentum. It is called the moment of momentum. The formula in physics for it takes the following form:

Here r¯ is a vector equal to the distance from the axis of rotation to a particle with momentum p¯ making circular motions around this axis. The quantity L¯ is also a vector, but it is somewhat more difficult to calculate than p¯, since we are talking about a cross product.

Conservation law L¯

The formula for L¯ given above is the definition of this quantity. In practice, they prefer to use a slightly different expression. We will not go into the details of obtaining it (it is not difficult, and everyone can do it on their own), but we will give it right away:

Here I is the moment of inertia (for a material point it is equal to m * r 2), which describes the inertial properties of a rotating object, ω¯ is the angular velocity. As you can see, this equation is similar in form to the one for the linear momentum p¯.

If no external forces act on the rotating system (in fact, the moment of forces), then the product of I and ω¯ will be preserved regardless of the processes occurring inside the system. That is, the conservation law for L¯ has the form:

An example of its manifestation is the performance of athletes in figure skating when they make rotations on ice.

They change, since interaction forces act on each of the bodies, but the sum of the impulses remains constant. This is called law of conservation of momentum.

Newton's second law expressed by the formula. It can be written in a different way, if we remember that acceleration is equal to the rate of change in the speed of the body. For uniformly accelerated motion, the formula will look like:

If we substitute this expression into the formula, we get:

,

This formula can be rewritten as:

The change in the product of the body's mass and its speed is written on the right side of this equation. The product of body mass and speed is physical quantity, which is called body momentum or amount of body movement.

body momentum is called the product of the mass of the body and its speed. This is a vector quantity. The direction of the momentum vector coincides with the direction of the velocity vector.

In other words, a body of mass m moving at a speed has momentum. The unit of momentum in SI is the momentum of a body with a mass of 1 kg moving at a speed of 1 m/s (kg m/s). When two bodies interact with each other, if the first acts on the second body with a force, then, according to Newton's third law, the second acts on the first with a force. Let us denote the masses of these two bodies as m 1 and m 2 , and their velocities relative to any frame of reference through and . Over time t as a result of the interaction of bodies, their velocities will change and become equal and . Substituting these values ​​into the formula, we get:

,

,

Hence,

Let us change the signs of both sides of the equality to opposite ones and write it in the form

On the left side of the equation - the sum of the initial impulses of two bodies, on the right side - the sum of the impulses of the same bodies after time t. The amounts are equal. So in spite of that. that the momentum of each body changes during the interaction, the total momentum (the sum of the momenta of both bodies) remains unchanged.

It is also valid when several bodies interact. However, it is important that these bodies interact only with each other and that they are not affected by forces from other bodies that are not included in the system (or that external forces are balanced). A group of bodies that does not interact with other bodies is called closed system valid only for closed systems.