When the law of conservation of momentum applies. Law of conservation of momentum, kinetic and potential energies, power of force

As a result of the interaction of bodies, their coordinates and velocities can change continuously. The forces acting between bodies can also change. Fortunately, along with the variability of the world around us, there is also an unchanging background, due to the so-called conservation laws, which assert the constancy in time of certain physical quantities that characterize the system of interacting bodies as a whole.

Let some constant force act on a body of mass m for time t. Let us find out how the product of this force and the time of its action associated with a change in the state of the body.

The momentum conservation law owes its existence to such a fundamental symmetry property as homogeneity of space.

From Newton's second law (2.8) we see that the temporal characteristic of the action of a force is associated with a change in momentum Fdt=dP

body momentum P is called the product of body mass and the speed of its movement:

(2.14)

The unit of momentum is a kilogram-meter per second (kg m/s).

The momentum is always directed in the same direction as the velocity.

In modern wording the law of conservation of momentum says : for any processes occurring in a closed system, its total momentum remains unchanged.

Let's prove the validity of this law. Consider the movement of two material points interacting only with each other (Fig. 2.4).

 Such a system can be called isolated in the sense that there is no interaction with other bodies. According to Newton's third law, the forces acting on these bodies are equal in magnitude and opposite in direction:

Using Newton's second law, this can be expressed as:

Combining these expressions, we get

Let us rewrite this relation using the concept of momentum:

Consequently,

If the change in any quantity is zero, then this physical quantity is conserved. Thus, we come to the conclusion: the sum of the momenta of two interacting isolated points remains constant, regardless of the type of interaction between them.

(2.15)

This conclusion can be generalized to an arbitrary isolated system of material points interacting with each other. If the system is not closed, i.e. the sum of external forces acting on the system is not equal to zero: F ≠ 0, the law of conservation of momentum is not satisfied.

center of gravity (the center of inertia) of the system is a point whose coordinates are given by the equations:

(2.16)

where x 1; at 1 ; z1; x 2; at 2 ; z2; …; xN; at N; z N - coordinates of the corresponding material points of the system.

§2.5 Energy. Mechanical work and power

quantitative measure various kinds movement is energy. When one form of motion is transformed into another, there is a change in energy. In the same way, when motion is transferred from one body to another, the energy of one body decreases and the energy of the other body increases. Such transitions and transformations of motion and, consequently, energy can occur either in the process of work, i.e. when a body moves under the influence of a force, or in the process of heat transfer.

To determine the work of the force F, consider a curvilinear trajectory (Fig. 2.5), along which a material point moves from position 1 to position 2. Let's break the trajectory into elementary, sufficiently small displacements dr; this vector coincides with the direction of motion of the material point. We denote the modulus of elementary displacement by dS: |dr| = dS. Since the elementary displacement is small enough, in this case the force F can be considered unchanged and the elementary work can be calculated using the formula for the work of a constant force:

dA = F cosα dS = F cosα|dr|, (2.17)

or as the dot product of vectors:

(2.18)

E elementary work orjust a work of strength is the scalar product of the vectors of force and elementary displacement.

Summing up all the elementary work, it is possible to determine the work of a variable force on the section of the trajectory from point 1 to point 2 (see Fig. 2.5). This problem is reduced to finding the following integral:

(2.19)

Let this dependence be presented graphically (Fig. 2.6), then the desired work is determined on the graph by the area of ​​the shaded figure.

Note that, unlike Newton's second law, in expressions (2.22) and (2.23) it is not at all necessary to understand F as the resultant of all forces, it can be one force or the resultant of several forces.

Work can be positive or negative. The sign of the elementary work depends on the value of cosα. So, for example, from Figure 2.7 it can be seen that when moving along a horizontal surface of a body, on which the forces F, F tr and mg act, the work of the force F is positive (α\u003e 0), the work of the friction force F tr is negative (α \u003d 180 °) , and the work of gravity mg is zero (α = 90°). Since the tangential component of the force is F t = F cos α, the elementary work is calculated as the product of F t and the elementary displacement module dS:

dA = F t dS (2.20)

Thus, only the tangential component of the force does the work, the normal component of the force (α = 90°) does not do the work.

The rate at which work is done is characterized by a quantity called power.

Power called a scalar physical quantity,equal to the ratio of work to the time in which it is completedwobbles:

(2.21)

Taking into account (2.22), we obtain

(2.22)

or N = Fυcosα (2.23) Power is equal to dot product vectors of force and speed.

From the formula obtained, it can be seen that at a constant engine power, the traction force is greater when the speed is less
. That is why the driver of the car, when climbing uphill, when the greatest traction force is needed, switches the engine to low speed.

Impulse(momentum) of the body is called a physical vector quantity, which is a quantitative characteristic forward movement tel. 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 the body system is not affected external forces from other 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.

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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:

,

,

Consequently,

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.

Lecture 10. Law of conservation of momentum and jet propulsion.

Movement in nature does not arise from nothing and does not disappear - it is transmitted from one object to another. Under certain conditions, the movement is able to accumulate, but, being released, it reveals its property to be preserved.

Have you ever wondered why:

• ball flying from high speed, a football player can stop with his foot or head, but a car moving along rails even very slowly cannot be stopped by a person (the mass of the car is much greater than the mass of the ball).
• A glass of water is on a long strip of strong paper. If you pull the strip slowly, then the glass moves along with the paper. and if you sharply pull a strip of paper - the glass remains motionless. (the glass will remain motionless due to inertia - the phenomenon of keeping the speed of the body constant in the absence of other bodies acting on it)
• A tennis ball, hitting a person, does no harm, however, a bullet that is smaller in mass and moves at high speed (600-800 m / s) turns out to be deadly (the speed of the bullet is much greater than the ball).

This means that the result of the interaction of bodies depends on the mass of the bodies and on their speed at the same time.

Still great French philosopher, mathematician, physicist and physiologist, founder of new European rationalism and one of the most influential metaphysicians of modern times, introduced such a concept as "quantity of motion". He also stated the law of conservation of momentum, gave the concept of the impulse of force.

"I accept that in the Universe ... there is a certain amount of motion that never increases, never decreases, and thus, if one body sets another in motion, it loses as much of its motion as it imparts." R. Descartes

Descartes, judging by his statements, understood the fundamental significance of the concept of momentum introduced by him in the 17th century - or the momentum of a body - as the product of a body's mass and its speed. And although he made the mistake of not considering momentum as a vector quantity, the law of conservation of momentum he formulated has withstood the test of time. At the beginning of the 18th century, the error was corrected, and the triumphal procession of this law in science and technology continues to this day.

As one of the fundamental laws of physics, it has given scientists an invaluable research tool, prohibiting some processes and opening the way for others. Explosion, jet propulsion, atomic and nuclear transformations - this law works excellently everywhere. And in how many of the most everyday situations the concept of impulse helps to understand, today, we hope, you will see for yourself.

Quantity of movement - measure mechanical movement, equal to material point the product of its mass m for speed v. Number of movement mv- vector quantity, directed in the same way as the velocity of a point. Sometimes the amount of motion is also called momentum. The amount of movement at any given time is characterized by speed object of a certain masses moving it from one point in space to another.

body momentum (or momentum) called a vector quantity equal to the product of the body's mass and its speed:

body momentum directed in the same direction as the speed of the body.

Unit of measurement momentum in SI is 1 kg m/s.

A change in the momentum of the body occurs when the bodies interact, for example, during impacts. (Video "Billiard balls") When bodies interact pulse one body can be partially or completely transferred to another body.

Types of collisions:

Absolutely inelastic impact- this is such a shock interaction in which the bodies are connected (stick together) with each other and move on as one body.

The bullet gets stuck in the bar and then they move as one. A piece of plasticine sticks to the wall.

Absolutely elastic impact- this is a collision in which the mechanical energy of a system of bodies is conserved.

The balls after the collision bounce off each other in different directions The ball bounces off the wall

Let a force F act on a body of mass m for some small time interval Δt.

Under the influence of this force, the speed of the body changed by

Therefore, during the time Δt the body moved with acceleration

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

A physical quantity equal to the product of a force and its duration, is called momentum of force:

The momentum of the force is also vector quantity.

The impulse of the force is equal to the change in the momentum of the body (Newton's II law in impulse form):

Denoting the momentum of the body with the letter p, 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.

To determine the change in momentum, it is convenient to use the momentum diagram, which depicts the momentum vectors, as well as the momentum sum vector, constructed according to the parallelogram rule.

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

In mechanics, there are often problems when it is necessary to simultaneously consider several bodies moving in different ways. Such, for example, are the problems of the motion of celestial bodies, the collision of bodies, the recoil firearms, where both the projectile and the gun begin to move after being fired, etc. In these cases, one speaks of the movement of a system of bodies: the solar system, the system of two colliding bodies, the “gun-projectile” system, etc. Some forces act between the bodies of the system . AT solar system these are forces gravity, in the system of colliding bodies - elastic forces, in the "gun - projectile" system - forces created by powder gases.

The impulse of the system of bodies will be equal to the sum of the impulses of each of the bodies. included in the system.

In addition to the forces acting from some bodies of the system on others (" internal forces”), forces can also act on bodies from bodies that do not belong to the system (“external” forces); for example, the force of gravity and the elasticity of the table also act on colliding billiard balls, the force of gravity also acts on the cannon and projectile, etc. However, in a number of cases all external forces can be neglected. Thus, when studying the impact of rolling balls, the forces of gravity are balanced for each ball separately and therefore do not affect their motion; when fired from a cannon, gravity will have its effect on the flight of the projectile only after it leaves the barrel, which will not affect the amount of recoil. Therefore, it is often possible to consider the motions of a system of bodies, assuming that there are no external forces.

If a system of bodies is not affected by external forces from other bodies, such a system is called closed.

CLOSED SYSTEM – THIS IS A SYSTEM OF BODIES THAT INTERACT ONLY WITH EACH OTHER.

Law of conservation of momentum.

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.

The law of conservation of momentum serves as the basis for explaining a wide range of natural phenomena, and is used in various sciences:

1. The law is strictly observed in the phenomena of recoil when fired, the phenomenon jet propulsion, explosive phenomena and collision phenomena of bodies.
2. The law of conservation of momentum is used: in calculating the velocities of bodies during explosions and collisions; when calculating jet vehicles; in military industry when designing weapons; in engineering - when driving piles, forging metals, etc.

As we have already said, exactly closed systems of bodies do not exist. Therefore, the question arises: in what cases can the law of conservation of momentum be applied to non-closed systems of bodies? Let's consider these cases.

1. External forces balance each other or they can be neglected

We have already met this case in the previous paragraph using the example of two interacting carts.

As a second example, consider a first grader and a tenth grader competing in a tug of war standing on skateboards (Figure 26.1). In this case, external forces also balance each other, and the friction force can be neglected. Therefore, the sum of the opponents' impulses is conserved.

Let the students be at rest at the initial moment. Then their total momentum at the initial moment is equal to zero. According to the law of conservation of momentum, it will remain equal to zero even when they move. Consequently,

where 1 and 2 are the speeds of schoolchildren at an arbitrary moment (until the actions of all other bodies are compensated).

1. Prove that the ratio of the modules of the boys' speeds is inverse to the ratio of their masses:

v 1 / v 2 \u003d m 2 / m 1. (2)

Note that this ratio will hold regardless of how the opponents interact. For example, it does not matter whether they pull the rope jerkily or smoothly, only one of them or both of them sorts out the rope with their hands.

2. There is a platform weighing 120 kg on the rails, and on it is a person weighing 60 kg (Fig. 26.2, a). Friction between platform wheels and rails can be neglected. The person begins to walk along the platform to the right at a speed of 1.2 m / s relative to the platform (Fig. 26.2, b).

The initial total momentum of the platform and the person is equal to zero in the reference frame associated with the earth. Therefore, we apply the law of conservation of momentum in this frame of reference.

a) What is the ratio of the person's speed to the platform's speed relative to the ground?
b) How are the modules of the speed of a person relative to the platform, the speed of a person relative to the ground and the speed of the platform relative to the ground?
c) At what speed and in what direction will the platform move relative to the ground?
d) What will be the speed of a person and a platform relative to the earth when he reaches its opposite end and stops?

2. The projection of external forces on some coordinate axis is zero

Let, for example, a trolley with sand with a mass of m tons roll along the rails at a speed. We assume that the friction between the wheels of the trolley and the rails can be neglected.

A load of mass m g falls into the cart (Fig. 26.3, a), and the cart rolls further with the load (Fig. 26.3, b). Let us denote the final speed of the loaded trolley as k.

Let's introduce the coordinate axes, as shown in the figure. Only vertically directed external forces acted on the bodies (gravity and normal reaction force from the side of the rails). These forces cannot change the horizontal projections of the momentum of the bodies. Therefore, the projection of the total momentum of the bodies on the horizontally directed x-axis remained unchanged.

3. Prove that the final speed of the loaded cart

v k \u003d v (m t / (m t + m g)).

We see that the speed of the trolley after the fall of the load has decreased.

The decrease in the speed of the cart is explained by the fact that it transferred part of its initial horizontally directed impulse to the load, accelerating it to speed k. When the cart accelerated the load, it, according to Newton's third law, slowed down the cart.

Please note that in the process under consideration, the total momentum of the trolley and the load was not conserved. Only the projection of the total momentum of the bodies on the horizontally directed x-axis remained unchanged.

The projection of the total momentum of the bodies on the vertically directed axis y changed in this process: before the load fell, it was different from zero (the load moved down), and after the load fell, it became equal to zero (both bodies move horizontally).

4. A load of 10 kg flies into a cart with sand with a mass of 20 kg standing on rails. The speed of the load just before it hits the cart is 6 m/s and is directed at an angle of 60º to the horizon (Fig. 26.4). The friction between the bogie wheels and the rails can be neglected.

a) What projection of the total momentum is preserved in this case?
b) What is the horizontal projection of the momentum of the load just before it hits the cart?
c) What is the speed of the cart with the load?

3. Impacts, collisions, breaks, shots

In these cases, there is a significant change in the speed of the bodies (and hence their momentum) in a very short period of time. As we already know (see the previous paragraph), this means that during this period of time the bodies act on each other with big forces. Usually these forces are much higher than the external forces acting on the bodies of the system.
Therefore, the system of bodies during such interactions can be considered closed with a good degree of accuracy, due to which the law of conservation of momentum can be used.

For example, when during cannon shot the cannonball moves inside the cannon barrel, the forces with which the cannon and the cannonball act on each other are much greater than the horizontally directed external forces acting on these bodies.

5. From a cannon with a mass of 200 kg, a cannonball with a mass of 10 kg was fired in a horizontal direction (Fig. 26.5). The core flew out of the cannon at a speed of 200 m/s. What is the gun's recoil speed?

In collisions, the bodies also act on each other with fairly large forces for a short period of time.

The simplest to study is the so-called absolutely inelastic collision (or absolutely inelastic impact). This is the name of the collision of bodies, as a result of which they begin to move as a whole. This is how the carts interacted in the first experiment (see Fig. 25.1), considered in the previous paragraph. Finding the total speed of the bodies after a completely inelastic collision is quite simple.

6. Two plasticine balls of mass m 1 and m 2 move with speeds 1 and 2. As a result of the collision, they began to move as a whole. Prove that their common speed can be found using the formula

Usually, cases are considered when the bodies before the collision move along one straight line. Let's direct the x-axis along this straight line. Then, in projections onto this axis, formula (3) takes the form

The direction of the total velocity of bodies after an absolutely inelastic collision is determined by the sign of the projection v x .

7. Explain why it follows from formula (4) that the velocity of the “united body” will be directed in the same way as the initial velocity of a body with a large momentum.

8. Two carts are moving towards each other. When they collide, they interlock and move as a single unit. Let us denote the mass and speed of the cart, which initially went to the right, m ​​p and p, and the mass and speed of the cart, which initially went to the left, m l and l. In what direction and at what speed will the coupled carts move if:
a) m p \u003d 1 kg, v p \u003d 2 m / s, m l \u003d 2 kg, v l \u003d 0.5 m / s?
b) m p \u003d 1 kg, v p \u003d 2 m / s, m l \u003d 4 kg, v l \u003d 0.5 m / s?
c) m p \u003d 1 kg, v p \u003d 2 m / s, m l \u003d 0.5 kg, v l \u003d 6 m / s?

Additional questions and tasks

In the tasks for this section, it is assumed that friction can be neglected (if the coefficient of friction is not specified).

9. A cart weighing 100 kg is on the rails. A schoolboy with a mass of 50 kg running along the rails jumped onto this cart with a running start, after which it, together with the schoolboy, began to move at a speed of 2 m/s. What was the student's speed just before the jump?

10. There are two carts of mass M each on the rails not far from each other. On the first of them stands a man of mass m. A person jumps from the first cart to the second.
a) Which cart will have the greater speed?
b) What will be the ratio of the velocities of the carts?

11. From anti-aircraft gun, installed on a railway platform, a projectile of mass m is fired at an angle α to the horizon. Initial projectile speed v0. What speed will the platform acquire if its mass together with the tool is equal to M? At the initial moment the platform was at rest.

12. A 160 g puck sliding on ice hits an ice floe. After the impact, the puck slides in the same direction, but the modulus of its speed has halved. The speed of the ice became equal initial speed washers. What is the mass of the ice?

13. At one end of a platform 10 m long and 240 kg in weight stands a man weighing 60 kg. What will be the movement of the platform relative to the ground when the person moves to its opposite end?
Clue. Assume that the person is walking at a constant speed v relative to the platform; Express in terms of v the speed of the platform relative to the ground.

14. A bullet of mass m, flying horizontally with a speed, hits a wooden block of mass M lying on a long table and gets stuck in it. How long after this will the bar slide on the table if the coefficient of friction between the table and the bar is μ?