The law of conservation of momentum is a short and clear definition. Body impulse

In this lesson, everyone will be able to study the topic “Impulse. Law of conservation of momentum." First, we will define the concept of momentum. Then we will determine what the law of conservation of momentum is - one of the main laws, the observance of which is necessary for a rocket to move and fly. Let's consider how it is written for two bodies and what letters and expressions are used in the recording. We will also discuss its application in practice.

Topic: Laws of interaction and motion of bodies

Lesson 24. Impulse. Law of conservation of momentum

Eryutkin Evgeniy Sergeevich

The lesson is devoted to the topic “Momentum and the “law of conservation of momentum”. To launch satellites, you need to build rockets. In order for rockets to move and fly, we must strictly observe the laws by which these bodies will move. The most important law in this sense is the law of conservation of momentum. To go directly to the law of conservation of momentum, let's first define what it is pulse.

is called the product of a body's mass and its speed: . Momentum is a vector quantity; it is always directed in the direction in which the speed is directed. The word “impulse” itself is Latin and is translated into Russian as “push”, “move”. Impulse is denoted by a small letter and the unit of impulse is .

The first person to use the concept of momentum was. He tried to use impulse as a quantity replacing force. The reason for this approach is obvious: measuring force is quite difficult, but measuring mass and speed is quite simple. This is why it is often said that momentum is the amount of movement. And since measuring impulse is an alternative to measuring force, it means that these two quantities need to be connected.

Rice. 1. Rene Descartes

These quantities - momentum and force - are interconnected by the concept. The impulse of a force is written as the product of a force and the time during which that force is applied: impulse of force. There is no special designation for force impulse.

Let's look at the relationship between momentum and force impulse. Let us consider such a quantity as the change in the momentum of a body, . It is the change in the momentum of the body that is equal to the impulse of the force. So we can write: .

Now let's move on to the next one important issue - law of conservation of momentum. This law is valid for a closed isolated system.

Definition: a closed isolated system is one in which bodies interact only with each other and do not interact with external bodies.

For a closed system, the law of conservation of momentum is valid: in a closed system, the momentum of all bodies remains constant.

Let us turn to how the law of conservation of momentum is written for a system of two bodies: .

We can write the same formula as follows: .

Rice. 2. The total momentum of a system of two balls is conserved after their collision

Note: this law makes it possible, avoiding consideration of the action of forces, to determine the speed and direction of motion of bodies. This law makes it possible to talk about such an important phenomenon as jet propulsion.

Derivation of Newton's second law

Using the law of conservation of momentum and the relationship between the momentum of a force and the momentum of a body, Newton's second and third laws can be obtained. The impulse of force is equal to the change in the momentum of the body: . Then we take the mass out of brackets, leaving . Let's move time from the left side of the equation to the right and write the equation as follows: .

Recall that acceleration is defined as the ratio of the change in speed to the time during which the change occurred. If we now substitute the acceleration symbol instead of the expression, we get the expression: - Newton’s second law.

Derivation of Newton's third law

Let's write down the law of conservation of momentum: . Let's move all the quantities associated with m 1 to the left side of the equation, and with m 2 - to right side: .

Let's take the mass out of brackets: . The interaction of bodies did not occur instantly, but over a certain period. And this period of time for the first and second bodies in a closed system was the same value: .

Dividing the right and left sides by time t, we get the ratio of the change in speed to time - this will be the acceleration of the first and second bodies, respectively. Based on this, we rewrite the equation as follows: . This is Newton’s third law, well known to us: . Two bodies interact with each other with forces equal in magnitude and opposite in direction.

List of additional literature:

Are you familiar with the quantity of motion? // Quantum. - 1991. - No. 6. — P. 40-41. Kikoin I.K., Kikoin A.K. Physics: Textbook. for 9th grade. avg. schools. - M.: Education, 1990. - P. 110-118 Kikoin A.K. Impulse and kinetic energy// Quantum. - 1985. - No. 5. - P. 28-29. Physics: Mechanics. 10th grade: Textbook. For in-depth study physicists / M.M. Balashov, A.I. Gomonova, A.B. Dolitsky and others; Ed. G.Ya. Myakisheva. - M.: Bustard, 2002. - P. 284-307.

Let's do some simple transformations with the formulas. According to Newton's second law, the force can be found: F=m*a. The acceleration is found as follows: a=v⁄t. Thus we get: F= m*v/t.

Determination of body momentum: formula

It turns out that force is characterized by a change in the product of mass and velocity over time. If we denote this product by a certain quantity, then we get the change in this quantity over time as a characteristic of force. This quantity is called the momentum of the body. The momentum of the body is expressed by the formula:

where p is the momentum of the body, m is the mass, v is the speed.

Momentum is a vector quantity, and its direction always coincides with the direction of velocity. The unit of impulse is kilogram per meter per second (1 kg*m/s).

What is body impulse: how to understand?

Let’s try to understand in a simple way, “on the fingers”, what a body impulse is. If the body is at rest, then its momentum is zero. Logical. If the speed of a body changes, then the body acquires a certain impulse, which characterizes the magnitude of the force applied to it.

If there is no impact on a body, but it moves at a certain speed, that is, it has a certain impulse, then its impulse means what impact this body can have when interacting with another body.

The impulse formula includes the mass of a body and its speed. That is, the more mass and/or speed a body has, the greater the impact it can have. This is clear from life experience.

To move a body of small mass, a small force is needed. The greater the body weight, the more effort will have to be applied. The same applies to the speed imparted to the body. In the case of the influence of the body itself on another, the impulse also shows the magnitude with which the body is capable of acting on other bodies. This value directly depends on the speed and mass of the original body.

Impulse during interaction of bodies

Another question arises: what will happen to the momentum of a body when it interacts with another body? The mass of a body cannot change if it remains intact, but the speed can easily change. In this case, the speed of the body will change depending on its mass.

In fact, it is clear that when bodies with very different masses collide, their speed will change differently. If a soccer ball flying at high speed hits an unprepared person, for example a spectator, then the spectator may fall, that is, it will acquire some small speed, but will certainly not fly like a ball.

And all because the mass of the spectator is much greater than the mass of the ball. But at the same time, the total momentum of these two bodies will remain unchanged.

Law of conservation of momentum: formula

This is the law of conservation of momentum: when two bodies interact, their total momentum remains unchanged. The law of conservation of momentum operates only in a closed system, that is, in a system in which there is no influence of external forces or their total action is zero.

In reality, a system of bodies is almost always subject to external influence, but the total impulse, like energy, does not disappear into nowhere and does not arise from nowhere; it is distributed among all participants in the interaction.

As we have already said, there are no exactly closed systems of bodies. Therefore, the question arises: in what cases can the law of conservation of momentum be applied to open systems of bodies? Let's consider these cases.

1. External forces balance each other or 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 while standing on skateboards (Figure 26.1). Wherein external forces also balance each other, and the friction force can be neglected. Therefore, the sum of the opponents' impulses is preserved.

Let in starting moment the schoolchildren were at rest. Then their total momentum at the initial moment is zero. According to the law of conservation of momentum, it will remain equal to zero even when they move. Hence,

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

1. Prove that the ratio of the boys’ velocity modules is inverse to the ratio of their masses:

v 1 /v 2 = m 2 /m 1. (2)

Please note that this relationship will hold no matter how the opponents interact. For example, it doesn’t matter whether they pull the rope jerkily or smoothly; only one of them or both of them moves 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). The friction between the platform wheels and the 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 zero in the reference frame associated with the ground. Therefore, we apply the law of conservation of momentum in this reference frame.

a) What is the ratio of the person’s speed to the speed of the platform 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 related?
c) At what speed and in what direction will the platform move relative to the ground?
d) What will be the speed of the person and the platform relative to the ground when he reaches its opposite end and stops?

2. The projection of external forces onto a certain coordinate axis is zero

Let, for example, let a cart with sand of mass mt roll along the rails at speed. We will assume that the friction between the wheels of the cart 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 cart with the load k.

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

3. Prove that the final speed of the loaded cart is

v k = v(m t /(m t + m g)).

We see that the speed of the cart decreased after the load fell.

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 cart and the load was not conserved. Only the projection of the total momentum of the bodies onto the horizontally directed x axis remained unchanged.

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

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

a) What is the projection of the total impulse in in this case is it preserved?
b) What is it equal to horizontal projection impulse of the load just before it hits the cart?
c) At what speed will the cart with the load move?

3. Impacts, collisions, explosions, shots

In these cases it happens significant change the speed of bodies (and therefore 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 great forces. Typically these forces are much greater 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 a cannonball moves inside a cannon barrel during a cannon shot, the forces exerted on each other by the cannon and the cannonball far exceed the horizontally directed external forces acting on these bodies.

5. A cannon weighing 200 kg fired a cannonball weighing 10 kg in a horizontal direction (Fig. 26.5). The cannonball flew out of the cannon at a speed of 200 m/s. What is the speed of the gun during recoil?

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

The easiest to study is the so-called absolutely inelastic collision (or absolutely inelastic impact). This is the name for the collision of bodies, as a result of which they begin to move as a single whole. This is exactly how the carts interacted in the first experiment (see Fig. 25.1), discussed 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 one. Prove that their total speed can be found using the formula

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

Direction overall speed 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 speed of the “united body” will be directed in the same way as the initial speed of a body with a large impulse.

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

Additional questions and tasks

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

9. A cart weighing 100 kg stands on the rails. A schoolboy weighing 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 immediately before the jump?

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

11. An anti-aircraft gun mounted on a railway platform fires a projectile of mass m at an angle α to the horizontal. The initial velocity of the projectile is v0. What speed will the platform acquire if its mass together with the gun is equal to M? At the initial moment the platform was at rest.

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

13. A person weighing 60 kg stands at one end of a platform 10 m long and weighing 240 kg. What will be the displacement of the platform relative to the ground when a person moves to its opposite end?
Clue. Accept that man walking With constant speed v relative to the platform; express in terms of v the speed of the platform relative to the ground.

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

As a result of the interaction of bodies, their coordinates and velocities can continuously change. The forces acting between bodies can also change. Fortunately, along with the variability of the world around us, there is also an unchanging background, determined by 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 during 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 this body.

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

From Newton’s second law (2.8) we see that the time characteristic of the force is related to the change in momentum Fdt=dP

Body impulse P is the product of a body’s mass and its speed of movement:

(2.14)

The unit of impulse is kilogram meter per second (kg m/s).

The impulse is always directed in the same direction as the speed.

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

Let us prove the validity of this law. Let's 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's rewrite this relationship using the concept of momentum:

Hence,

If the change in any quantity is zero, then this physical quantity is conserved. Thus, we come to the conclusion: the sum of the impulses 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 mass (center of inertia) of a system is a point whose coordinates are given by the equations:

(2.16)

where x 1; y 1; z 1 ; x 2; y 2; z 2 ; ...; xN; y N ; z N - coordinates of the corresponding material points of the system.

§2.5 Energy. Mechanical work and power

Quantitative measure various types movement is energy. When one form of motion is transformed into another, a change in energy occurs. In the same way, when motion is transferred from one body to another, the energy of one body decreases and the energy of another 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 force, or during the process of heat exchange.

To determine the work of force F, consider a curvilinear trajectory (Fig. 2.5) along which a material point moves from position 1 to position 2. Let us divide the trajectory into elementary, sufficiently small movements dr; this vector coincides with the direction of motion of the material point. Let us denote the module of elementary displacement by dS: |dr| = dS. Since the elementary displacement is quite small, 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 a scalar product of vectors:

(2.18)

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

Summing it all up basic work, you can determine the work of a variable force on the trajectory section from point 1 to point 2 (see Fig. 2.5). This problem boils down to finding the following integral:

(2.19)

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

Note that, in contrast to Newton’s second law in expressions (2.22) and (2.23), F does not necessarily mean 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 is clear that when moving along a horizontal surface of a body on which forces F, F tr and mg act, the work of the force F is positive (α > 0), the work of the friction force F tr is negative (α = 180°) , and the work done by gravity mg is zero (α = 90°). Since the tangential component of the force F t = F cos α, the elementary work is calculated as the product of F t by the modulus of elementary displacement 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 speed of work is characterized by a quantity called power.

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

(2.21)

Taking (2.22) into account, we obtain

(2.22)

or N = Fυcosα (2.23) Power equal to scalar product force and velocity vectors.

From the resulting formula it is clear that with constant engine power, the traction force is greater when the speed is lower
. That is why the driver of a car, when going uphill, when the greatest traction force is needed, switches the engine to low speed.

Impulse(amount of motion) of a body is a physical vector quantity that is quantitative characteristics forward movement tel. The impulse is designated R. Body impulse equal to the product body mass to its speed, i.e. it is calculated by the formula:

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

Total momentum of a system of bodies equals vector the sum of the impulses of all bodies in 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 – body impulse at the initial moment of time, p k – to the final one. 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 other interactions other than direct contact are taken into account. When absolutely elastic impact On a fixed surface, the speed of the object after the impact is modulo equal to the speed of the object before the impact, that is, the magnitude of the impulse does not change. Only its direction can change. In this case, the angle of incidence equal to angle reflections.

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

Law of conservation of momentum

When bodies interact, the impulse of one body can be partially or completely transferred to another body. If a system of bodies is not acted upon by 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 law of conservation of momentum (LCM). Its consequences are Newton's laws. Newton's second law pulse form can be written as follows:

As follows from this formula, if there is no external force acting on a system of bodies, 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 conserved:

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

Similar records can be made for other coordinate axes. One way or another, you need to understand that 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 momentum projection

Situations are possible when the law of conservation of momentum is only partially satisfied, that is, only when projecting onto 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 force on this axis is equal to zero. Then the projection of the impulse onto 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 bodies do not move along one straight line, then in general case, in order to apply the law of conservation of momentum, you need to describe it for all coordinate axes involved in the task. But the solution to such a problem can be greatly simplified if you use vector method. It is used if one of the bodies is at rest before or after the impact. Then the law of conservation of momentum is written in one of the following ways:

From the rules for adding vectors it follows that the three vectors in these formulas must form a triangle. For triangles, the cosine theorem applies.

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How to successfully prepare for the CT in physics and mathematics?

In order to successfully prepare for the CT in physics and mathematics, among other things, it is necessary to fulfill three most important conditions:

1. Study all topics and complete all tests and assignments given in the educational materials on this site. To do this, you need nothing at all, namely: devote three to four hours every day to preparing for the CT in physics and mathematics, studying theory and solving problems. The fact is that CT is an exam where it is not enough just to know physics or mathematics, you also need to be able to solve it quickly and without failures a large number of tasks for different topics and of varying complexity. The latter can only be learned by solving thousands of problems.
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3. Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to decide on both options. Again, on the CT, in addition to the ability to quickly and efficiently solve problems, and knowledge of formulas and methods, you must also be able to properly plan time, distribute forces, and most importantly, correctly fill out the answer form, without confusing the numbers of answers and problems, or your own last name. Also, during RT, it is important to get used to the style of asking questions in problems, which may seem very unusual to an unprepared person at the DT.

Successful, diligent and responsible implementation of these three points will allow you to show up on the CT excellent result, the maximum of what you are capable of.

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