Newton's second law \ (~ m \ vec a = \ vec F \) can be written in a different form, which is given by Newton himself in his main work "Mathematical Principles of Natural Philosophy".

If a constant force acts on a body (material point), then acceleration is also constant

\ (~ \ vec a = \ frac (\ vec \ upsilon_2 - \ vec \ upsilon_1) (\ Delta t) \),

where \ (~ \ vec \ upsilon_1 \) and \ (~ \ vec \ upsilon_2 \) are the initial and final values ​​of the body's velocity.

Substituting this acceleration value into Newton's second law, we get:

\ (~ \ frac (m \ cdot (\ vec \ upsilon_2 - \ vec \ upsilon_1)) (\ Delta t) = \ vec F \) or \ (~ m \ vec \ upsilon_2 - m \ vec \ upsilon_1 = \ vec F \ Delta t \). (one)

A new physical quantity appears in this equation - the momentum of a material point.

The impulse of the material points are called a value equal to the product of the mass of a point by its speed.

Let us denote momentum (sometimes also called momentum) by the letter \ (~ \ vec p \). Then

\ (~ \ vec p = m \ vec \ upsilon \). (2)

It can be seen from formula (2) that momentum is a vector quantity. As m> 0, then the impulse has the same direction as the velocity.

The unit of momentum has no specific name. Its name is derived from the definition of this quantity:

[p] = [m] · [ υ ] = 1 kg · 1 m / s = 1 kg · m / s.

Another form of writing Newton's second law

We denote by \ (~ \ vec p_1 = m \ vec \ upsilon_1 \) the momentum of a material point at the initial moment of the interval Δ t, and after \ (~ \ vec p_2 = m \ vec \ upsilon_2 \) - the impulse at the end of this interval. Then \ (~ \ vec p_2 - \ vec p_1 = \ Delta \ vec p \) is change in momentum in time Δ t... Now equation (1) can be written as follows:

\ (~ \ Delta \ vec p = \ vec F \ Delta t \). (3)

Since Δ t> 0, then the directions of the vectors \ (~ \ Delta \ vec p \) and \ (~ \ vec F \) coincide.

According to formula (3)

the change in the momentum of a material point is proportional to the force applied to it and has the same direction as the force.

This is how it was first formulated Newton's second law.

The product of force by the time of its action is called impulse of power... Do not confuse the momentum \ (~ m \ vec \ upsilon \) of a material point and the impulse of force \ (\ vec F \ Delta t \). These are completely different concepts.

Equation (3) shows that the same changes in the momentum of a material point can be obtained as a result of the action of a large force during a short time interval or a small force over a long time interval. When you jump from a certain height, then the stop of your body occurs due to the action of force from the side of the ground or floor. The shorter the duration of the collision, the greater the braking force. To reduce this force, it is necessary that the braking occurs gradually. This is why when jumping high, athletes land on soft mats. Sagging, they gradually slow down the athlete. Formula (3) can be generalized to the case when the force changes over time. For this, the entire time interval Δ t the action of the force must be divided into such small intervals Δ t i, so that on each of them the value of the force can be considered constant without a large error. For each small time interval, formula (3) is valid. Summing up the changes in impulses over small time intervals, we get:

\ (~ \ Delta \ vec p = \ sum ^ (N) _ (i = 1) (\ vec F_i \ Delta t_i) \). (4)

The symbol Σ (Greek sigma) stands for sum. Indexes i= 1 (bottom) and N(top) means that it is summed N terms.

To find the impulse of the body, they do the following: mentally break the body into separate elements (material points), find the impulses of the received elements, and then sum them up as vectors.

The momentum of a body is equal to the sum of the impulses of its individual elements.

Changing the impulse of the system of bodies. Momentum conservation law

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

Changing the momentum of a system of bodies

Consider a three-body system. These can be three stars, which are influenced by neighboring cosmic bodies. External forces act on the bodies of the system \ (~ \ vec F_i \) ( i- body number; for example, \ (~ \ vec F_2 \) is the sum of external forces acting on body number two). Forces \ (~ \ vec F_ (ik) \), called internal forces, act between the bodies (Fig. 1). Here is the first letter i in the index means the number of the body on which the force \ (~ \ vec F_ (ik) \) acts, and the second letter k means the number of the body from which the given force acts. Based on Newton's third law

\ (~ \ vec F_ (ik) = - \ vec F_ (ki) \). (five)

Due to the action of forces on the bodies of the system, their impulses change. If the force does not change noticeably over a short period of time, then for each body of the system it is possible to write down the change in momentum in the form of equation (3):

\ (~ \ Delta (m_1 \ vec \ upsilon_1) = (\ vec F_ (12) + \ vec F_ (13) + \ vec F_1) \ Delta t \), \ (~ \ Delta (m_2 \ vec \ upsilon_2) = (\ vec F_ (21) + \ vec F_ (23) + \ vec F_2) \ Delta t \), (6) \ (~ \ Delta (m_3 \ vec \ upsilon_3) = (\ vec F_ (31) + \ vec F_ (32) + \ vec F_3) \ Delta t \).

Here, on the left side of each equation, there is a change in the momentum of the body \ (~ \ vec p_i = m_i \ vec \ upsilon_i \) in a short time Δ t... More details \ [~ \ Delta (m_i \ vec \ upsilon_i) = m_i \ vec \ upsilon_ (ik) - m_i \ vec \ upsilon_ (in) \] where \ (~ \ vec \ upsilon_ (in) \) - speed in beginning, and \ (~ \ vec \ upsilon_ (ik) \) - at the end of the time interval Δ t.

Let us add the left and right sides of equations (6) and show that the sum of the changes in the momenta of individual bodies is equal to the change in the total momentum of all bodies in the system, which is equal to

\ (~ \ vec p_c = m_1 \ vec \ upsilon_1 + m_2 \ vec \ upsilon_2 + m_3 \ vec \ upsilon_3 \). (7)

Really,

\ (~ \ Delta (m_1 \ vec \ upsilon_1) + \ Delta (m_2 \ vec \ upsilon_2) + \ Delta (m_3 \ vec \ upsilon_3) = m_1 \ vec \ upsilon_ (1k) - m_1 \ vec \ upsilon_ (1n) + m_2 \ vec \ upsilon_ (2k) - m_2 \ vec \ upsilon_ (2n) + m_3 \ vec \ upsilon_ (3k) - m_3 \ vec \ upsilon_ (3n) = \) \ (~ = (m_1 \ vec \ upsilon_ ( 1k) + m_2 \ vec \ upsilon_ (2k) + m_3 \ vec \ upsilon_ (3k)) - (m_1 \ vec \ upsilon_ (1n) + m_2 \ vec \ upsilon_ (2n) + m_3 \ vec \ upsilon_ (3n)) = \ vec p_ (ck) - \ vec p_ (cn) = \ Delta \ vec p_c \).

Thus,

\ (~ \ Delta \ vec p_c = (\ vec F_ (12) + \ vec F_ (13) + \ vec F_ (21) + \ vec F_ (23) + \ vec F_ (31) + \ vec F_ (32 ) + \ vec F_1 + \ vec F_2 + \ vec F_3) \ Delta t \). (eight)

But the forces of interaction of any pair of bodies add up to zero, since according to formula (5)

\ (~ \ vec F_ (12) = - \ vec F_ (21); \ vec F_ (13) = - \ vec F_ (31); \ vec F_ (23) = - \ vec F_ (32) \).

Therefore, the change in the momentum of the system of bodies is equal to the momentum of external forces:

\ (~ \ Delta \ vec p_c = (\ vec F_1 + \ vec F_2 + \ vec F_3) \ Delta t \). (nine)

We came to an important conclusion:

the momentum of a system of bodies can only be changed by external forces, and the change in the momentum of the system is proportional to the sum of external forces and coincides with it in direction. Internal forces, changing the impulses of individual bodies of the system, do not change the total impulse of the system.

Equation (9) is valid for any time interval if the sum of external forces remains constant.

Momentum conservation law

An extremely important consequence follows from equation (9). If the sum of the external forces acting on the system is equal to zero, then the change in the momentum of the system \ [~ \ Delta \ vec p_c = 0 \] is also equal to zero. This means that no matter what time interval we take, the total impulse at the beginning of this interval \ (~ \ vec p_ (cn) \) and at its end \ (~ \ vec p_ (ck) \) is the same \ [~ \ vec p_ (cn) = \ vec p_ (ck) \]. The momentum of the system remains unchanged, or, as they say, persists:

\ (~ \ vec p_c = m_1 \ vec \ upsilon_1 + m_2 \ vec \ upsilon_2 + m_3 \ vec \ upsilon_3 = \ operatorname (const) \). (10)

Momentum conservation law is formulated as follows:

if the sum of external forces acting on the bodies of the system is equal to zero, then the momentum of the system is conserved.

The bodies can only exchange impulses, the total value of the impulse does not change. It is only necessary to remember that the vector sum of the impulses is saved, and not the sum of their modules.

As can be seen from our conclusion, the law of conservation of momentum is a consequence of Newton's second and third laws. A system of bodies that is not acted upon by external forces is called closed or isolated. In a closed system of bodies, momentum is conserved. But the field of application of the law of conservation of momentum is wider: even if external forces act on the bodies of the system, but their sum is equal to zero, the momentum of the system is still conserved.

The result obtained can be easily generalized to the case of a system containing an arbitrary number N of bodies:

\ (~ m_1 \ vec \ upsilon_ (1n) + m_2 \ vec \ upsilon_ (2n) + m_3 \ vec \ upsilon_ (3n) + \ ldots + m_N \ vec \ upsilon_ (Nn) = m_1 \ vec \ upsilon_ (1k) + m_2 \ vec \ upsilon_ (2k) + m_3 \ vec \ upsilon_ (3k) + \ ldots + m_N \ vec \ upsilon_ (Nk) \). (eleven)

Here \ (~ \ vec \ upsilon_ (in) \) are the velocities of the bodies at the initial moment of time, and \ (~ \ vec \ upsilon_ (ik) \) - at the final one. Since the momentum is a vector quantity, equation (11) is a compact record of three equations for the projections of the momentum of the system on the coordinate axes.

When is the momentum conservation law satisfied?

All real systems, of course, are not closed, the sum of external forces can rarely be equal to zero. Nevertheless, in very many cases the law of conservation of momentum can be applied.

If the sum of the external forces is not zero, but the sum of the projections of the forces on some direction is equal to zero, then the projection of the momentum of the system on this direction is preserved. For example, a system of bodies on the Earth or near its surface cannot be closed, since gravity acts on all bodies, which changes the vertical momentum according to equation (9). However, along the horizontal direction, the force of gravity cannot change the momentum, and the sum of the projections of the impulses of the bodies on the horizontally directed axis will remain unchanged if the action of the resistance forces can be neglected.

In addition, during fast interactions (explosion of a projectile, a shot from a weapon, collisions of atoms, etc.), the change in the momenta of individual bodies will actually be caused only by internal forces. In this case, the momentum of the system is preserved with great accuracy, because such external forces as the force of gravity and the force of friction, which depends on the speed, do not noticeably change the momentum of the system. They are small compared to the internal forces. So, the speed of shell fragments during an explosion, depending on the caliber, can vary within 600 - 1000 m / s. The time interval for which the force of gravity could impart such a velocity to bodies is

\ (~ \ Delta t = \ frac (m \ Delta \ upsilon) (mg) \ approx 100 c \)

The internal forces of gas pressure impart such velocities in 0.01 s, i.e. 10,000 times faster.

Jet propulsion. Meshchersky's equation. Reactive force

Under jet propulsion understand the movement of a body that occurs when some part of it is separated at a certain speed relative to the body,

for example, when the combustion products flow out from the nozzle of a jet aircraft. In this case, the so-called reactive force appears, imparting acceleration to the body.

Observing jet propulsion is very simple. Inflate the baby's rubber ball and release it. The ball will rapidly rise upward (Fig. 2). The movement, however, will be short-lived. The reactive force acts only as long as the flow of air continues.

The main feature of the reactive force is that it arises without any interaction with external bodies. There is only interaction between the rocket and the stream of matter flowing out of it.

The force that imparts acceleration to a car or a pedestrian on the ground, a steamer on water or a propeller plane in the air, arises only due to the interaction of these bodies with the earth, water or air.

When the products of fuel combustion flow out, due to the pressure in the combustion chamber, they acquire a certain speed relative to the rocket and, therefore, a certain momentum. Therefore, in accordance with the law of conservation of momentum, the rocket itself receives the same pulse in modulus, but directed in the opposite direction.

The mass of the rocket decreases over time. A rocket in flight is a body of variable mass. To calculate its motion, it is convenient to apply the law of conservation of momentum.

Meshchersky's equation

Let's derive the equation of motion of the rocket and find an expression for the reactive force. We will assume that the velocity of gases flowing out of the rocket relative to the rocket is constant and equal to \ (~ \ vec u \). External forces do not act on the rocket: it is in outer space far from stars and planets.

Let at some point in time the rocket speed relative to the inertial system associated with the stars is \ (~ \ vec \ upsilon \) (Fig. 3), and the rocket mass is M... After a short time interval Δ t the mass of the rocket will be equal

\ (~ M_1 = M - \ mu \ Delta t \),

where μ - fuel consumption ( fuel consumption is called the ratio of the mass of the burned fuel to the time of its combustion).

During the same period of time, the rocket speed will change to \ (~ \ Delta \ vec \ upsilon \) and become equal to \ (~ \ vec \ upsilon_1 = \ vec \ upsilon + \ Delta \ vec \ upsilon \). The gas outflow velocity relative to the selected inertial reference system is \ (~ \ vec \ upsilon + \ vec u \) (Fig. 4), since before the start of combustion the fuel had the same speed as the rocket.

Let us write the law of conservation of momentum for the rocket - gas system:

\ (~ M \ vec \ upsilon = (M - \ mu \ Delta t) (\ vec \ upsilon + \ Delta \ vec \ upsilon) + \ mu \ Delta t (\ vec \ upsilon + \ vec u) \).

Expanding the brackets, we get:

\ (~ M \ vec \ upsilon = M \ vec \ upsilon - \ mu \ Delta t \ vec \ upsilon + M \ Delta \ vec \ upsilon - \ mu \ Delta t \ Delta \ vec \ upsilon + \ mu \ Delta t \ vec \ upsilon + \ mu \ Delta t \ vec u \).

The term \ (~ \ mu \ Delta t \ vec \ upsilon \) can be neglected in comparison with the others, since it contains the product of two small quantities (this is a quantity, as they say, of the second order of smallness). After bringing similar terms, we will have:

\ (~ M \ Delta \ vec \ upsilon = - \ mu \ Delta t \ vec u \) or \ (~ M \ frac (\ Delta \ vec \ upsilon) (\ Delta t) = - \ mu \ vec u \ ). (12)

This is one of Meshchersky's equations for the motion of a body of variable mass, obtained by him in 1897.

If we enter the notation \ (~ \ vec F_r = - \ mu \ vec u \), then equation (12) coincides in the form of notation with Newton's second law. However, body weight M here it is not constant, but decreases with time due to the loss of matter.

The value \ (~ \ vec F_r = - \ mu \ vec u \) is called reactive force... It appears due to the outflow of gases from the rocket, is applied to the rocket and is directed opposite to the speed of the gases relative to the rocket. The reactive force is determined only by the rate of outflow of gases relative to the rocket and the fuel consumption. It is essential that it does not depend on the details of the engine device. It is only important that the engine provides the outflow of gases from the rocket at a speed \ (~ \ vec u \) with a fuel consumption μ ... The reactive force of space rockets reaches 1000 kN.

If external forces act on the rocket, then its movement is determined by the reactive force and the sum of the external forces. In this case, equation (12) will be written as follows:

\ (~ M \ frac (\ Delta \ vec \ upsilon) (\ Delta t) = \ vec F_r + \ vec F \). (13)

Jet engines

Jet engines are now widely used in connection with the exploration of outer space. They are also used for meteorological and military rockets of various ranges. In addition, all modern high-speed aircraft are powered by jet engines.

In outer space, it is impossible to use any other engines besides jet ones: there is no support (solid, liquid or gaseous), pushing off from which the spacecraft could get acceleration. The use of jet engines for airplanes and rockets that do not leave the atmosphere is due to the fact that it is jet engines that are capable of providing the maximum flight speed.

Jet engines are divided into two classes: missile and air-jet.

In rocket engines, the fuel and the oxidizer necessary for its combustion are located directly inside the engine or in its fuel tanks.

Figure 5 shows a schematic of a solid propellant rocket engine. Gunpowder or some other solid fuel capable of burning in the absence of air is placed inside the combustion chamber of the engine.

When the fuel burns, gases are formed that have a very high temperature and exert pressure on the walls of the chamber. The force of pressure on the front wall of the chamber is greater than on the back, where the nozzle is located. The gases flowing out through the nozzle do not encounter a wall on their way on which they could exert pressure. The result is a force that propels the rocket forward.

The narrowed part of the chamber - the nozzle serves to increase the speed of the outflow of combustion products, which in turn increases the reactive force. The narrowing of the gas jet causes an increase in its velocity, since in this case the same gas mass must pass through the smaller cross section per unit time as with the larger cross section.

Liquid propellant rocket engines are also used.

In liquid jet engines (LRE), kerosene, gasoline, alcohol, aniline, liquid hydrogen, etc. can be used as fuel, and liquid oxygen, nitric acid, liquid fluorine, hydrogen peroxide, etc. can be used as an oxidizing agent required for combustion. The fuel and the oxidizer are stored separately in special tanks and are pumped into the chamber, where, during fuel combustion, a temperature of up to 3000 ° C and a pressure of up to 50 atm develop (Fig. 6). Otherwise, the engine operates in the same way as a solid fuel engine.

The hot gases (combustion products) exiting through the nozzle rotate the gas turbine that drives the compressor. Turbocharger engines are installed in our Tu-134, Il-62, Il-86, etc.

Jet engines are equipped not only with rockets, but also with most modern aircraft.

Advances in space exploration

The foundations of the theory of a jet engine and scientific proof of the possibility of flights in interplanetary space were first expressed and developed by the Russian scientist K.E. Tsiolkovsky in his work "Exploration of world spaces by jet devices".

K.E. Tsiolkovsky also owns the idea of ​​using multistage rockets. The individual stages that make up the rocket are supplied with their own engines and fuel reserves. As the fuel burns out, each successive stage is separated from the rocket. Therefore, in the future, no fuel is consumed to accelerate its body and engine.

Tsiolkovsky's idea of ​​building a large satellite station in orbit around the Earth, from which rockets to other planets of the solar system will be launched, has not yet been implemented, but there is no doubt that sooner or later such a station will be created.

At present, the prophecy of Tsiolkovsky is becoming a reality: "Humanity will not remain forever on Earth, but in the pursuit of light and space, at first it timidly penetrates beyond the atmosphere, and then conquers the entire solar space."

Our country has the great honor of launching the first artificial earth satellite on October 4, 1957. Also for the first time in our country on April 12, 1961, a spacecraft flight with cosmonaut Yu.A. Gagarin on board.

These flights were carried out on rockets designed by Russian scientists and engineers under the leadership of S.P. Queen. American scientists, engineers and astronauts are of great service in space exploration. Two American astronauts from the crew of the Apollo 11 spacecraft - Neil Armstrong and Edwin Aldrin - landed on the moon for the first time on July 20, 1969. The first steps were taken by man on the cosmic body of the solar system.

With the advent of man into space, not only the possibilities of exploring other planets were opened up, but truly fantastic opportunities for studying the natural phenomena and resources of the Earth were presented, which could only be dreamed of. Space science arose. Previously, the general map of the Earth was compiled bit by bit, like a mosaic panel. Now images from orbit, covering millions of square kilometers, allow you to select the most interesting areas of the earth's surface for research, thereby saving forces and resources. From space, it was possible to discover a new type of geological formations, ring structures similar to the craters of the Moon and Mars,

Now on orbital complexes have developed technologies for obtaining materials that cannot be produced on Earth, but only in a state of prolonged weightlessness in space. The cost of these materials (ultrapure single crystals, etc.) is close to the cost of launching spacecraft.

Literature

1. Physics: Mechanics. 10th grade: Textbook. for in-depth study of physics / M.M. Balashov, A.I. Gomonova, A.B. Dolitsky and others; Ed. G.Ya. Myakisheva. - M .: Bustard, 2002 .-- 496 p.

Newton's laws make it possible to solve various practically important problems concerning the interaction and motion of bodies. A large number of such problems are associated, for example, with finding the acceleration of a moving body if all forces acting on this body are known. And then other quantities are determined by acceleration (instantaneous speed, displacement, etc.).

But it is often very difficult to determine the forces acting on the body. Therefore, to solve many problems, one more important physical quantity is used - the momentum of the body.

• The momentum of a body p is a vector physical quantity equal to the product of the body's mass by its velocity

Momentum is a vector quantity. The direction of the vector of the momentum of the body always coincides with the direction of the vector of the speed of movement.

The unit of impulse in SI is the impulse of a body weighing 1 kg, moving at a speed of 1 m / s. This means that the unit of momentum of a body in SI is 1 kg m / s.

In the calculations, the equation for the vector projections is used: p x = mv x.

Depending on the direction of the velocity vector with respect to the selected X-axis, the projection of the impulse vector can be either positive or negative.

The word "impulse" (impulsus) in translation from Latin means "push". Some books use the term momentum instead of momentum.

This value was introduced into science at about the same time period when Newton discovered the laws that were later named after him (that is, at the end of the 17th century).

When bodies interact, their impulses can change. This can be verified by simple experience.

Two balls of the same mass are suspended on thread loops from a wooden ruler attached to the tripod ring, as shown in Figure 44, a.

Rice. 44. Demonstration of the law of conservation of momentum

Ball 2 is deflected from the vertical at an angle a (Fig. 44, b) and released. Returning to the previous position, he hits the ball 1 and stops. In this case, the ball 1 starts to move and is deflected by the same angle a (Fig. 44, c).

In this case, it is obvious that as a result of the interaction of the balls, the momentum of each of them has changed: by how much the momentum of ball 2 has decreased, the momentum of ball 1 has increased by the same amount.

If two or more bodies interact only with each other (that is, they are not exposed to external forces), then these bodies form a closed system.

The momentum of each of the bodies included in a closed system can change as a result of their interaction with each other. But

• the vector sum of the impulses of the bodies that make up a closed system does not change over time for any motions and interactions of these bodies

This is the law of conservation of momentum.

The law of conservation of momentum is also fulfilled if external forces act on the bodies of the system, the vector sum of which is equal to zero. Let us show this using the second and third Newton's laws to derive the law of conservation of momentum. For simplicity, consider a system consisting only of two bodies - balls of masses m 1 and m 2, which move rectilinearly towards each other with velocities v 1 and v 2 (Fig. 45).

Rice. 45. A system of two bodies - balls moving in a straight line towards each other

The forces of gravity acting on each of the balls are balanced by the forces of elasticity of the surface on which they roll. This means that the action of these forces can be ignored. The forces of resistance to movement in this case are small, so we will not take their influence into account either. Thus, we can assume that the balls interact only with each other.

Figure 45 shows that after a while the balls will collide. During the collision, which lasts for a very short time interval t, the forces of interaction F 1 and F 2 will arise, applied to the first and second balls, respectively. As a result of the action of the forces, the speeds of the balls will change. Let us denote the velocities of the balls after collision by the letters v 1 and v 2.

In accordance with Newton's third law, the forces of interaction of balls are equal in magnitude and directed in opposite directions:

According to Newton's second law, each of these forces can be replaced by the product of mass and acceleration obtained by each of the balls during interaction:

m 1 a 1 = -m 2 a 2.

Accelerations, as you know, are determined from the equalities:

Replacing the acceleration forces in the equation with the corresponding expressions, we get:

As a result of reducing both sides of the equality by t, we get:

m1 (v "1 - v 1) = -m 2 (v" 2 - v 2).

We group the terms of this equation as follows:

m 1 v 1 "+ m 2 v 2" = m 1 v 1 = m 2 v 2. (one)

Taking into account that mv = p, we write equation (1) in the following form:

P "1 + P" 2 = P 1 + P 2. (2)

The left-hand sides of equations (1) and (2) represent the total momentum of the balls after their interaction, and the right-hand sides - the total momentum before the interaction.

This means that in spite of the fact that the momentum of each of the balls changed during the interaction, the vector sum of their momenta after the interaction remained the same as before the interaction.

Equations (1) and (2) are a mathematical representation of the momentum conservation law.

Since in this course only interactions of bodies moving along one straight line are considered, then to write the law of conservation of momentum in scalar form, one equation is sufficient, which includes the projections of vector quantities on the X axis:

m 1 v "1x + m 2 v" 2x = m 1 v 1x + m 2 v 2x.

Questions

1. What is called the impulse of the body?
2. What can be said about the directions of the momentum vectors and the velocity of a moving body?
3. Tell us about the course of the experiment shown in Figure 44. What does it testify to?
4. What does the statement that several bodies form a closed system mean?
5. Formulate the law of conservation of momentum.
6. For a closed system consisting of two bodies, write down the law of conservation of momentum in the form of an equation that would include the masses and velocities of these bodies. Explain what each symbol in this equation means.

Exercise # 20

1. Two toy clockwork machines, each weighing 0.2 kg, move in a straight line towards each other. The speed of each machine relative to the ground is 0.1 m / s. Are the vectors of the impulses of the machines equal; modules of the vectors of impulses? Determine the projection of the momentum of each of the machines on the X-axis, parallel to their path.
2. How much will the impulse of a car weighing 1 ton change (in modulus) when its speed changes from 54 to 72 km / h?
3. A man sits in a boat resting on the surface of the lake. At some point, he gets up and walks from the stern to the bow. What happens to the boat? Explain the phenomenon based on the law of conservation of momentum.
4. A railway car weighing 35 tons drives up to a stationary car weighing 28 tons standing on the same track and automatically interlocks with it. After coupling, the cars move in a straight line at a speed of 0.5 m / s. What was the speed of a 35-ton car in front of the hitch?

Themes of the 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 body's mass by its velocity:

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

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 impulse form

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

Taking into account 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 derivative sign:

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

. ( 1 )

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

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

You can also say this: the resulting force acting on the body is equal to the rate of change in the body's momentum.

The derivative in 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 ratio is to the derivative, and the closer the average force is to its instantaneous value at a given moment in time.

In tasks, as a rule, the time interval is rather short. For example, it can be the time the ball hits the wall, and then the average force acting on the ball from the side of the wall during the strike.

The vector on the left-hand side of relation (2) is called change of momentum during . The change in momentum is the difference between the final and initial vectors of the momentum. 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 again that the change in momentum is the difference of vectors (Fig. 1):

For example, let the ball fly perpendicular to the wall (the impulse before the impact is equal) and bounces back without losing speed (the impulse after the impact is equal). Despite the fact that the modulus of the impulse has not changed (), there is a change in the impulse:

Geometrically, this situation is shown in Fig. 2:

The modulus of the impulse change, as we can see, is equal to the doubled modulus of the initial impulse of the ball:.

Let's rewrite formula (2) as follows:

, ( 3 )

or, describing the change in momentum, as above:

The quantity is called impulse of power. There is no special unit of measure for force impulse; the dimension of the impulse of force 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 impulse form.

Force calculation example

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

A task. A ball of mass g, flying horizontally at a speed of m / s, hits a smooth vertical wall and bounces off it without losing speed. The angle of incidence of the ball (that is, the angle between the direction of movement of the ball and the perpendicular to the wall) is equal to. Strike lasts for. Find the average strength,
acting on the ball during impact.

Solution. Let us show first of all 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:. Hence it follows that the vector of change in momentum co-directional with a vector, that is, directed perpendicular to the wall in the direction of the ball's rebound (Fig. 5).

Rice. 5. To the task

Vectors and
equal in modulus
(since the speed of the ball has not changed). Therefore, a triangle composed of vectors and is isosceles. This means that the angle between the vectors and is equal, that is, the angle of reflection is really equal to the angle of incidence.

Now note, in addition, that our isosceles triangle has an angle (this is the angle of incidence); therefore, this triangle is equilateral. Hence:

And then the required average force acting on the ball:

The impulse of the system of bodies

Let's start with a simple situation for a two-body system. Namely, let there be body 1 and body 2 with impulses and, respectively. The momentum of the system of these bodies 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 deduce this formula.

All other objects with which the bodies 1 and 2 we are considering interact, we will call external bodies. The forces with which external bodies act on bodies 1 and 2 are called external forces. Let be 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 magnitude and opposite in direction:. Forces and is internal forces, operating in the system.

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

, ( 4 )

. ( 5 )

Let us add equalities (4) and (5):

On the left side of the obtained equality is the sum of derivatives, equal to the derivative of the sum of 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. Let's also designate - this is the resultant of external forces acting on the system. We get:

. ( 6 )

Thus, the rate of change of the 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 a system of bodies, is what we wanted to obtain.

Formula (6) was derived for the case of two bodies. Now let us 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:

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 down equalities similar to (4) and (5), and then add all these equalities, then on the left side we will again get the derivative of the momentum of the system, and on the right side there will be only the sum of external forces (internal forces, adding in pairs, will give zero in view of Newton's third law). Therefore, equality (6) remains valid in the general case.

Momentum conservation law

The system of bodies is called closed, if the actions of external bodies on the bodies of a given system are either negligible or cancel each other out. 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 the closed system is zero:. In this case, from (6) we obtain:

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:

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

Solution. The situation is shown in Fig. 7. The axis is directed towards the movement of the first body.

Rice. 7. To the task

Since the surface is smooth, there is no friction. Since the surface is horizontal and 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, the law of conservation of momentum is fulfilled for it:

. ( 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 required speed:

From the law of conservation of momentum (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 stuck together bodies move in the direction opposite to the axis. Seeking speed: m / s.

Impulse projection conservation law

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

Let us show this more rigorously. Let's project equality (6) onto the axis:

If the projection of the resultant external forces vanishes, then

Therefore, the projection is a constant:

Impulse projection conservation law. If the projection onto the axis of the sum of external forces acting on the system is zero, then the projection of the momentum of the system does not change over 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, standing on skates on smooth ice, throws a mass stone at a speed at an angle to the horizon. Find the speed with which the boy rolls back after being thrown.

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

Rice. 8. To the task

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

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

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

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

Determination of body impulse: formula

It turns out that force is characterized by a change in the product of mass and speed in time. If we designate this product by a certain value, then we will receive the change in this value over time as a characteristic of force. This value was called the momentum of the body. The body impulse 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, while its direction always coincides with the direction of the 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 in a simple way, "on the fingers" to figure out what a body impulse is. If the body is at rest, then its momentum is zero. It is logical. If the speed of the body changes, then a certain impulse appears in the body, which characterizes the magnitude of the force applied to it.

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

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

A small force is needed to move a body of small mass. The more body weight, the more effort will have to be made. The same goes for the speed that is imparted to the body. In the case of the action of the body itself on another, the impulse also shows the amount 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 in the 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 change easily. In this case, the speed of the body will change depending on its mass.

Indeed, it is clear that when bodies with very different masses collide, their speed will change in different ways. If a soccer ball flying at high speed crashes into an unprepared person, for example a spectator, then the viewer may fall, that is, gain some low speed, but will definitely 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 impulse of these two bodies will remain unchanged.

The 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 acts only in a closed system, that is, in a system in which there is no external force or their total action is zero.

In reality, there is almost always an outside influence on the system of bodies, but the total impulse, like energy, does not disappear into anywhere and does not arise out of nowhere, it is distributed among all participants in the interaction.

Impulse in physics

In translation from Latin "impulse" means "push". This physical quantity is also called "quantity of motion". It was introduced into science at about the same time as Newton's laws were discovered (at the end of the 17th century).

The branch of physics that studies the movement and interaction of material bodies is mechanics. Momentum in mechanics is a vector quantity equal to the product of the mass of a body by its velocity: p = mv. The directions of the momentum and velocity vectors always coincide.

In the SI system, the unit of impulse is taken as the impulse of a body weighing 1 kg, which moves at a speed of 1 m / s. Therefore, the SI unit of momentum is 1 kg ∙ m / s.

In computational problems, the projections of the velocity and momentum vectors on any axis are considered and equations for these projections are used: for example, if the x axis is selected, then the projections v (x) and p (x) are considered. By definition of momentum, these quantities are related by the relationship: p (x) = mv (x).

Depending on which axis is selected and where it is directed, the projection of the impulse vector onto it can be either positive or negative.

Momentum conservation law

The impulses of material bodies during their physical interaction can change. For example, when two balls, suspended on threads, collide, their impulses mutually change: one ball can move from a stationary state or increase its speed, while the other, on the contrary, can decrease its speed or stop. However, in a closed system, i.e. when the bodies interact only with each other and are not subject to external forces, the vector sum of the impulses of these bodies remains constant for any of their interactions and movements. This is the law of conservation of momentum. Mathematically, it can be deduced from Newton's laws.

The law of conservation of momentum is also applicable to such systems where some external forces act on bodies, but their vector sum is equal to zero (for example, the force of gravity is balanced by the force of elasticity of the surface). Conventionally, such a system can also be considered closed.

In mathematical form, the law of conservation of momentum is written as follows: p1 + p2 +… + p (n) = p1 ’+ p2’ +… + p (n) ’(momenta p are vectors). For a two-body system, this equation looks like p1 + p2 = p1 ’+ p2’, or m1v1 + m2v2 = m1v1 ’+ m2v2’. For example, in the considered case with balls, the total momentum of both balls before interaction will be equal to the total momentum after interaction.