The momentum of a body is a quantity equal to. Law of conservation of momentum, kinetic and potential energies, force power

1. As you know, the result of a force depends on its magnitude, point of application and direction. Indeed, the greater the force acting on the body, the greater the acceleration it acquires. The direction of the acceleration also depends on the direction of the force. So, by applying a small force to the handle, we can easily open the door, but if we apply the same force near the hinges on which the door hangs, then it may not be possible to open it.

Experiments and observations indicate that the result of a force (interaction) depends not only on the modulus of the force, but also on the time of its action. Let's do an experiment. We hang a load on a thread from the tripod, to which another thread is tied from below (Fig. 59). If you pull the lower thread sharply, it will break, and the load will remain hanging on the upper thread. If you now slowly pull the bottom thread, the top thread will break.

The impulse of force is a vector physical quantity equal to the product of force and the time of its action F t .

The SI unit of impulse of force is newton second (1 N s): [Ft] = 1 N s.

The force impulse vector coincides in direction with the force vector.

2. You also know that the result of a force depends on the mass of the body on which the force acts. Thus, the greater the mass of a body, the less acceleration it acquires under the action of the same force.

Let's look at an example. Let's imagine that there is a loaded platform on the rails. A carriage moving at some speed collides with it. As a result of the collision, the platform will acquire acceleration and move a certain distance. If a carriage moving at the same speed collides with a light trolley, then as a result of the interaction it will move a significantly greater distance than a loaded platform.

Another example. Let's assume that a bullet approaches the target at a speed of 2 m/s. The bullet will most likely bounce off the target, leaving only a small dent in it. If the bullet flies at a speed of 100 m/s, then it will pierce the target.

Thus, the result of the interaction of bodies depends on their mass and speed of movement.

The momentum of a body is a vector physical quantity equal to the product of the mass of the body and its speed.

The SI unit of momentum of a body is kilogram-meter per second(1 kg m/s): [ p] = [m][v] = 1 kg 1m/s = 1 kg m/s.

The direction of the body's momentum coincides with the direction of its speed.

Momentum is a relative quantity; its value depends on the choice of reference system. This is understandable, since speed is a relative quantity.

3. Let us find out how the impulse of force and the impulse of the body are related.

According to Newton's second law:

F = ma.

Substituting the expression for acceleration into this formula a= , we get:

F= , or
Ft = mv – mv 0 .

On the left side of the equation is the impulse of force; on the right side of the equality is the difference between the final and initial impulses of the body, i.e. e. change in the momentum of the body.

Thus,

the impulse of force is equal to the change in the momentum of the body.

This is a different formulation of Newton's second law. This is exactly how Newton formulated it.

4. Let's assume that two balls moving on a table collide. Any interacting bodies, in this case balls, form system. Forces act between the bodies of the system: action force F 1 and counter force F 2. At the same time, the force of action F 1 according to Newton's third law is equal to the reaction force F 2 and is directed opposite to it: F 1 = –F 2 .

The forces with which the bodies of the system interact with each other are called internal forces.

In addition to internal forces, external forces act on the bodies of the system. Thus, the interacting balls are attracted to the Earth and are acted upon by the support reaction force. These forces are in this case external forces. During movement, the balls are subject to air resistance and friction. They are also external forces in relation to the system, which in this case consists of two balls.

External forces are forces that act on the bodies of a system from other bodies.

We will consider a system of bodies that is not affected by external forces.

A closed system is a system of bodies that interact with each other and do not interact with other bodies.

In a closed system, only internal forces act.

5. Let us consider the interaction of two bodies that make up a closed system. Mass of the first body m 1, its speed before interaction v 01, after interaction v 1 . Mass of the second body m 2, its speed before interaction v 02 , after interaction v 2 .

The forces with which bodies interact, according to the third law: F 1 = –F 2. The time of action of the forces is the same, therefore

F 1 t = –F 2 t.

For each body we write Newton’s second law:

F 1 t = m 1 v 1 – m 1 v 01 , F 2 t = m 2 v 2 – m 2 v 02 .

Since the left sides of the equalities are equal, then their right sides are equal, i.e.

m 1 v 1 – m 1 v 01 = –(m 2 v 2 – m 2 v 02).

Transforming this equality, we get:

On the left side of the equation is the sum of the momenta of the bodies before the interaction, on the right is the sum of the momenta of the bodies after the interaction. As can be seen from this equality, the momentum of each body changed during interaction, but the sum of the impulses remained unchanged.

The geometric sum of the momenta of the bodies that make up a closed system remains constant for any interactions of the bodies of this system.

This is law of conservation of momentum.

6. A closed system of bodies is a model of a real system. There are no systems in nature that are not affected by external forces. However, in a number of cases, systems of interacting bodies can be considered closed. This is possible in the following cases: internal forces are much greater than external forces, interaction time is short, external forces compensate each other. In addition, the projection of external forces to any direction may be equal to zero, and then the law of conservation of momentum is satisfied for the projections of the impulses of interacting bodies to this direction.

7. Example of problem solution

Two railway platforms are moving towards each other at speeds of 0.3 and 0.2 m/s. The masses of the platforms are respectively 16 and 48 tons. At what speed and in what direction will the platforms move after automatic coupling?

and apply the law of conservation of momentum to it.

m 1 v 01 + m 2 v 02 = (m 1 + m 2)v.

In projections onto the axis X can be written:

m 1 v 01x + m 2 v 02x = (m 1 + m 2)v x.

Because v 01x = v 01 ; v 02x = –v 02 ; v x = – v, That m 1 v 01 – m 2 v 02 = –(m 1 + m 2)v.

Where v = – .

v= – = 0.75 m/s.

After coupling, the platforms will move in the direction in which the platform with the larger mass moved before the interaction.

Answer: v= 0.75 m/s; directed in the direction of movement of the cart with the greater mass.

Self-test questions

1. What is the impulse of a body?

2. What is called a force impulse?

3. How are the impulse of a force and the change in the momentum of a body related?

4. What system of bodies is called closed?

5. Formulate the law of conservation of momentum.

6. What are the limits of applicability of the law of conservation of momentum?

Task 17

1. What is the momentum of a body weighing 5 kg moving at a speed of 20 m/s?

2. Determine the change in momentum of a body weighing 3 kg in 5 s under the influence of a force of 20 N.

3. Determine the momentum of a car with a mass of 1.5 tons moving at a speed of 20 m/s in a reference frame associated with: a) a car stationary relative to the Earth; b) with a car moving in the same direction at the same speed; c) with a car moving at the same speed, but in the opposite direction.

4. A boy weighing 50 kg jumped from a stationary boat weighing 100 kg located in the water near the shore. At what speed did the boat move away from the shore if the boy’s speed is directed horizontally and is equal to 1 m/s?

5. A projectile weighing 5 kg, flying horizontally, exploded into two fragments. What is the speed of the projectile if a fragment weighing 2 kg at the explosion acquired a speed of 50 m/s, and a second fragment weighing 3 kg acquired a speed of 40 m/s? The velocities of the fragments are directed horizontally.

Impulse(quantity of motion) of a body is a physical vector quantity, which is a quantitative characteristic of the translational motion of bodies. The impulse is designated 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 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. With an absolutely elastic impact on a fixed surface, the speed of the object after the impact is equal in magnitude 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 is equal to the angle of reflection.

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 in momentum 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 ​​of the acting forces are 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 the general case, in order to apply the law of conservation of momentum, it is necessary to describe it along all coordinate axes involved in the problem. But solving such a problem can be greatly simplified if you use the 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.

Having studied Newton's laws, we see that with their help it is possible to solve the basic problems of mechanics if we know all the forces acting on the body. There are situations in which it is difficult or even impossible to determine these values. Let's consider several such situations.When two billiard balls or cars collide, we can assert about the forces at work that this is their nature; elastic forces act here. However, we will not be able to accurately determine either their modules or their directions, especially since these forces have an extremely short duration of action.With the movement of rockets and jet planes, we also can say little about the forces that set these bodies in motion.In such cases, methods are used that allow one to avoid solving the equations of motion and immediately use the consequences of these equations. In this case, new physical quantities are introduced. Let's consider one of these quantities, called the momentum of the body

An arrow fired from a bow. The longer the contact of the string with the arrow continues (∆t), the greater the change in the arrow's momentum (∆), and therefore, the higher its final speed.

Two colliding balls. While the balls are in contact, they act on each other with forces equal in magnitude, as Newton’s third law teaches us. This means that the changes in their momenta must also be equal in magnitude, even if the masses of the balls are not equal.

After analyzing the formulas, two important conclusions can be drawn:

1. Identical forces acting for the same period of time cause the same changes in momentum in different bodies, regardless of the mass of the latter.

2. The same change in the momentum of a body can be achieved either by acting with a small force over a long period of time, or by acting briefly with a large force on the same body.

According to Newton's second law, we can write:

∆t = ∆ = ∆ / ∆t

The ratio of the change in the momentum of a body to the period of time during which this change occurred is equal to the sum of the forces acting on the body.

Having analyzed this equation, we see that Newton's second law allows us to expand the class of problems to be solved and include problems in which the mass of bodies changes over time.

If we try to solve problems with variable mass of bodies using the usual formulation of Newton’s second law:

then attempting such a solution would lead to an error.

An example of this is the already mentioned jet plane or space rocket, which burn fuel while moving, and the products of this combustion are released into the surrounding space. Naturally, the mass of an aircraft or rocket decreases as fuel is consumed.

Despite the fact that Newton’s second law in the form “the resultant force is equal to the product of the mass of a body and its acceleration” allows us to solve a fairly wide class of problems, there are cases of motion of bodies that cannot be fully described by this equation. In such cases, it is necessary to apply another formulation of the second law, connecting the change in the momentum of the body with the impulse of the resultant force. In addition, there are a number of problems in which solving the equations of motion is mathematically extremely difficult or even impossible. In such cases, it is useful for us to use the concept of momentum.

Using the law of conservation of momentum and the relationship between the momentum of a force and the momentum of a body, we can derive Newton's second and third laws.

Newton's second law is derived from the relationship between the impulse of a force and the momentum of a body.

The impulse of force is equal to the change in the momentum of the body:

Having made the appropriate transfers, we obtain the dependence of force on acceleration, because acceleration is defined as the ratio of the change in speed to the time during which this change occurred:

Substituting the values ​​into our formula, we get the formula for Newton’s second law:

To derive Newton's third law, we need the law of conservation of momentum.

Vectors emphasize the vector nature of speed, that is, the fact that speed can change in direction. After transformations we get:

Since the period of time in a closed system was a constant value for both bodies, we can write:

We have obtained Newton's third law: two bodies interact with each other with forces equal in magnitude and opposite in direction. The vectors of these forces are directed towards each other, respectively, the modules of these forces are equal in value.

Bibliography

1. Tikhomirova S.A., Yavorsky B.M. Physics (basic level) - M.: Mnemosyne, 2012.
2. Gendenshtein L.E., Dick Yu.I. Physics 10th grade. - M.: Mnemosyne, 2014.
3. Kikoin I.K., Kikoin A.K. Physics - 9, Moscow, Education, 1990.

Homework

1. Define the impulse of a body, the impulse of force.
2. How are the impulse of a body related to the impulse of force?
3. What conclusions can be drawn from the formulas for body impulse and force impulse?

1. Internet portal Questions-physics.ru ().
2. Internet portal Frutmrut.ru ().
3. Internet portal Fizmat.by ().

Instructions

Find the mass of the moving body and measure its motion. After its interaction with another body, the speed of the body under study will change. In this case, subtract the initial speed from the final (after interaction) and multiply the difference by the body mass Δp=m∙(v2-v1). Measure the instantaneous speed with a radar and the body mass with a scale. If, after the interaction, the body begins to move in the direction opposite to that in which it moved before the interaction, then the final speed will be negative. If it is positive, it has increased, if negative, it has decreased.

Since the cause of a change in the speed of any body is force, it is also the cause of a change in momentum. To calculate the change in momentum of any body, it is enough to find the momentum of the force acting on this body at some time. Using a dynamometer, measure the force that causes a body to change speed, giving it acceleration. At the same time, use a stopwatch to measure the time that this force acts on the body. If a force causes a body to move, then consider it positive, but if it slows down its movement, consider it negative. An impulse of force equal to the change in impulse will be the product of the force and the time of its action Δp=F∙Δt.

Determining instantaneous speed with a speedometer or radar If a moving body is equipped with a speedometer (), then instantaneous speed will be continuously displayed on its scale or electronic display speed at a given moment in time. When observing a body from a fixed point (), send a radar signal to it, an instantaneous signal will be displayed on its display speed bodies at a given moment in time.

Video on the topic

Force is a physical quantity acting on a body, which, in particular, imparts some acceleration to it. To find pulse strength, you need to determine the change in momentum, i.e. pulse but the body itself.

Instructions

The movement of a material point under the influence of some strength or forces that give it acceleration. Application result strength a certain amount for a certain amount is the corresponding quantity. Impulse strength the measure of its action over a certain period of time is called: Pc = Fav ∆t, where Fav is the average force acting on the body; ∆t is the time interval.

Thus, pulse strength equal to change pulse and the body: Pc = ∆Pt = m (v – v0), where v0 is the initial speed; v is the final speed of the body.

The resulting equality reflects Newton's second law in relation to the inertial reference system: the derivative of the function of a material point with respect to time is equal to the magnitude of the constant force acting on it: Fav ∆t = ∆Pt → Fav = dPt/dt.

Total pulse a system of several bodies can change only under the influence of external forces, and its value is directly proportional to their sum. This statement is a consequence of Newton's second and third laws. Let there be three interacting bodies, then it is true: Pс1 + Pc2 + Pc3 = ∆Pт1 + ∆Pт2 + ∆Pт3, where Pci – pulse strength, acting on the body i;Pтi – pulse bodies i.

This equality shows that if the sum of external forces is zero, then the total pulse closed system of bodies is always constant, despite the fact that the internal strength

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

If a toy cart is rolling towards you, you can stop it with your toe. If a truck is rolling towards you, you should move your feet out of its path.

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

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

Change in body momentum

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

1) During a strike, two forces act on the ball: ground reaction force, gravity.

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

2) Change in momentum body shown in the picture

3) From Newton's second law

The main thing to remember

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

Derivation of Newton's second law in general form

Graph F(t). Variable force

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

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

Average resultant force

LAW OF CONSERVATION OF MOMENTUM

Testing online

Closed system of bodies

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

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

Law of conservation of momentum

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

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

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

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

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

Since the bodies move in opposite directions, we obtain a vector sum of multidirectional impulses. Read more about vector sum.

The main thing to remember

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