In classical mechanics, there are two conservation laws: the law of conservation of momentum and the law of conservation of energy.
Body impulse
The concept of momentum was first introduced by a French mathematician, physicist, and mechanic. and the philosopher Descartes, who called impulse amount of movement .
From Latin, “impulse” is translated as “push, move.”
Any body that moves has momentum.
Let's imagine a cart standing still. Its momentum is zero. But as soon as the cart starts moving, its momentum will no longer be zero. It will begin to change as the speed changes.
Momentum of a material point, or momentum, – a vector quantity equal to the product of the mass of a point and its speed. The direction of the point's momentum vector coincides with the direction of the velocity vector.
If we are talking about a solid physical body, then the momentum of such a body is called the product of the mass of this body and the speed of the center of mass.
How to calculate the momentum of a body? One can imagine that the body consists of many material points, or systems of material points.
If - the impulse of one material point, then the impulse of a system of material points
That is, momentum of a system of material points is the vector sum of the momenta of all material points included in the system. It is equal to the product of the masses of these points and their speed.
The unit of impulse in the international system of units SI is kilogram-meter per second (kg m/sec).
Impulse force
In mechanics, there is a close connection between the momentum of a body and force. These two quantities are connected by a quantity called impulse of force .
If a constant force acts on a bodyF over a period of time t , then according to Newton's second law
This formula shows the relationship between the force that acts on the body, the time of action of this force and the change in the speed of the body.
The quantity equal to the product of the force acting on a body and the time during which it acts is called impulse of force .
As we see from the equation, the impulse of force is equal to the difference in the impulses of the body at the initial and final moments of time, or the change in impulse over some time.
Newton's second law in momentum form is formulated as follows: the change in the momentum of a body is equal to the momentum of the force acting on it. It must be said that Newton himself originally formulated his law in exactly this way.
Force impulse is also a vector quantity.
The law of conservation of momentum follows from Newton's third law.
It must be remembered that this law operates only in a closed, or isolated, physical system. A closed system is a system in which bodies interact only with each other and do not interact with external bodies.
Let us imagine a closed system of two physical bodies. The forces of interaction of bodies with each other are called internal forces.
The force impulse for the first body is equal to
According to Newton's third law, the forces that act on bodies during their interaction are equal in magnitude and opposite in direction.
Therefore, for the second body the momentum of the force is equal to
By simple calculations we obtain a mathematical expression for the law of conservation of momentum:
Where m 1 And m 2 – body masses,
v 1 And v 2 – velocities of the first and second bodies before interaction,
v 1" And v 2" – velocities of the first and second bodies after interaction .
p 1 = m 1 · v 1 - momentum of the first body before interaction;
p 2 = m 2 · v 2 - momentum of the second body before interaction;
p 1 "= m 1 · v 1" - momentum of the first body after interaction;
p 2 "= m 2 · v 2" - momentum of the second body after interaction;
That is
p 1 + p 2 = p 1" + p 2"
In a closed system, bodies only exchange impulses. And the vector sum of the momenta of these bodies before their interaction is equal to the vector sum of their momenta after the interaction.
So, as a result of firing a gun, the momentum of the gun itself and the momentum of the bullet will change. But the sum of the impulses of the gun and the bullet in it before the shot will remain equal to the amount impulses of a gun and a flying bullet after a shot.
When firing a cannon, there is recoil. The projectile flies forward, and the gun itself rolls back. The projectile and the gun are a closed system in which the law of conservation of momentum operates.
The momentum of each body in a closed system can change as a result of their interaction with each other. But the vector sum of the impulses of bodies included in a closed system does not change when these bodies interact over time, that is, it remains constant. That's what it is law of conservation of momentum.
More precisely, the law of conservation of momentum is formulated as follows: the vector sum of the impulses of all bodies of a closed system is a constant value if there are no external forces acting on it, or their vector sum is equal to zero.
The momentum of a system of bodies can change only as a result of the action of external forces on the system. And then the law of conservation of momentum will not apply.
It must be said that closed systems do not exist in nature. But, if the time of action of external forces is very short, for example, during an explosion, shot, etc., then in this case the influence of external forces on the system is neglected, and the system itself is considered as closed.
In addition, if external forces act on the system, but the sum of their projections onto one of the coordinate axes is zero (that is, the forces are balanced in the direction of this axis), then the law of conservation of momentum is satisfied in this direction.
The law of conservation of momentum is also called law of conservation of momentum .
Most shining example application of the law of conservation of momentum - reactive motion.
Jet propulsion
Reactive motion is the movement of a body that occurs when some part of it is separated from it at a certain speed. The body itself receives an oppositely directed impulse.
The simplest example of jet propulsion is the flight of a balloon from which air escapes. If we inflate a balloon and release it, it will begin to fly in the direction opposite to the movement of the air coming out of it.
An example of jet propulsion in nature is the release of liquid from the fruit of a crazy cucumber when it bursts. At the same time, the cucumber itself flies in the opposite direction.
Jellyfish, cuttlefish and other inhabitants of the deep sea move by taking in water and then throwing it out.
Jet thrust is based on the law of conservation of momentum. We know that when a rocket with a jet engine moves, as a result of fuel combustion, a jet of liquid or gas is ejected from the nozzle ( jet stream ). As a result of the interaction of the engine with the escaping substance, Reactive force . Since the rocket, together with the emitted substance, is a closed system, the momentum of such a system does not change with time.
Reactive force arises from the interaction of only parts of the system. External forces have no influence on its appearance.
Before the rocket began to move, the sum of the impulses of the rocket and the fuel was zero. Consequently, according to the law of conservation of momentum, after the engines are turned on, the sum of these impulses is also zero.
where is the mass of the rocket
Gas flow rate
Changing rocket speed
∆m f - fuel consumption
Suppose the rocket operated for a period of time t .
Dividing both sides of the equation by ∆ t, we get the expression
According to Newton's second law, the reactive force is equal to
Reaction force, or jet thrust, ensures the movement of the jet engine and the object associated with it in the direction opposite to the direction of the jet stream.
Jet engines are used in modern aircraft and various missiles, military, space, etc.
Momentum is one of the most fundamental characteristics of a physical system. The momentum of a closed system is conserved during any processes occurring in it.
Let's start getting acquainted with this quantity with the simplest case. The momentum of a material point of mass moving with speed is the product
Law of momentum change. From this definition, using Newton's second law, we can find the law of change in the momentum of a particle as a result of the action of some force on it. By changing the speed of a particle, the force also changes its momentum: . In the case of a constant acting force, therefore
The rate of change of momentum of a material point is equal to the resultant of all forces acting on it. With a constant force, the time interval in (2) can be taken by anyone. Therefore, for the change in momentum of a particle during this interval, it is true
In the case of a force that changes over time, the entire period of time should be divided into small intervals during each of which the force can be considered constant. The change in particle momentum over a separate period is calculated using formula (3):
The total change in momentum over the entire time period under consideration is equal to the vector sum of changes in momentum over all intervals
If we use the concept of derivative, then instead of (2), obviously, the law of change in particle momentum is written as
Impulse of force. The change in momentum over a finite period of time from 0 to is expressed by the integral
The quantity on the right side of (3) or (5) is called the impulse of force. Thus, the change in the momentum Dr of a material point over a period of time is equal to the impulse of the force acting on it during this period of time.
Equalities (2) and (4) are essentially another formulation of Newton's second law. It was in this form that this law was formulated by Newton himself.
The physical meaning of the concept of impulse is closely related to the intuitive idea that each of us has, or one drawn from everyday experience, about whether it is easy to stop a moving body. What matters here is not the speed or mass of the body being stopped, but both together, i.e., precisely its momentum.
System impulse. The concept of momentum becomes especially meaningful when it is applied to a system of interacting material points. The total momentum P of a system of particles is the vector sum of the momenta of individual particles at the same moment in time:
Here the summation is performed over all particles included in the system, so that the number of terms is equal to the number of particles in the system.
Internal and external forces. It is easy to come to the law of conservation of momentum of a system of interacting particles directly from Newton’s second and third laws. We will divide the forces acting on each of the particles included in the system into two groups: internal and external. Internal force is the force with which a particle acts on the External force is the force with which all bodies that are not part of the system under consideration act on the particle.
The law of change in particle momentum in accordance with (2) or (4) has the form
Let us add equation (7) term by term for all particles of the system. Then on the left side, as follows from (6), we obtain the rate of change
total momentum of the system Since the internal forces of interaction between particles satisfy Newton’s third law:
then when adding equations (7) on the right side, where internal forces occur only in pairs, their sum will go to zero. As a result we get
The rate of change of total momentum is equal to the sum of the external forces acting on all particles.
Let us pay attention to the fact that equality (9) has the same form as the law of change in the momentum of one material point, and the right side includes only external forces. In a closed system, where there are no external forces, the total momentum P of the system does not change regardless of what internal forces act between the particles.
The total momentum does not change even in the case when the external forces acting on the system are equal to zero in total. It may turn out that the sum of external forces is zero only along a certain direction. Although the physical system in this case is not closed, the component of the total momentum along this direction, as follows from formula (9), remains unchanged.
Equation (9) characterizes the system of material points as a whole, but refers to a certain point in time. From it it is easy to obtain the law of change in the momentum of the system over a finite period of time. If the acting external forces are constant during this interval, then from (9) it follows
If external forces change with time, then on the right side of (10) there will be a sum of integrals over time from each of the external forces:
Thus, the change in the total momentum of a system of interacting particles over a certain period of time is equal to the vector sum of the impulses of external forces over this period.
Comparison with the dynamic approach. Let us compare approaches to solving mechanical problems based on dynamic equations and based on the law of conservation of momentum using the following simple example.
A railway car of mass taken from a hump, moving at a constant speed, collides with a stationary car of mass and is coupled with it. At what speed do the coupled cars move?
We know nothing about the forces with which the cars interact during a collision, except for the fact that, based on Newton's third law, they are equal in magnitude and opposite in direction at each moment. With a dynamic approach, it is necessary to specify some kind of model for the interaction of cars. The simplest possible assumption is that the interaction forces are constant throughout the entire time the coupling occurs. In this case, using Newton’s second law for the speeds of each of the cars, after the start of the coupling, we can write
Obviously, the coupling process ends when the speeds of the cars become the same. Assuming that this happens after time x, we have
From here we can express the impulse of force
Substituting this value into any of formulas (11), for example into the second, we find the expression for the final speed of the cars:
Of course, the assumption made about the constancy of the force of interaction between the cars during the process of their coupling is very artificial. The use of more realistic models leads to more cumbersome calculations. However, in reality, the result for the final speed of the cars does not depend on the interaction pattern (of course, provided that at the end of the process the cars are coupled and moving at the same speed). The easiest way to verify this is to use the law of conservation of momentum.
Since no external forces in the horizontal direction act on the cars, the total momentum of the system remains unchanged. Before the collision, it is equal to the momentum of the first car. After coupling, the momentum of the cars is equal. Equating these values, we immediately find
which, naturally, coincides with the answer obtained on the basis of the dynamic approach. The use of the law of conservation of momentum made it possible to find the answer to the question posed using less cumbersome mathematical calculations, and this answer is more general, since it was obtained without using any specific model interactions.
Let us illustrate the application of the law of conservation of momentum of a system using an example of more difficult task, where choosing a model for a dynamic solution is already difficult.
Task
Shell explosion. The projectile explodes at the top point of the trajectory, located at a height above the surface of the earth, into two identical fragments. One of them falls to the ground exactly below the point of explosion after a time. How many times will the horizontal distance from this point at which the second fragment will fly away change, compared to the distance at which an unexploded shell would fall?
Solution: First of all, let's write an expression for the distance over which an unexploded shell would fly. Since the speed of the projectile at the top point (we denote it by is directed horizontally), then the distance is equal to the product of the time of falling from a height without an initial speed, equal to which an unexploded projectile would fly away. Since the speed of the projectile at the top point (we denote it by is directed horizontally, then the distance is equal to the product of the time of falling from a height without an initial speed, equal to the body considered as a system of material points:
The bursting of a projectile into fragments occurs almost instantly, i.e., the internal forces tearing it apart act within a very short period of time. It is obvious that the change in the velocity of the fragments under the influence of gravity over such a short period of time can be neglected in comparison with the change in their speed under the influence of these internal forces. Therefore, although the system under consideration, strictly speaking, is not closed, we can assume that its total momentum when the projectile ruptures remains unchanged.
From the law of conservation of momentum one can immediately identify some features of the movement of fragments. Momentum is a vector quantity. Before the explosion, it lay in the plane of the projectile trajectory. Since, as stated in the condition, the speed of one of the fragments is vertical, i.e. its momentum remained in the same plane, then the momentum of the second fragment also lies in this plane. This means that the trajectory of the second fragment will remain in the same plane.
Further, from the law of conservation of the horizontal component of the total impulse it follows that the horizontal component of the velocity of the second fragment is equal because its mass is equal to half the mass of the projectile, and the horizontal component of the impulse of the first fragment is equal to zero by condition. Therefore, the horizontal flight range of the second fragment is from
the location of the rupture is equal to the product of the time of its flight. How to find this time?
To do this, remember that the vertical components of the impulses (and therefore the velocities) of the fragments must be equal in magnitude and directed in opposite directions. The flight time of the second fragment of interest to us depends, obviously, on whether the vertical component of its speed is directed upward or downward at the moment the projectile explodes (Fig. 108).
Rice. 108. Trajectory of fragments after a shell burst
This is easy to find out by comparing the time of the vertical fall of the first fragment given in the condition with the time of free fall from height A. If then the initial speed of the first fragment is directed downward, and the vertical component of the speed of the second is directed upward, and vice versa (cases a and in Fig. 108). At an angle a to the vertical, a bullet flies into the box at speed u and almost instantly gets stuck in the sand. The box starts to move and then stops. How long did it take for the box to move? The ratio of the mass of the bullet to the mass of the box is equal to y. Under what conditions will the box not move at all?
2. During the radioactive decay of an initially resting neutron, a proton, electron and antineutrino are formed. The momenta of the proton and electron are equal and the angle between them is a. Determine the momentum of the antineutrino.
What is called the momentum of one particle and the momentum of a system of material points?
Formulate the law of change in momentum of one particle and a system of material points.
Rice. 109. To determine the impulse of force from the graph
Why are internal forces not explicitly included in the law of changes in the momentum of a system?
In what cases can the law of conservation of momentum of a system be used in the presence of external forces?
What are the advantages of using the law of conservation of momentum compared to the dynamic approach?
When a variable force acts on a body, its momentum is determined by the right-hand side of formula (5) - the integral of over the period of time during which it acts. Let us be given a dependence graph (Fig. 109). How to determine the force impulse from this graph for each of cases a and
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. At the same time, new physical quantities. 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 obtain 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
- Tikhomirova S.A., Yavorsky B.M. Physics ( a basic level of) - M.: Mnemosyne, 2012.
- Gendenshtein L.E., Dick Yu.I. Physics 10th grade. - M.: Mnemosyne, 2014.
- Kikoin I.K., Kikoin A.K. Physics - 9, Moscow, Education, 1990.
Homework
- Define the impulse of a body, the impulse of force.
- How are the impulse of a body related to the impulse of force?
- What conclusions can be drawn from the formulas for body impulse and force impulse?
- Internet portal Questions-physics.ru ().
- Internet portal Frutmrut.ru ().
- Internet portal Fizmat.by ().
Force impulse and body impulse
As has been shown, Newton's second law can be written as
Ft=mv-mv o =p-p o =D p.
Vector quantity Ft, equal to the product force for the duration of its action is called impulse of force. The vector quantity p=mv, equal to the product of the mass of a body and its speed, is called body impulse.
In SI, the unit of impulse is taken to be the impulse of a body weighing 1 kg moving at a speed of 1 m/s, i.e. The unit of impulse is the kilogrammeter per second (1 kg m/s).
The change in the momentum of the body D p over time t is equal to the impulse of the force Ft acting on the body during this time.
The concept of momentum is one of the fundamental concepts of physics. The momentum of a body is one of the quantities capable of maintaining its value unchanged under certain conditions.(but in modulus and in direction).
Conservation of total momentum of a closed-loop system
Closed system call a group of bodies that do not interact with any other bodies that are not part of this group. The forces of interaction between bodies included in a closed system are called internal. (Internal forces are usually denoted by the letter f).
Let's consider the interaction of bodies inside a closed system. Let two balls of the same diameter, made of different substances (i.e., having different masses), roll along a perfectly smooth horizontal surface and collide with each other. During an impact, which we will consider central and absolutely elastic, the velocities and impulses of the balls change. Let the mass of the first ball m 1, its speed before the impact V 1, and after the impact V 1 "; the mass of the second ball m 2, its speed before the impact v 2, after the impact v 2". According to Newton's third law, the interaction forces between the balls are equal in magnitude and opposite in direction, i.e. f 1 = -f 2 .
According to Newton’s second law, the change in the impulses of the balls as a result of their collision is equal to the impulses of the interaction forces between them, i.e.
m 1 v 1 "-m 1 v 1 =f 1 t (3.1)
m 2 v 2 "-m 2 v 2 =f 2 t (3.2)
where t is the interaction time of the balls.
Adding expressions (3.1) and (3.2) term by term, we find that
m 1 v 1 "-m 1 v 1 +m 2 v 2 "-m 2 v 2 =0.
Hence,
m 1 v 1 "+m 2 v 2 "=m 1 v 1 +m 2 v 2
or else
p 1 "+p 2 "=p 1 +p 2 . (3.3)
Let us denote p 1 "+p 2 "=p" and p 1 +p 2 =p.
The vector sum of the momenta of all bodies included in the system is called full impulse of this system. From (3.3) it is clear that p"=p, i.e. p"-p=D p=0, therefore,
p=p 1 +p 2 =const.
Formula (3.4) expresses law of conservation of momentum in a closed system, which is formulated as follows: the total momentum of a closed system of bodies remains constant during any interactions of the bodies of this system with each other.
In other words, internal forces cannot change the total momentum of the system, either in magnitude or in direction.
Change in total momentum of an open-loop system
A group of bodies that interact not only with each other, but also with bodies that are not part of this group is called open system. The forces with which bodies not included in this system act on the bodies of a given system are called external (usually external forces are denoted by the letter F).
Let us consider the interaction of two bodies in an open system. Changes in the impulses of these bodies occur both under the influence of internal forces and under the influence of external forces.
According to Newton's second law, the changes in the momenta of the bodies under consideration for the first and second bodies are
D р 1 =f 1 t+F 1 t (3.5)
D р 2 =f 2 t+F 2 t (3.6)
where t is the time of action of external and internal forces.
Adding expressions (3.5) and (3.6) term by term, we find that
D (p 1 +p 2)=(f 1 +f 2)t +(F 1 +F 2)t (3.7)
In this formula, p=p 1 +p 2 is the total impulse of the system, f 1 +f 2 =0 (since according to Newton’s third law (f 1 = -f 2), F 1 +F 2 =F is the resultant of all external forces , acting on the bodies of this system. Taking into account the above, formula (3.7) takes the form
D р=Ft. (3.8)
From (3.8) it is clear that the total momentum of the system changes only under the influence of external forces. If the system is closed, i.e. F=0, then D р=0 and, therefore, р=const. Thus, formula (3.4) is a special case of formula (3.8), which shows under what conditions the total momentum of the system is conserved and under what conditions it changes.
Jet propulsion.
The significance of Tsiolkovsky’s work for astronautics
The movement of a body resulting from the separation of part of its mass from it at a certain speed is called reactive.
All types of motion, except reactive, are impossible without the presence of forces external to a given system, i.e., without the interaction of the bodies of a given system with environment, A to achieve jet propulsion, no interaction of the body with the environment is required. Initially the system is at rest, i.e. its total momentum is zero. When part of its mass begins to be ejected from the system at a certain speed, then (since the total momentum of a closed system, according to the law of conservation of momentum, must remain unchanged) the system receives a speed directed in the opposite direction. Indeed, since m 1 v 1 +m 2 v 2 =0, then m 1 v 1 =-m 2 v 2, i.e.
v 2 = -v 1 m 1 / m 2 .
From this formula it follows that the speed v 2 obtained by a system with mass m 2 depends on the ejected mass m 1 and the speed v 1 of its ejection.
A heat engine in which the traction force arising from the reaction of a jet of escaping hot gases is applied directly to its body is called a reactive engine. Unlike other vehicles, a jet-powered device can move in outer space.
The founder of the theory of space flight is the outstanding Russian scientist Tsiolkovsky (1857 - 1935). He gave the general principles of the theory of jet propulsion, developed the basic principles and designs of jet aircraft, and proved the need for using a multi-stage rocket for interplanetary flights. Tsiolkovsky's ideas were successfully implemented in the USSR during the construction of artificial Earth satellites and spacecraft.
The founder of practical cosmonautics is the Soviet scientist Academician Korolev (1906 - 1966). Under his leadership, the world's first artificial Earth satellite was created and launched, and the first human flight into space took place in the history of mankind. The first cosmonaut on Earth was soviet man Yu.A. Gagarin (1934 - 1968).
Questions for self-control:
- How is Newton's second law written in impulse form?
- What is called a force impulse? body impulse?
- What system of bodies is called closed?
- What forces are called internal?
- Using the example of the interaction of two bodies in a closed system, show how the law of conservation of momentum is established. How is it formulated?
- What is the total momentum of a system?
- Can internal forces change the total momentum of a system?
- What system of bodies is called unclosed?
- What forces are called external?
- Establish a formula showing under what conditions the total momentum of the system changes and under what conditions it is conserved.
- What kind of movement is called reactive?
- Can it occur without interaction of a moving body with the environment?
- What law is jet propulsion based on?
- What is the significance of Tsiolkovsky’s work for astronautics?
In some cases, it is possible to study the interaction of bodies without using expressions for the forces acting between the bodies. This is possible due to the fact that there are physical quantities that remain unchanged (conserved) when bodies interact. In this chapter we will look at two such quantities - momentum and mechanical energy.
Let's start with momentum.
A physical quantity equal to the product of a body’s mass m and its speed is called the body’s momentum (or simply impulse):
Momentum is a vector quantity. The magnitude of the impulse is p = mv, and the direction of the impulse coincides with the direction of the body's velocity. The unit of impulse is 1 (kg * m)/s.
1. A truck weighing 3 tons is driving along a highway in the north direction at a speed of 40 km/h. In what direction and at what speed should a passenger car weighing 1 ton travel so that its momentum is equal to the impulse of the truck?
2. A ball with a mass of 400 g falls freely without an initial speed from a height of 5 m. After the impact, the ball bounces up, and the modulus of the ball’s velocity does not change as a result of the impact.
a) What is the magnitude and direction of the ball’s momentum immediately before impact?
b) What is the magnitude and direction of the ball’s momentum immediately after impact?
c) What is the change in momentum of the ball as a result of the impact and in what direction? Find the change in momentum graphically.
Clue. If the momentum of the body was equal to 1, and became equal to 2, then the change in momentum ∆ = 2 – 1.
2. Law of conservation of momentum
The most important property of momentum is that, under certain conditions, the total momentum of interacting bodies remains unchanged (conserved).
Let's put experience
Two identical carts can roll along a table along the same straight line with virtually no friction. (This experiment can be carried out with modern equipment.) No friction - important condition our experience!
We will install latches on the carts, thanks to which the carts move as one body after a collision. Let the right cart initially be at rest, and with the left push we impart speed 0 (Fig. 25.1, a). 
After the collision, the carts move together. Measurements show that their total speed is 2 times less than the initial speed of the left cart (25.1, b).
Let us denote the mass of each cart as m and compare the total impulses of the carts before and after the collision. 
We see that the total momentum of the carts remained unchanged (preserved).
Maybe this is only true when the bodies move as a single unit after interaction?
Let's put experience
Let's replace the latches with an elastic spring and repeat the experiment (Fig. 25.2). 
This time the left cart stopped, and the right one acquired a speed equal to the initial speed of the left cart.
3. Prove that in this case the total momentum of the carts is conserved.
Maybe this is true only when the masses of the interacting bodies are equal?
Let's put experience
Let's attach another similar cart to the right cart and repeat the experiment (Fig. 25.3). 
Now, after the collision, the left cart began to move in the opposite direction (that is, to the left) at a speed equal to -/3, and the double cart began to move to the right at a speed of 2/3.
4. Prove that in this experiment the total momentum of the carts was conserved.
To determine under what conditions the total momentum of bodies is conserved, let us introduce the concept of a closed system of bodies. This is the name given to a system of bodies that interact only with each other (that is, they do not interact with bodies that are not part of this system).
Exactly closed systems of bodies do not exist in nature, if only because it is impossible to “turn off” the forces of universal gravity.
But in many cases, a system of bodies can be considered closed with good accuracy. For example, when external forces (forces acting on the bodies of the system from other bodies) balance each other or can be neglected.
This is exactly what happened in our experiments with carts: external forces acting on them (gravity and force normal reaction) balanced each other, and the friction force could be neglected. Therefore, the speeds of the carts changed only as a result of their interaction with each other.
The experiments described, as well as many others like them, indicate that
law of conservation of momentum: the vector sum of the momenta of the bodies that make up a closed system does not change during any interactions between the bodies of the system:
The law of conservation of momentum is satisfied only in inertial frames of reference.
Law of conservation of momentum as a consequence of Newton's laws
Let us show, using the example of a closed system of two interacting bodies, that the law of conservation of momentum is a consequence of Newton’s second and third laws.
Let us denote the masses of the bodies as m 1 and m 2, and their initial velocities as 1 and 2. Then the vector sum of the momenta of the bodies
Let the interacting bodies move with accelerations 1 and 2 during a period of time ∆t.
5. Explain why the change in the total momentum of bodies can be written in the form
Clue. Use the fact that for each body ∆ = m∆, and also the fact that ∆ = ∆t.
6. Let us denote 1 and 2 forces acting on the first and second bodies, respectively. Prove that
Clue. Take advantage of Newton's second law and the fact that the system is closed, as a result of which the accelerations of bodies are caused only by the forces with which these bodies act on each other.
7. Prove that
Clue. Use Newton's third law.
So, the change in the total momentum of the interacting bodies is zero. And if the change in a certain quantity is zero, then this means that this quantity is conserved.
8. Why does it follow from the above reasoning that the law of conservation of momentum is satisfied only in inertial frames of reference?
3. Force impulse
There is a saying: “If only I knew where you would fall, I would lay down straws.” Why do you need a “straw”? Why do athletes fall or jump on soft mats during training and competitions rather than on the hard floor? Why after a jump should you land on bent legs and not straightened ones? Why do cars need seat belts and airbags?
We can answer all these questions by becoming familiar with the concept of “force impulse”.
The impulse of a force is the product of a force and the time interval ∆t during which this force acts.
It is no coincidence that the name “impulse of force” “echoes” the concept of “impulse”. Let us consider the case when a body of mass m is acted upon by a force during a period of time ∆t.
9. Prove that the change in the momentum of the body ∆ is equal to the momentum of the force acting on this body:
Clue. Use the fact that ∆ = m∆ and Newton's second law.
Let us rewrite formula (6) in the form
This formula is another form of writing Newton's second law. (It was in this form that Newton himself formulated this law.) It follows from it that a large force acts on a body if its momentum changes significantly in a very short period of time ∆t.
This is why large forces arise during impacts and collisions: impacts and collisions are characterized by precisely a short interaction time interval.
To weaken the force of an impact or reduce the forces arising when bodies collide, it is necessary to lengthen the period of time during which the impact or collision occurs.
10. Explain the meaning of the saying given at the beginning of this section, and also answer the other questions placed in the same paragraph.
11. A ball with a mass of 400 g hit a wall and bounced off it with the same absolute speed, equal to 5 m/s. Just before impact, the ball's speed was directed horizontally. What is the average force exerted by the ball on the wall if it was in contact with the wall for 0.02 s?
12. A cast iron block weighing 200 kg falls from a height of 1.25 m into sand and sinks 5 cm into it.
a) What is the momentum of the blank immediately before the impact?
b) What is the change in momentum of the blank during the impact?
c) How long did the blow last?
d) What is the average impact force?
Additional questions and tasks
13. A ball with a mass of 200 g moves at a speed of 2 m/s to the left. How should another ball of mass 100 g move so that the total momentum of the balls is zero?
14. A ball with a mass of 300 g moves uniformly in a circle of radius 50 cm at a speed of 2 m/s. What is the modulus of change in the momentum of the ball:
a) for one full period appeals?
b) for half the circulation period?
c) in 0.39 s?
15. The first board lies on the asphalt, and the second board is the same - on loose sand. Explain why it is easier to hammer a nail into the first board than into the second?
16. A bullet weighing 10 g, flying at a speed of 700 m/s, pierced the board, after which the bullet speed became equal to 300 m/s. Inside the board, the bullet moved for 40 μs.
a) What is the change in momentum of the bullet due to passing through the board?
b) What average force did the bullet exert on the board as it passed through it?
