Determination of changes in body momentum. What is body impulse

Basic dynamic quantities: force, mass, body impulse, moment of force, angular momentum.

Force is a vector quantity, which is a measure of the action of other bodies or fields on a given body.

Strength is characterized by:

· Module

Direction

Application point

In the SI system, force is measured in newtons.

In order to understand what a force of one Newton is, we need to remember that a force applied to a body changes its speed. In addition, let us remember the inertia of bodies, which, as we remember, is associated with their mass. So,

One newton is a force that changes the speed of a body weighing 1 kg by 1 m/s every second.

Examples of forces include:

· Gravity– a force acting on a body as a result of gravitational interaction.

· Elastic force- the force with which a body resists an external load. Its cause is the electromagnetic interaction of body molecules.

· Archimedes' force- a force associated with the fact that a body displaces a certain volume of liquid or gas.

· Ground reaction force- the force with which the support acts on the body located on it.

· Friction force– the force of resistance to the relative movement of the contacting surfaces of bodies.

· Surface tension is a force that occurs at the interface between two media.

· Body weight- the force with which the body acts on a horizontal support or vertical suspension.

And other forces.

Strength is measured using a special device. This device is called a dynamometer (Fig. 1). The dynamometer consists of spring 1, the stretching of which shows us the force, arrow 2, sliding along scale 3, limiter bar 4, which prevents the spring from stretching too much, and hook 5, from which the load is suspended.

Rice. 1. Dynamometer (Source)

Many forces can act on the body. In order to correctly describe the movement of a body, it is convenient to use the concept of resultant forces.

The resultant force is a force whose action replaces the action of all forces applied to the body (Fig. 2).

Knowing the rules for working with vector quantities, it is easy to guess that the resultant of all forces applied to a body is the vector sum of these forces.

Rice. 2. Resultant of two forces acting on a body

In addition, since we are considering the movement of a body in some coordinate system, it is usually advantageous for us to consider not the force itself, but its projection onto the axis. The projection of force on the axis can be negative or positive, because the projection is a scalar quantity. So, in Figure 3 the projections of forces are shown, the projection of force is negative, and the projection of force is positive.

Rice. 3. Projections of forces onto the axis

So, from this lesson we have deepened our understanding of the concept of strength. We remembered the units of measurement of force and the device with which force is measured. In addition, we looked at what forces exist in nature. Finally, we learned how to act when several forces act on the body.

Weight, a physical quantity, one of the main characteristics of matter, determining its inertial and gravitational properties. Accordingly, a distinction is made between inertial Mass and gravitational Mass (heavy, gravitating).

The concept of Mass was introduced into mechanics by I. Newton. In classical Newtonian mechanics, Mass is included in the definition of momentum (amount of motion) of a body: momentum R proportional to the speed of the body v, p = mv(1). The proportionality coefficient is a constant value for a given body m- and is the Mass of the body. The equivalent definition of Mass is obtained from the equation of motion of classical mechanics f = ma(2). Here Mass is the coefficient of proportionality between the force acting on the body f and the acceleration of the body caused by it a. The mass defined by relations (1) and (2) is called inertial mass, or inertial mass; it characterizes the dynamic properties of a body, is a measure of the inertia of the body: with a constant force, the greater the mass of the body, the less acceleration it acquires, i.e., the slower the state of its motion changes (the greater its inertia).

By acting on different bodies with the same force and measuring their accelerations, we can determine the relationship between the mass of these bodies: m 1: m 2: m 3 ... = a 1: a 2: a 3 ...; if one of the Masses is taken as a unit of measurement, the Mass of the remaining bodies can be found.

In Newton's theory of gravity, Mass appears in a different form - as a source of the gravitational field. Each body creates a gravitational field proportional to the Mass of the body (and is affected by the gravitational field created by other bodies, the strength of which is also proportional to the Mass of the bodies). This field causes the attraction of any other body to this body with a force determined by Newton’s law of gravity:

(3)

Where r- distance between bodies, G is the universal gravitational constant, a m 1 And m 2- Masses of attracting bodies. From formula (3) it is easy to obtain the formula for weight R body mass m in the Earth's gravitational field: P = mg (4).

Here g = G*M/r 2- acceleration of free fall in the gravitational field of the Earth, and r » R- the radius of the Earth. The mass determined by relations (3) and (4) is called the gravitational mass of the body.

In principle, it does not follow from anywhere that the Mass that creates the gravitational field also determines the inertia of the same body. However, experience has shown that inertial Mass and gravitational Mass are proportional to each other (and with the usual choice of units of measurement, they are numerically equal). This fundamental law of nature is called the principle of equivalence. Its discovery is associated with the name of G. Galileo, who established that all bodies on Earth fall with the same acceleration. A. Einstein put this principle (formulated by him for the first time) into the basis of the general theory of relativity. The equivalence principle has been established experimentally with very high accuracy. For the first time (1890-1906), a precision test of the equality of inertial and gravitational Masses was carried out by L. Eotvos, who found that the Masses coincide with an error of ~ 10 -8. In 1959-64, American physicists R. Dicke, R. Krotkov and P. Roll reduced the error to 10 -11, and in 1971, Soviet physicists V.B. Braginsky and V.I. Panov - to 10 -12.

The principle of equivalence allows us to most naturally determine body weight by weighing.

Initially, Mass was considered (for example, by Newton) as a measure of the amount of matter. This definition has a clear meaning only for comparing homogeneous bodies built from the same material. It emphasizes the additivity of Mass - the Mass of a body is equal to the sum of the Mass of its parts. The mass of a homogeneous body is proportional to its volume, so we can introduce the concept of density - Mass of a unit volume of a body.

In classical physics it was believed that the mass of a body does not change in any processes. This corresponded to the law of conservation of Mass (matter), discovered by M.V. Lomonosov and A.L. Lavoisier. In particular, this law stated that in any chemical reaction the sum of the Masses of the initial components is equal to the sum of the Masses of the final components.

The concept of Mass acquired a deeper meaning in the mechanics of A. Einstein’s special theory of relativity, which considers the movement of bodies (or particles) at very high speeds - comparable to the speed of light with ~ 3 10 10 cm/sec. In new mechanics - it is called relativistic mechanics - the relationship between momentum and velocity of a particle is given by the relation:

(5)

At low speeds ( v << c) this relation goes into the Newtonian relation p = mv. Therefore the value m 0 is called rest mass, and the mass of a moving particle m is defined as the speed-dependent proportionality coefficient between p And v:

(6)

Bearing in mind, in particular, this formula, they say that the mass of a particle (body) grows with an increase in its speed. Such a relativistic increase in the mass of a particle as its speed increases must be taken into account when designing accelerators of high-energy charged particles. Rest mass m 0(Mass in the reference frame associated with the particle) is the most important internal characteristic of the particle. All elementary particles have strictly defined meanings m 0, inherent in a given type of particle.

It should be noted that in relativistic mechanics, the definition of Mass from the equation of motion (2) is not equivalent to the definition of Mass as a coefficient of proportionality between the momentum and the speed of the particle, since the acceleration ceases to be parallel to the force that caused it and the Mass turns out to depend on the direction of the particle’s speed.

According to the theory of relativity, Particle mass m connected to her energy E ratio:

(7)

The rest mass determines the internal energy of the particle - the so-called rest energy E 0 = m 0 s 2. Thus, energy is always associated with Mass (and vice versa). Therefore, there is no separate law (as in classical physics) of the conservation of Mass and the law of conservation of energy - they are merged into a single law of conservation of total (i.e., including the rest energy of particles) energy. An approximate division into the law of conservation of energy and the law of conservation of mass is possible only in classical physics, when particle velocities are small ( v << c) and particle transformation processes do not occur.

In relativistic mechanics, Mass is not an additive characteristic of a body. When two particles combine to form one compound stable state, an excess of energy (equal to the binding energy) is released D E, which corresponds to Mass D m = D E/s 2. Therefore, the Mass of a composite particle is less than the sum of the Masses of the particles forming it by the amount D E/s 2(the so-called mass defect). This effect is especially pronounced in nuclear reactions. For example, deuteron mass ( d) is less than the sum of proton masses ( p) and neutron ( n); defect Mass D m associated with energy E g gamma quantum ( g), born during the formation of a deuteron: p + n -> d + g, E g = Dmc 2. The Mass defect that occurs during the formation of a composite particle reflects the organic connection between Mass and energy.

The unit of mass in the CGS system of units is gram, and in International System of Units SI - kilogram. The mass of atoms and molecules is usually measured in atomic mass units. The mass of elementary particles is usually expressed either in units of electron mass m e, or in energy units, indicating the rest energy of the corresponding particle. Thus, the mass of an electron is 0.511 MeV, the mass of a proton is 1836.1 m e, or 938.2 MeV, etc.

The nature of Mass is one of the most important unsolved problems of modern physics. It is generally accepted that the mass of an elementary particle is determined by the fields that are associated with it (electromagnetic, nuclear and others). However, a quantitative theory of Mass has not yet been created. There is also no theory that explains why the mass of elementary particles forms a discrete spectrum of values, much less allows us to determine this spectrum.

In astrophysics, the mass of a body creating a gravitational field determines the so-called gravitational radius of the body R gr = 2GM/s 2. Due to gravitational attraction, no radiation, including light, can escape beyond the surface of a body with a radius R=< R гр . Stars of this size will be invisible; That's why they were called "black holes". Such celestial bodies must play an important role in the Universe.

Impulse of force. Body impulse

The concept of momentum was introduced in the first half of the 17th century by Rene Descartes, and then refined by Isaac Newton. According to Newton, who called momentum the quantity of motion, this is a measure of it, proportional to the speed of a body and its mass. Modern definition: The momentum of a body is a physical quantity equal to the product of the mass of the body and its speed:

First of all, from the above formula it is clear that impulse is a vector quantity and its direction coincides with the direction of the body’s speed; the unit of measurement for impulse is:

= [kg m/s]

Let us consider how this physical quantity is related to the laws of motion. Let's write down Newton's second law, taking into account that acceleration is the change in speed over time:

There is a connection between the force acting on the body, or more precisely, the resultant force, and the change in its momentum. The magnitude of the product of a force and a period of time is called the impulse of force. From the above formula it is clear that the change in the momentum of the body is equal to the impulse of the force.

What effects can be described using this equation (Fig. 1)?

Rice. 1. Relationship between force impulse and body impulse (Source)

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.

MOMENT OF POWER- quantity characterizing the rotational effect of the force; has the dimension of the product of length and force. Distinguish moment of power relative to the center (point) and relative to the axis.

M. s. relative to the center ABOUT called vector quantity M 0 equal to the vector product of the radius vector r , carried out from O to the point of application of force F , to strength M 0 = [rF ] or in other notations M 0 = r F (rice.). Numerically M. s. equal to the product of the modulus of force and the arm h, i.e. by the length of the perpendicular lowered from ABOUT on the line of action of the force, or twice the area

triangle built on the center O and strength:

Directed vector M 0 perpendicular to the plane passing through O And F . Side to which it is heading M 0, selected conditionally ( M 0 - axial vector). With a right-handed coordinate system, the vector M 0 is directed in the direction from which the rotation made by the force is visible counterclockwise.

M. s. relative to the z-axis called scalar quantity M z, equal to the projection onto the axis z vector M. s. relative to any center ABOUT, taken on this axis; size M z can also be defined as a projection onto a plane xy, perpendicular to the z axis, the area of the triangle OAB or as a moment of projection Fxy strength F to the plane xy, taken relative to the point of intersection of the z axis with this plane. T. o.,

In the last two expressions of M. s. is considered positive when the rotation force Fxy visible from positive the end of the z axis counterclockwise (in the right coordinate system). M. s. relative to coordinate axes Oxyz can also be calculated analytically. f-lam:

Where Fx, Fy, Fz- force projections F on the coordinate axes, x, y, z- point coordinates A application of force. Quantities M x , M y , M z are equal to the projections of the vector M 0 on the coordinate axes.

They change because interaction forces act on each of the bodies, but the sum of the impulses remains constant. This is called law of conservation of momentum.

Newton's second law is expressed by the formula. It can be written in another way, if we remember that acceleration is equal to the rate of change in the speed of a body. For uniformly accelerated motion, the formula will look like:

If we substitute this expression into the formula, we get:

This formula can be rewritten as:

The right-hand side of this equality records the change in the product of a body’s mass and its speed. The product of body mass and speed is a physical quantity called body impulse or amount of body movement.

Body impulse is called the product of a body's mass and its speed. This is a vector quantity. The direction of the momentum vector coincides with the direction of the velocity vector.

In other words, a body of mass m, moving with speed has momentum. The SI unit of impulse is the impulse of a body weighing 1 kg moving at a speed of 1 m/s (kg m/s). When two bodies interact with each other, if the first acts on the second body with a force, then, according to Newton’s third law, the second acts on the first with a force. Let us denote the masses of these two bodies by m 1 and m 2, and their speeds relative to any reference system through and. Over time t as a result of the interaction of bodies, their velocities will change and become equal and . Substituting these values into the formula, we get:

Hence,

Let us change the signs of both sides of the equality to their opposites and write them in the form

On the left side of the equation is the sum of the initial impulses of two bodies, on the right side is the sum of the impulses of the same bodies over time t. The amounts are equal. So, despite that. that the impulse of each body changes during interaction, the total impulse (the sum of the impulses of both bodies) remains unchanged.

Valid also when several bodies interact. However, it is important that these bodies interact only with each other and are not affected by forces from other bodies not included in the system (or that external forces are balanced). A group of bodies that does not interact with other bodies is called closed system valid only for closed systems.

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.

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.

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

Tikhomirova S.A., Yavorsky B.M. Physics (basic level) - 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 ().

Problems with moving bodies in physics, when the speed is much less than light, are solved using the laws of Newtonian or classical mechanics. One of the important concepts in it is impulse. The basic ones in physics are given in this article.

Impulse or momentum?

Before giving the formulas for the momentum of a body in physics, let’s get acquainted with this concept. For the first time, the quantity called impeto (impulse) was used in the description of his works by Galileo at the beginning of the 17th century. Subsequently, Isaac Newton used another name for it - motus (motion). Since the figure of Newton had a greater influence on the development of classical physics than the figure of Galileo, it was initially customary to speak not about the momentum of a body, but about the quantity of motion.

The quantity of motion is understood as the product of the speed of movement of a body by the inertial coefficient, that is, by mass. The corresponding formula is:

Here p¯ is a vector whose direction coincides with v¯, but the module is m times greater than the module v¯.

Change in p¯ value

The concept of momentum is currently used less often than impulse. And this fact is directly related to the laws of Newtonian mechanics. Let's write it in the form given in school physics textbooks:

Let's replace the acceleration a¯ with the corresponding expression for the speed derivative, we get:

Transferring dt from the denominator of the right side of the equality to the numerator of the left, we get:

We got an interesting result: in addition to the fact that the acting force F¯ leads to the acceleration of the body (see the first formula of this paragraph), it also changes the amount of its motion. The product of force and time, which is on the left side, is called the impulse of force. It turns out to be equal to the change in p¯. Therefore, the last expression is also called the momentum formula in physics.

Note that dp¯ is also but, unlike p¯, it is directed not as speed v¯, but as force F¯.

A striking example of a change in the vector of momentum (impulse) is the situation when a football player hits the ball. Before the hit the ball moved towards the player, after the hit it moved away from him.

Law of conservation of momentum

Formulas in physics that describe the conservation of the value p¯ can be given in several versions. Before writing them down, let's answer the question of when momentum is conserved.

Let's return to the expression from the previous paragraph:

It says that if the sum of external forces acting on the system is zero (closed system, F¯= 0), then dp¯= 0, that is, no change in momentum will occur:

This expression is common to the momentum of a body and the law of conservation of momentum in physics. Let us note two important points that you should know about in order to successfully apply this expression in practice:

The momentum is conserved along each coordinate, that is, if before some event the value of p x of the system was 2 kg*m/s, then after this event it will be the same.
Momentum is conserved regardless of the nature of collisions of solid bodies in the system. There are two ideal cases of such collisions: absolutely elastic and absolutely plastic impacts. In the first case, kinetic energy is also conserved, in the second, part of it is spent on plastic deformation of bodies, but the momentum is still conserved.

Elastic and inelastic interaction of two bodies

A special case of using the momentum formula in physics and its conservation is the motion of two bodies that collide with each other. Let's consider two fundamentally different cases, which were mentioned in the paragraph above.

If the impact is absolutely elastic, that is, the transfer of momentum from one body to another is carried out through elastic deformation, then the conservation formula p will be written as follows:

m 1 *v 1 + m 2 *v 2 = m 1 *u 1 + m 2 *u 2

It is important to remember here that the sign of the speed must be substituted taking into account its direction along the axis under consideration (opposite speeds have different signs). This formula shows that, given the known initial state of the system (values m 1, v 1, m 2, v 2), in the final state (after the collision) there are two unknowns (u 1, u 2). You can find them if you use the corresponding law of conservation of kinetic energy:

m 1 *v 1 2 + m 2 *v 2 2 = m 1 *u 1 2 + m 2 *u 2 2

If the impact is absolutely inelastic or plastic, then after the collision the two bodies begin to move as a single whole. In this case, the expression takes place:

m 1 *v 1 + m 2 *v 2 = (m 1 + m 2)*u

As you can see, we are talking about only one unknown (u), so this one equality is enough to determine it.

Momentum of a body while moving in a circle

Everything that was said above about momentum applies to linear movements of bodies. What to do if objects rotate around an axis? For this purpose, another concept has been introduced in physics, which is similar to linear momentum. It is called angular momentum. The formula in physics for it takes the following form:

Here r¯ is a vector equal to the distance from the axis of rotation to a particle with momentum p¯, performing circular motion around this axis. The quantity L¯ is also a vector, but it is somewhat more difficult to calculate than p¯, since we are talking about a vector product.

Conservation law L¯

The formula for L¯, which is given above, is the definition of this quantity. In practice, they prefer to use a slightly different expression. We won’t go into the details of how to obtain it (it’s not difficult, and everyone can do it on their own), but let’s give it right away:

Here I is the moment of inertia (for a material point it is equal to m*r 2), which describes the inertial properties of a rotating object, ω¯ is the angular velocity. As you can see, this equation is similar in form to that for linear momentum p¯.

If no external forces act on the rotating system (in fact, torque), then the product of I and ω¯ will be preserved regardless of the processes occurring inside the system. That is, the conservation law for L¯ has the form:

An example of its manifestation is the performance of figure skating athletes when they perform spins on the ice.

Topics of the Unified State Examination codifier: momentum of a body, momentum of a system of bodies, law of conservation of momentum.

Pulse of a body is a vector quantity equal to the product of the body’s mass and its speed:

There are no special units for measuring 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 with its help you can give Newton's second law a slightly different, also extremely useful form.

Newton's second law in impulse form

Let be the resultant of forces applied to a body of mass . We start with the usual notation 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 side:

. ( 1 )

Relationship (1) is a new form of writing Newton’s second law.

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

We can say this: the resulting force acting on a body is equal to the rate of change of 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 time.

In tasks, as a rule, the time interval is quite small. For example, this could be the time of impact of the ball with the wall, and then - the average force acting on the ball from the wall during the impact.

The vector on the left side of relation (2) is called change in impulse during . The change in momentum is the difference between the final and initial momentum vectors. 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:

Let us emphasize once again that the change in momentum is the difference between vectors (Fig. 1):

Let, for example, the ball fly perpendicular to the wall (the momentum before the impact is equal to ) and bounce back without losing speed (the momentum after the impact is equal to ). Despite the fact that the impulse has not changed in absolute value (), there is a change in the impulse:

Geometrically, this situation is shown in Fig. 2:

The modulus of change in momentum, as we see, is equal to twice the modulus of the initial impulse of the ball: .

Let us 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 measurement for force impulse; the dimension of the force impulse is simply the product of the dimensions of force and time:

(Note that this turns out to be another possible unit of measurement for a body's momentum.)

The verbal formulation of equality (3) is as follows: the change in the momentum of a body is equal to the momentum of the force acting on the body over a given period of time. This, of course, is again Newton's second law in momentum form.

Example of force calculation

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

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 . The blow lasts for s. Find the average force,
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: . It follows that the vector of momentum change co-directed with 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 to , that is, the angle of reflection is really equal to the angle of incidence.

Now notice in addition that in our isosceles triangle there is an angle (this is the angle of incidence); therefore, this triangle is equilateral. From here:

And then the desired average force acting on the ball is:

Impulse of a system of bodies

Let's start with a simple situation of a two-body system. Namely, let there be body 1 and body 2 with impulses and, respectively. The impulse 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 derive this formula.

We will call all other objects with which the bodies 1 and 2 we are considering interact external bodies. The forces with which external bodies act on bodies 1 and 2 are called by external forces. Let be the resultant external force acting on body 1. Similarly, let be the resultant 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 a force. Then body 1 acts on body 2 with a force. According to Newton's third law, the forces are equal in magnitude and opposite in direction: . Forces and are 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's add equalities (4) and (5):

On the left side of the resulting equality there is a sum of derivatives equal to the derivative of the sum of the 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 us also denote - this is the resultant of external forces acting on the system. We get:

. ( 6 )

Thus, the rate of change of momentum of a system of bodies is the resultant of external forces applied to the system. We wanted to obtain equality (6), which plays the role of Newton’s second law for a system of bodies.

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.

Impulse of the system of bodies bodies is the vector sum of the momenta of all bodies included in the system. If a system consists of bodies, then the momentum of this system is equal to:

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 again obtain the derivative of the momentum of the system, and on the right side there remains only the sum of external forces (internal forces, adding in pairs, will give zero due to Newton's third law). Therefore, equality (6) will remain valid in the general case.

Law of conservation of momentum

The system of bodies is called closed, if the actions of external bodies on the bodies of a given system are either negligible or compensate each other. Thus, in the case of a closed system of bodies, only the interaction of these bodies with each other, but not with any other bodies, is essential.

The resultant of external forces applied to a closed system is equal to zero: . In this case, from (6) we obtain:

But if the derivative of a vector goes to zero (the rate of change of the vector is zero), then the vector itself does not change over time:

Law of conservation of momentum. 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.

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

Solution. The situation is shown in Fig. 7. Let's direct the axis in the direction of movement of the first body.

Rice. 7. To the task

Because 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 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 satisfied for it:

. ( 7 )

The impulse of the system before the impact is the sum of the impulses of the bodies:

After the inelastic impact, one body of mass is obtained, which moves with the desired 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 projections onto the axis:

By condition we have: m/s, m/s, so

The minus sign indicates that the stuck together bodies move in the direction opposite to the axis. Required speed: m/s.

Law of conservation of momentum projection

The following situation often occurs in problems. The system of bodies is not closed (the vector sum of external forces acting on the system is not equal to zero), but there is such an axis, the sum of the projections of external forces onto the axis is zero at any given time. Then we can say that along this axis our system of bodies behaves as closed, and the projection of the system’s momentum onto the axis is preserved.

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

If the projection of the resultant external forces vanishes, then

Therefore, the projection is a constant:

Law of conservation of momentum projection. If the projection onto the axis of the sum of external forces acting on the system is equal to zero, then the projection of the system’s momentum does not change over time.

Let's look at an example of a specific problem to see how the law of conservation of momentum projection works.

Task. A mass boy, standing on skates on smooth ice, throws a mass stone at an angle to the horizontal. Find the speed with which the boy rolls back after the throw.

Solution. The situation is shown schematically in Fig. 8 . The boy is depicted as straight-laced.

Rice. 8. To the task

The momentum of the “boy + stone” system is not conserved. This can be seen from the fact that after the throw, a vertical component of the system’s momentum appears (namely, the vertical component of the stone’s momentum), which was not there before the throw.

Therefore, the system that the boy and the stone form is not closed. Why? The fact is that the vector sum of 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 system’s momentum appears.

However, external forces act only vertically (there is no friction). Therefore, the projection of the impulse onto the horizontal axis is preserved. Before the throw, this projection was zero. Directing the axis in the direction of the throw (so that the boy went in the direction of the negative semi-axis), we get.