In everyday life, in order to characterize a person who commits spontaneous actions, the epithet "impulsive" is sometimes used. At the same time, some people do not even remember, and a significant part do not even know at all with what physical quantity this word is associated. What is hidden under the concept of "body impulse" and what properties does it possess? Such great scientists as René Descartes and Isaac Newton were looking for answers to these questions.
Like any science, physics operates with clearly formulated concepts. At the moment, the following definition has been adopted for a quantity called the impulse of a body: it is a vector quantity, which is a measure (amount) of the mechanical movement of a body.
Suppose that the question is considered within the framework of classical mechanics, i.e., it is believed that the body moves with ordinary, and not with relativistic speed, which means that it is at least an order of magnitude less than the speed of light in vacuum. Then the pulse module of the body is calculated using formula 1 (see photo below).
Thus, by definition, this value is equal to the product of the body's mass by its velocity, with which its vector is co-directed.
In SI (International System of Units), 1 kg / m / s is taken as the unit of measurement for impulse.
Where did the term "impulse" come from?
Several centuries before the concept of the amount of mechanical motion of a body appeared in physics, it was believed that the cause of any movement in space is a special force - impetus.
In the 14th century, Jean Buridan made adjustments to this concept. He suggested that the flying cobblestone has an impetus directly proportional to its speed, which would be unchanged if there were no air resistance. At the same time, according to this philosopher, bodies with greater weight had the ability to "contain" more of such a driving force.
Further development of the concept, later called impulse, was given by Rene Descartes, who designated it with the words "momentum". However, he did not take into account that speed has a direction. That is why the theory put forward by him in some cases contradicted experience and did not find recognition.
The English scientist John Wallis was the first to guess that the momentum should also have a direction. It happened in 1668. However, it took another couple of years for him to formulate the well-known law of conservation of momentum. A theoretical proof of this fact, established empirically, was given by Isaac Newton, who used the third and second laws of classical mechanics discovered by him and named after him.
The momentum of the system of material points
Let us first consider the case when we are talking about speeds much lower than the speed of light. Then, according to the laws of classical mechanics, the total momentum of a system of material points is a vector quantity. It is equal to the sum of the products of their masses at speed (see formula 2 in the picture above).
In this case, the momentum of one material point is taken as a vector quantity (formula 3), which is codirectional with the speed of the particle.
If we are talking about a body of finite size, then first it is mentally broken into small parts. Thus, the system of material points is considered again, but its momentum is calculated not by ordinary summation, but by integration (see formula 4).
As you can see, there is no time dependence, therefore, the impulse of the system, which is not affected by external forces (or their influence is mutually compensated), remains unchanged over time.
Proof of the conservation law
Let's continue to consider a body of finite size as a system of material points. For each of them, Newton's Second Law is formulated according to Formula 5.
Let's pay attention to the fact that the system is closed. Then, summing over all points and applying Newton's Third Law, we obtain expression 6.
Thus, the impulse of a closed system is constant.
The conservation law is also valid in those cases when the total amount of forces that act on the system from the outside is equal to zero. One important particular statement follows from this. It says that the impulse of the body is constant if there is no external influence or the influence of several forces is compensated. For example, in the absence of friction after hitting with a stick, the puck must retain its momentum. Such a situation will be observed even in spite of the fact that the body is affected by the force of gravity and the reaction of the support (ice), since, although they are equal in magnitude, they are directed in opposite directions, i.e., they compensate each other.
Properties
The momentum of a body or material point is an additive quantity. What does it mean? Everything is simple: the impulse of a mechanical system of material points consists of the impulses of all material points included in the system.
The second property of this quantity is that it remains unchanged during interactions that change only the mechanical characteristics of the system.
In addition, the momentum is invariant with respect to any rotation of the frame of reference.
Relativistic case
Suppose that we are talking about non-interacting material points with speeds of the order of 10 to the 8th power or slightly less in the SI system. The three-dimensional impulse is calculated by formula 7, where c is understood as the speed of light in a vacuum.
In the case when it is closed, the law of conservation of momentum is true. At the same time, the three-dimensional momentum is not a relativistically invariant quantity, since there is its dependence on the frame of reference. There is also a 4D option. For one material point, it is determined by formula 8.
Impulse and energy
These quantities, as well as mass, are closely related to each other. In practical problems, relations (9) and (10) are usually used.
Definition through de Broglie waves
In 1924, it was hypothesized that not only photons, but also any other particles (protons, electrons, atoms) possess wave-particle duality. Its author was the French scientist Louis de Broglie. If we translate this hypothesis into the language of mathematics, then we can assert that with any particle that has energy and momentum, a wave is associated with a frequency and length expressed by formulas 11 and 12, respectively (h is Planck's constant).
From the last relation, we find that the pulse modulus and the wavelength denoted by the letter "lambda" are inversely proportional to each other (13).
If a particle with a relatively low energy is considered, which moves at a speed incommensurate with the speed of light, then the modulus of the momentum is calculated in the same way as in classical mechanics (see formula 1). Therefore, the wavelength is calculated according to expression 14. In other words, it is inversely proportional to the product of the mass and velocity of the particle, ie, its momentum.
Now you know that the impulse of a body is a measure of mechanical movement, and you have become familiar with its properties. Among them, in practical terms, the Law of Conservation is especially important. Even people far from physics observe it in everyday life. For example, everyone knows that firearms and artillery pieces give recoil when fired. The law of conservation of momentum is clearly demonstrated by the game of billiards. With its help, you can predict the direction of expansion of the balls after impact.
The law has found application in the calculations necessary to study the consequences of possible explosions, in the creation of jet vehicles, in the design of firearms and in many other areas of life.
1. As you know, the result of the action of a force depends on its modulus, 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, applying a small force to the handle, we easily open the door, if the same force is applied near the hinges on which the door hangs, then it may not be open.
Experiments and observations indicate that the result of the action of the force (interaction) depends not only on the modulus of the force, but also on the time of its action. Let's make an experiment. To the tripod on a thread we hang a weight, to which another thread is tied from below (Fig. 59). If the bobbin thread is pulled sharply, it will break off, and the weight will remain hanging on the upper thread. Now if you slowly pull on the bobbin thread, the bobbin thread will break.
The impulse of force is called a vector physical quantity equal to the product of the force by the time of its action F t .
The unit of momentum of force in SI 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 the action of a force depends on the mass of the body on which this force acts. So, the greater the mass of the 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 car moving at a certain speed collides with it. As a result of the collision, the platform will gain acceleration and move a certain distance. If, however, a car moving at the same speed collides with a light trolley, then as a result of interaction it will move a significantly greater distance than a loaded platform.
Another example. Suppose that a bullet flies up to the target at a speed of 2 m / s. The bullet will most likely bounce off the target, leaving only a small dent on the target. If the bullet travels at a speed of 100 m / s, 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 body's mass and its velocity.
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p = m v. |
The unit of momentum of a body in SI is kilogram-meter per second(1 kg m / s): [ p] = [m][v] = 1 kg 1 m / s = 1 kg m / s.
The direction of the body's impulse coincides with the direction of its velocity.
Impulse is a relative value, its value depends on the choice of the frame of reference. This is understandable, since speed is a relative value.
3. Let us find out how the impulse of force and the impulse of the body are connected.
According to Newton's second law:
F = ma.
Substituting into this formula the expression for acceleration a=, we get:
F=, or
Ft = mv – mv 0 .
On the left side of the equality is the impulse of power; on the right side of the equality is the difference between the final and initial impulses of the body, i.e. that is, a change in the impulse of the body.
Thus,
the impulse of the force is equal to the change in the impulse of the body.
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F t = D ( m v). |
This is a different formulation of Newton's second law. This is how Newton formulated it.
4. Suppose that two balls are colliding moving on the table. Any interacting bodies, in this case balls, form the system... Forces act between the bodies of the system: the force of action F 1 and the force of reaction F 2. In this case, 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. So, the interacting balls are attracted to the Earth, they are affected by the reaction force of the support. These forces are, in this case, external forces. During movement, the force of air resistance and friction force act on the balls. They are also external forces in relation to the system, which in this case consists of two balls.
External forces are called forces that act on the bodies of the system from the side of 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. Consider the interaction of two bodies that make up a closed system. First body mass m 1, its speed before interaction v 01, after interaction v 1 . Second body weight m 2, its speed before interaction v 02, after interaction v 2 .
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 down 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-hand sides of the equalities are equal, their right-hand sides are also equal, i.e.
m 1 v 1 – m 1 v 01 = –(m 2 v 2 – m 2 v 02).
Transforming this equality, we get:
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m 1 v 01 + m 1 v 02 = m 2 v 1 + m 2 v 2 . |
On the left side of the equality is the sum of the impulses of the bodies before the interaction, in the right - the sum of the impulses of the bodies after the interaction. As can be seen from this equality, the impulse of each body changed during the interaction, and the sum of impulses remained unchanged.
The geometric sum of the impulses of the bodies that make up a closed system remains constant for any interactions of the bodies of this system.
This is momentum conservation law.
6. A closed system of bodies is a model of a real system. There are no such systems in nature that would not be acted upon by external forces. However, in a number of cases, systems of interacting bodies can be regarded as 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 on some direction can be equal to zero, and then the law of conservation of momentum is fulfilled for the projections of the impulses of interacting bodies to this direction.
7. An example of solving the problem
Two railway platforms move towards each other at speeds of 0.3 and 0.2 m / s. The platform weights are 16 and 48 tons, respectively. At what speed and in what direction will the platforms move after the automatic coupling?
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Given: |
SI |
Solution |
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v 01 = 0.3 m / s v 02 = 0.2 m / s m 1 = 16 t m 2 = 48 t v 1 = v 2 = v |
v 02 = v 02 = 1,6104kg 4,8104kg |
Let us depict in the figure the direction of movement of the platforms before and after interaction (Fig. 60). The forces of gravity acting on the platforms and the reaction forces of the support compensate each other. The system of two platforms can be considered closed |
|
vx? |
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 on the axis X you can write:
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, then m 1 v 01 – m 2 v 02 = –(m 1 + m 2)v.
Where v = – .
v= - = 0.75 m / s.
After hitching, the platforms will move in the direction in which the platform with the greater mass moved before the interaction.
Answer: v= 0.75 m / s; directed towards the movement of the trolley with a greater mass.
Self-test questions
1. What is called the impulse of the body?
2. What is called the impulse of force?
3. How are the impulse of force and the change in impulse of the 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 impulse of a 5 kg body moving at a speed of 20 m / s?
2. Determine the change in impulse of a body weighing 3 kg in 5 s under the action of a force of 20 N.
3. Determine the impulse of a car with a mass of 1.5 t moving at a speed of 20 m / s in the frame of reference associated: a) with 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. With 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, was torn into two fragments. What is the speed of the projectile, if a fragment with a mass of 2 kg, upon rupture, acquired a speed of 50 m / s, and with a second mass of 3 kg - 40 m / s? The speed of the fragments is directed horizontally.
Instructions
Find the mass of a moving body and measure its movements. After its interaction with another body, the speed of the investigated body will change. In this case, subtract the initial velocity from the final (after interaction) and multiply the difference by the body mass Δp = m ∙ (v2-v1). Measure instantaneous speed with radar, body weight - with scales. If, after the interaction, the body began to move in the direction opposite to the one that was moving before the interaction, then the final speed will be negative. If it is positive, it has grown, if it is negative, it has decreased.
Since the cause of the change in the speed of any body is force, it is also the cause of the change in impulse. To calculate the change in the momentum of any body, it is enough to find the momentum of the force acting on the given body at some time. Use a dynamometer to measure the force that causes the body to change speed, giving it acceleration. At the same time, use a stopwatch to measure the time that this force acted on the body. If the force makes the body move, then consider it positive, but if it slows down its movement, consider it negative. The impulse of the force equal to the change in the impulse will be the product of the force by the time of its action Δp = F ∙ Δt.
Determination of instantaneous speed with a speedometer or radar If a moving body is equipped with a speedometer (), then on its scale or electronic board, speed at this point in time. When observing a body from a stationary point (), direct the radar signal at it, its display will display an instantaneous speed body at a given time.
Related Videos
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 the amount of motion, i.e. pulse but the body itself.
Instructions
The movement of a material point by the influence of some strength or forces that give it acceleration. The result of the application strength a certain amount for a certain amount is the corresponding amount. Impulse strength the measure of its action for 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 velocity; v is the final velocity 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 value 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 from three interacting bodies, then it is true: Pc1 + Pc2 + Pc3 = ∆Pt1 + ∆Pt2 + ∆Pt3, where Pci - pulse strength acting on body i; Pтi - pulse body i.
This equality shows that if the sum of the external forces is zero, then the total pulse a closed system of bodies is always constant, despite the fact that the internal strength
Impulse(amount of motion) of a body is called a physical vector quantity, which is a quantitative characteristic of the translational motion of bodies. The impulse is indicated by R... The momentum of the body is equal to the product of the body's mass by its velocity, 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 tangentially to the trajectory). The unit of measurement of impulse is kg ∙ m / s.
General impulse of the system of bodies is equal to vector the sum of impulses of all bodies of the system:
Changing the momentum of one body is found by the formula (note that the difference between the final and initial impulses is vector):
where: p n - momentum of the body at the initial moment of time, p to - in the final. The main thing is not to confuse the last two concepts.
Absolutely resilient impact- an abstract model of collision, which does not take into account energy losses due to friction, deformation, etc. No other interactions other than direct contact are counted. With an absolutely elastic impact on a fixed surface, the velocity of the object after impact on the modulus is equal to the velocity of the object before 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 blow- a blow, as a result of which the bodies are connected 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, an automatic coupler is triggered and they also continue to move on together.
Momentum conservation law
When bodies interact, the impulse of one body can be partially or completely transferred to another body. If external forces from other bodies do not act on a system of bodies, such a system is called closed.
In a closed system, the vector sum of the momenta of all bodies included in the system remains constant for any interactions between the bodies of this system. This fundamental law of nature is called momentum conservation law (MMP)... Its consequence is Newton's laws. Newton's second law in impulse form can be written as follows:
As follows from this formula, if external forces do not act on the system of bodies, or the effect of external forces is compensated (the resultant force is equal to zero), then the change in momentum is equal to zero, which means that the total momentum of the system is conserved:
Similarly, you can reason for the equality to zero of the projection of the force on the selected axis. If external forces do not act only along one of the axes, then the projection of the impulse onto this axis is preserved, for example:
Similar records can be made for the rest of the coordinate axes. One way or another, you need to understand that in this case 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.
Storing the projection of the momentum
Situations are possible when the law of conservation of momentum is fulfilled only partially, that is, only when projecting onto one axis. If a force acts on the body, then its momentum is not conserved. But you can always choose an axis so that the projection of the force on this axis is 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 FID. Vector method
In cases where the bodies do not move along one straight line, then in the general case, in order to apply the law of conservation of momentum, you need to paint it along all coordinate axes involved in the problem. But the solution to such a problem can be greatly simplified by using the vector method. It is applied 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 vector addition rules it follows that the three vectors in these formulas must form a triangle. For triangles, the cosine theorem applies.
If on a body of mass m for a certain period of time Δ t the force F → acts, then there follows a change in the speed of the body ∆ v → = v 2 → - v 1 →. We get that during the time Δ t the body continues to move with acceleration:
a → = ∆ v → ∆ t = v 2 → - v 1 → ∆ t.
Based on the basic law of dynamics, that is, Newton's second law, we have:
F → = m a → = m v 2 → - v 1 → ∆ t or F → ∆ t = m v 2 → - m v 1 → = m ∆ v → = ∆ m v →.
Definition 1Body impulse, or amount of movement Is a physical quantity equal to the product of the body's mass by the speed of its movement.
The momentum of a body is considered a vector quantity, which is measured in kilogram-meter per second (to gm / s).
Definition 2
Impulse of force- This is a physical quantity equal to the product of force by the time of its action.
The impulse is referred to as vector quantities. There is another formulation of the definition.
Definition 3
The change in the momentum of the body is equal to the momentum of the force.
When denoting momentum p → Newton's second law is written as:
F → ∆ t = ∆ p →.
This form allows you to formulate Newton's second law. Force F → is the resultant of all forces acting on the body. Equality is written as a projection on the coordinate axes of the form:
F x Δ t = Δ p x; F y Δ t = Δ p y; F z Δ t = Δ p z.
Picture 1 . 16 . 1 . Body impulse model.
The change in the projection of the momentum of the body on any of the three mutually perpendicular axes is equal to the projection of the impulse of the force on the same axis.
Definition 4
One-dimensional movement Is the movement of the body along one of the coordinate axes.
Example 1
For example, consider the free fall of a body with an initial velocity v 0 under the action of gravity over a time interval t. With the direction of the axis O Y vertically downward, the impulse of gravity F t = mg, acting during time t, is equal to m g t... Such an impulse is equal to a change in the impulse of the body:
F t t = m g t = Δ p = m (v - v 0), whence v = v 0 + g t.
The record coincides with the kinematic formula for determining the speed of uniformly accelerated motion. The modulus of the force does not change from the entire interval t. When it is variable in magnitude, then the impulse formula requires substitution of the average value of the force F with p from the time interval t. Picture 1 . 16 . 2 shows how the momentum of the force is determined, which depends on time.
Picture 1 . 16 . 2. Calculation of the impulse of force according to the graph of the dependence F (t)
It is necessary to select the interval Δt on the time axis, it is seen that the force F (t) practically unchanged. Force impulse F (t) Δ t for a time interval Δ t will equal the area of the shaded figure. When dividing the time axis into intervals by Δ t i in the interval from 0 to t, add up the impulses of all acting forces from these intervals Δ t i , then the total impulse of force will be equal to the area of formation using the stepwise and time axes.
Applying the limit (Δ t i → 0), you can find the area that will be limited by the graph F (t) and the t-axis. Using the definition of the momentum of force from the graph is applicable with any laws where there are changing forces and time. This solution leads to the integration of the function F (t) from the interval [0; t].
Picture 1 . 16 . 2 shows the impulse of force located in the interval from t 1 = 0 s to t 2 = 10.
From the formula we obtain that F with p (t 2 - t 1) = 1 2 F m a x (t 2 - t 1) = 100 N · s = 100 k g · m / s.
That is, the example shows F with p = 1 2 F m a x = 10 N.
There are cases when the determination of the average force F with p is possible with known time and data on the impulse reported. With a strong impact on a ball with a mass of 0.415 kg, a velocity equal to v = 30 m / s can be reported. The approximate impact time is 8 · 10 - 3 s.
Then the impulse formula takes the form:
p = m v = 12.5 kg m / s.
To determine the average force F with p during impact, you need F with p = p ∆ t = 1.56 · 10 3 N.
Got a very high value, which is equal to a body with a mass of 160 to g.
When movement occurs along a curvilinear trajectory, then the initial value p 1 → and the final
p 2 → can be different in absolute value and direction. To determine the momentum ∆ p →, a pulse diagram is used, where there are vectors p 1 → and p 2 →, and ∆ p → = p 2 → - p 1 → is constructed according to the parallelogram rule.
Example 2
Figure 1 is shown as an example. 16 . 2 for a diagram of the impulses of a ball bouncing off a wall. When serving, a ball with mass m with a speed v 1 → hits the surface at an angle α to the normal and rebounds with a speed v 2 → with an angle β. When hitting the wall, the ball was subjected to the action of the force F →, directed in the same way as the vector ∆ p →.
Picture 1 . 16 . 3. Ball bouncing off a rough wall and momentum diagram.
If there is a normal fall of a ball with mass m onto an elastic surface with a velocity v 1 → = v →, then upon rebound it will change to v 2 → = - v →. This means that for a certain period of time the impulse will change and will be equal to ∆ p → = - 2 m v →. Using projections on O X, the result is written as Δ p x = - 2 m v x. From the picture 1 . 16 . 3 it can be seen that the axis O X is directed from the wall, then v x< 0 и Δ p x >0. From the formula we obtain that the modulus Δ p is related to the modulus of velocity, which takes the form Δ p = 2 m v.
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