Often in physics they talk about the momentum of a body, implying amount of movement. In fact, this concept is closely connected with a completely different quantity - with force. The impulse of force - what is it, how is it introduced into physics, and what is its meaning: all these issues are covered in detail in the article.
Number of movement
The impulse of the body and the impulse of the force are two interrelated quantities, moreover, they practically mean the same thing. First, let's look at the concept of momentum.
The momentum as a physical quantity first appeared in scientific papers scientists of modern times, in particular in the 17th century. It is important to note two figures here: Galileo Galilei, the famous Italian, who called the quantity under discussion impeto (impulse), and Isaac Newton, the great Englishman, who, in addition to the magnitude of motus (movement), also used the concept of vis motrix (driving force).
So, the above-named scientists understood the product of the mass of an object and the speed of its linear movement in space as the amount of motion. This definition in the language of mathematics is written as follows:
Let us note that we are talking about the value of the vector (p¯), directed towards the movement of the body, which is proportional to the speed modulus, and the role of the proportionality coefficient is played by the body mass.
Relation between the momentum of force and the change in p¯
As mentioned above, in addition to the momentum, Newton also introduced the concept of driving force. He defined this as follows:
This is the familiar law of the appearance of acceleration a¯ on a body as a result of some external force F¯ acting on it. This important formula allows us to derive the law of momentum of force. Note that a¯ is the time derivative of the rate (the rate of change of v¯), which means:
F¯ = m*dv¯/dt or F¯*dt = m*dv¯ =>
F¯*dt = dp¯, where dp¯ = m*dv¯
The first formula in the second line is the impulse of the force, that is, the value equal to the product of the force and the time interval during which it acts on the body. It is measured in newtons per second.
Formula Analysis
The expression for the momentum of the force in the previous paragraph also reveals physical meaning this value: it shows how much the amount of motion changes over a period of time dt. Note that this change (dp¯) is completely independent of general meaning amount of body movement. The impulse of force is the cause of the change in the momentum, which can lead to both an increase in the latter (when the angle between the force F¯ and the speed v¯ is less than 90 o) and its decrease (the angle between F¯ and v¯ is greater than 90 o).
An important conclusion follows from the analysis of the formula: the units of measurement of the impulse of force are the same as those for p¯ (newton per second and kilogram per meter per second), moreover, the first value is equal to the change in the second, therefore, instead of the impulse of force, the phrase "body impulse" is often used, although it is more correct to say "change in momentum".
Forces that depend and do not depend on time
Above, the law of force impulse was presented in differential form. To calculate the value of this quantity, it is necessary to carry out integration over the action time. Then we get the formula:
∫ t1 t2 F¯(t)*dt = Δp¯
Here, the force F¯(t) acts on the body during the time Δt = t2-t1, which leads to a change in the momentum by Δp¯. As you can see, the impulse of a force is a quantity determined by a force that depends on time.
Now consider more simple situation, which is realized in a number of experimental cases: we assume that the force does not depend on time, then we can easily take the integral and obtain a simple formula:
F¯*∫ t1 t2 dt = Δp¯ => F¯*(t2-t1) = Δp¯
When solving real problems of changing the momentum, despite the fact that the force generally depends on the time of action, it is assumed to be constant and some effective average value F¯ is calculated.
Examples of manifestation in practice of an impulse of force
What role this value plays is easiest to understand with specific examples from practice. Before we give them, we write out the corresponding formula once again:
Note that if Δp¯ is a constant value, then the momentum modulus of the force is also a constant, so the larger Δt, the smaller F¯, and vice versa.
Now let's give concrete examples of the momentum of force in action:
- A person who jumps from any height to the ground tries to bend his knees when landing, thereby increasing the time Δt of the impact of the ground surface (support reaction force F¯), thereby reducing its strength.
- The boxer, deflecting his head from the blow, prolongs the contact time Δt of the opponent's glove with his face, reducing the impact force.
- Modern cars are trying to design in such a way that in the event of a collision, their body is deformed as much as possible (deformation is a process that develops over time, which leads to a significant reduction in the force of a collision and, as a result, a decrease in the risk of injury to passengers).
The concept of the moment of force and its momentum
And the impulse of this moment is other quantities different from those considered above, since they no longer concern linear, but rotary motion. So, the moment of force M¯ is defined as vector product shoulder (distance from the axis of rotation to the point of action of the force) on the force itself, that is, the formula is valid:
The moment of force reflects the ability of the latter to perform torsion of the system around the axis. For example, if you hold the wrench away from the nut (large lever d¯), you can create a large moment M¯, which will allow you to unscrew the nut.
By analogy with the linear case, the momentum M¯ can be obtained by multiplying it by the time interval during which it acts on a rotating system, that is:
The value ΔL¯ is called the change angular momentum, or angular momentum. The last equation has importance to consider systems with an axis of rotation, because it shows that the angular momentum of the system will be conserved if there are no external forces, creating the moment M¯, which is mathematically written as follows:
If M¯= 0 then L¯ = const
Thus, both momentum equations (for linear and circular motion) turn out to be similar in terms of their physical meaning and mathematical consequences.
Bird and Airplane Collision Challenge
This problem is not something fantastic. Such collisions do occur quite often. So, according to some data, in 1972 on the territory airspace Israel (the zone of the most dense migration of birds) was registered about 2.5 thousand collisions of birds with combat and transport aircraft, as well as with helicopters.
The task is as follows: it is necessary to approximately calculate what impact force falls on a bird if an airplane flying at a speed of v = 800 km / h is encountered on its path.
Before proceeding to the solution, let's assume that the length of the bird in flight is l = 0.5 meters, and its mass is m = 4 kg (it can be, for example, a drake or a goose).
We will neglect the speed of the bird (it is small compared to that of the aircraft), and we will also consider the mass of the aircraft to be much greater than that of the birds. These approximations allow us to say that the change in the momentum of the bird is equal to:
To calculate the impact force F, you need to know the duration of this incident, it is approximately equal to:
Combining these two formulas, we get the desired expression:
F \u003d Δp / Δt \u003d m * v 2 / l.
Substituting the numbers from the condition of the problem into it, we get F = 395062 N.
It will be more visual to translate this figure into an equivalent mass using the formula for body weight. Then we get: F = 395062/9.81 ≈ 40 tons! In other words, a bird perceives a collision with an airplane as if 40 tons of cargo had fallen on it.
Momentum... A concept quite often used in physics. What is meant by this term? If we ask this question to a simple layman, in most cases we will get the answer that the momentum of the body is a certain impact (push or blow) exerted on the body, due to which it gets the opportunity to move in a given direction. All in all, a pretty good explanation.
The momentum of a body is a definition that we first encounter at school: in a physics lesson, we were shown how a small cart rolled down an inclined surface and pushed a metal ball off the table. It was then that we reasoned about what could affect the strength and duration of this. From such observations and conclusions many years ago, the concept of the body's momentum was born as a characteristic of movement, directly dependent on the speed and mass of the object.
The term itself was introduced into science by the Frenchman René Descartes. It happened in early XVII century. The scientist explained the momentum of the body only as the "quantity of motion." As Descartes himself said, if one moving body collides with another, it loses as much of its energy as it gives to another object. The potential of the body, according to the physicist, did not disappear anywhere, but was only transferred from one object to another.
The main characteristic that a body's momentum possesses is its directionality. In other words, it represents itself. Hence, such a statement follows that any body in motion has a certain momentum.
The formula for the impact of one object on another: p = mv, where v is the speed of the body (vector value), m is the mass of the body.
However, the momentum of the body is not the only quantity that determines the movement. Why do some bodies, unlike others, do not lose it for a long time?
The answer to this question was the emergence of another concept - the impulse of force, which determines the magnitude and duration of the impact on the object. It is he who allows us to determine how the momentum of the body changes over a certain period of time. The impulse of force is the product of the magnitude of the impact (actual force) and the duration of its application (time).
One of the most remarkable features of IT is its preservation in an unchanged form under the condition of a closed system. In other words, in the absence of other influences on two objects, the momentum of the body between them will remain stable for an arbitrarily long time. The principle of conservation can also be taken into account in a situation where there is an external effect on the object, but its vector effect is 0. Also, the momentum will not change even if the effect of these forces is insignificant or acts on the body for a very short period of time (as, for example, when shot).
It is this conservation law that has been haunting inventors who have been puzzling over the creation of the notorious “perpetual motion machine” for hundreds of years, since it is precisely this law that underlies such a concept as
As for the application of knowledge about such a phenomenon as body momentum, they are used in the development of missiles, weapons and new, albeit not eternal, mechanisms.
Any problems on moving bodies in classical mechanics require knowledge of the concept of momentum. This article discusses this concept, gives an answer to the question of where the momentum vector of the body is directed, and also provides an example of solving the problem.
Number of movement
To find out where the momentum vector of the body is directed, it is necessary, first of all, to understand its physical meaning. The term was first explained by Isaac Newton, but it is important to note that the Italian scientist Galileo Galilei already used a similar concept in his works. To characterize a moving object, he introduced a quantity called aspiration, onslaught, or impulse proper (impeto in Italian). The merit of Isaac Newton lies in the fact that he was able to connect this characteristic with the forces acting on the body.
So, initially and more correctly, what most people understand by the momentum of the body, call the momentum. Really, mathematical formula for the quantity under consideration is written as:
Here m is the mass of the body, v¯ is its speed. As can be seen from the formula, we are not talking about any impulse, there is only the speed of the body and its mass, that is, the amount of motion.
It is important to note that this formula does not follow from mathematical proofs or expressions. Its occurrence in physics has an exclusively intuitive, everyday character. So, any person is well aware that if a fly and a truck move at the same speed, then the truck is much harder to stop, since it has much more movement than an insect.
The origin of the concept of the momentum vector of the body is discussed below.
The impulse of force is the cause of the change in momentum
Newton was able to connect the intuitively introduced characteristic with the second law bearing his last name.
The impulse of force is a known physical quantity, which is equal to the product of the applied external force to some body by the time of its action. Using the well-known Newton's law and assuming that the force does not depend on time, we can come to the expression:
F¯ * Δt = m * a¯ * Δt.
Here Δt is the time of action of the force F, a is the linear acceleration imparted by the force F to a body of mass m. As you know, multiplying the acceleration of a body by the period of time that it acts, gives an increase in speed. This fact allows us to rewrite the formula above in a slightly different form:
F¯ * Δt = m * Δv¯, where Δv¯= a¯ * Δt.
The right side of the equation represents the change in momentum (see the expression in the previous paragraph). Then it will turn out:
F¯ * Δt = Δp¯, where Δp¯ = m * Δv¯.
Thus, using Newton's law and the concept of the momentum of a force, one can come to an important conclusion: the impact of an external force on an object for some time leads to a change in its momentum.
Now it becomes clear why the amount of motion is usually called the impulse, because its change coincides with the impulse of the force (the word "force", as a rule, is omitted).
The vector quantity p¯
Some quantities (F¯, v¯, a¯, p¯) have a bar above them. This means that we are talking about a vector characteristic. That is, the amount of motion is the same as the speed, force and acceleration, in addition to absolute value(module) is also described by direction.
Since each vector can be decomposed into separate components, then, using the Cartesian rectangular coordinate system, we can write the following equalities:
1) p¯ = m * v¯;
2) p x \u003d m * v x; p y = m * v y ; p z = m * v z ;
3) |p¯| = √(p x 2 + p y 2 + p z 2).
Here, the 1st expression is the vector form of the momentum representation, the 2nd set of formulas allows you to calculate each of the momentum components p¯, knowing the corresponding velocity components (indices x, y, z indicate the projection of the vector onto the corresponding coordinate axis). Finally, the 3rd formula allows you to calculate the length of the momentum vector (the absolute value of the quantity) through its components.
Where is the body's momentum vector directed?
Having considered the concept of momentum p¯ and its basic properties, one can easily answer the question posed. The momentum vector of the body is directed in the same way as the vector linear speed. Indeed, it is known from mathematics that the multiplication of the vector a¯ by the number k leads to the formation of a new vector b¯ with the following properties:
- its length is equal to the product of the number and the modulus of the original vector, i.e. |b¯| = k * |a¯|;
- it is directed in the same way as the original vector if k > 0, otherwise it will be directed opposite to a¯.
In this case, the role of the vector a¯ is played by the velocity v¯, the momentum p¯ is the new vector b¯, and the number k is the mass of the body m. Since the latter is always positive (m>0), then, answering the question: what is the direction of the momentum vector of the body p¯, it should be said that it is co-directed to the velocity v¯.
Momentum change vector
It is interesting to consider another similar question: where is the vector of change in the momentum of the body directed, that is, Δp¯. To answer it, you should use the formula obtained above:
F¯ * Δt = m * Δv¯ = Δp¯.
Based on the considerations in the previous paragraph, we can say that the direction of change in momentum Δp¯ coincides with the direction of the force vector F¯ (Δt > 0) or with the direction of the vector of change in velocity Δv¯ (m > 0).
It is important not to confuse here that we are talking about a change in values. In general, the vectors p¯ and Δp¯ do not coincide, since they are not related to each other in any way. For example, if the force F¯ will act against the speed v¯ of the object, then p¯ and Δp¯ will be directed in opposite directions.
Where is it important to take into account the vector nature of the momentum?
The questions discussed above: where the momentum vector of the body and the vector of its change are directed, are not due to simple curiosity. The point is that the momentum conservation law p¯ holds for each of its components. That is, in its most complete form, it is written as follows:
p x = m * v x ; p y = m * v y ; p z = m * v z .
Each component of the vector p¯ retains its value in the system of interacting objects that are not affected by external forces (Δp¯ = 0).
How to use this law and vector representations of p¯ to solve problems on the interaction (collision) of bodies?
Problem with two balls
The figure below shows two balls of different masses that fly at different angles to a horizontal line. Let the masses of the balls be m 1 = 1 kg, m 2 = 0.5 kg, their speeds v 1 = 2 m/s, v 2 = 3 m/s. It is necessary to determine the direction of the momentum after the impact of the balls, assuming that the latter is absolutely inelastic.
Starting to solve the problem, one should write down the law of invariance of momentum in vector form, that is:
p 1 ¯ + p 2 ¯ = const.
Since each momentum component must be conserved, this expression must be rewritten, also taking into account that after the collision, the two balls will begin to move as a single object (perfectly inelastic impact):
m 1 * v 1x + m 2 * v 2x = (m 1 + m 2) * u x ;
M 1 * v 1y + m 2 * v 2y = (m 1 + m 2) * u y .
The minus sign for the projection of the momentum of the first body onto the y-axis appeared due to its direction against the chosen vector of the y-axis (see Fig.).
Now we need to express the unknown velocity components u, and then substitute the known values into the expressions (the corresponding velocity projections are determined by multiplying the absolute values of the vectors v 1 ¯ and v 2 ¯ by trigonometric functions):
u x = (m 1 * v 1x + m 2 * v 2x) / (m 1 + m 2), v 1x = v 1 * cos(45 o); v 2x = v 2 * cos(30o);
u x \u003d (1 * 2 * 0.7071 + 0.5 * 3 * 0.866) / (1 + 0.5) \u003d 1.8088 m / s;
u y = (-m 1 * v 1y + m 2 * v 2y) / (m 1 + m 2), v 1y = v 1 * sin(45 o); v 2y = v 2 * sin(30o);
u y = (-1 * 2 * 0.7071 + 0.5 * 3 * 0.5) / (1 + 0.5) = -0.4428 m/s.
These are two components of the speed of the body after the impact and "sticking" of the balls. Since the direction of the velocity coincides with the momentum vector p¯, then the question of the problem can be answered if we define u¯. Its angle relative to the horizontal axis will be equal to the arc tangent of the ratio of the components u y and u x:
α \u003d arctg (-0.4428 / 1.8088) \u003d -13.756 o.
The minus sign indicates that the momentum (velocity) after the impact will be directed downward from the x-axis.
Details Category: Mechanics Published on 21.04.2014 14:29 Views: 55846There are two conservation laws in classical mechanics: the law of conservation of momentum and the law of conservation of energy.
body momentum
For the first time the concept of momentum was introduced by a French mathematician, physicist, mechanic and the philosopher Descartes, who called the impulse amount of movement .
From the Latin "impulse" is translated as "push, move."
Any body that moves has momentum.
Imagine a cart standing still. Its momentum is zero. But as soon as the cart starts moving, its momentum will cease to be zero. It will start to change as the speed will change.
momentum of a material point, or amount of movement is a vector quantity equal to the product of the mass of a point and its speed. The direction of the momentum vector of the point coincides with the direction of the velocity vector.
If we talk about a solid physical body, then the product of the mass of this body and the speed of the center of mass is called the impulse of such a body.
How to calculate the momentum of a body? One can imagine that the body is made up of many material points, or systems of material points.
If - the momentum of one material point, then the momentum of the system of material points
I.e, momentum of a system of material points is the vector sum of the impulses 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 momentum in the international SI system of units is kilogram-meter per second (kg m/s).
Impulse of force
In mechanics, there is a close relationship between the momentum of a body and force. These two quantities are connected by a quantity called momentum of force .
If a constant force acts on the 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 value equal to the product of the force acting on the body and the time during which it acts is called momentum of force .
As we see from the equation, the momentum of the force is equal to the difference between the momentum of the body at the initial and final moment of time, or the change in momentum over some time.
Newton's second law in impulsive form is formulated as follows: the change in the momentum of the body is equal to the momentum of the force acting on it. It must be said that Newton himself formulated his law in exactly this way.
The momentum of a force 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 such a system in which the bodies interact only with each other and do not interact with external bodies.
Imagine a closed system of two physical bodies. The forces of interaction of bodies with each other are called internal forces.
The impulse of force 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
By simple calculations, we get mathematical expression momentum conservation law:
where m 1 And m2 - masses of bodies,
v1 And v2 are the speeds of the first and second bodies before interaction,
v1" And v2" – speeds of the first and second bodies after interaction .
p 1 = m 1 · v 1 - momentum of the first body before interaction;
p 2 \u003d m 2 · v2 - momentum of the second body before interaction;
p 1 "= m 1 · v1" - momentum of the first body after interaction;
p 2 "= m 2 · v2" - momentum of the second body after interaction;
I.e
p 1 + p 2 = p1" + p2"
In a closed system, bodies only exchange impulses. And the vector sum of the impulses of these bodies before their interaction is equal to the vector sum of their impulses after the interaction.
So, as a result of a shot from 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 sum of the impulses of the gun and the flying bullet after the shot.
When firing a cannon, recoil occurs. The projectile flies forward, and the gun itself rolls back. A projectile and a 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 during the interaction of these bodies over time, that is, remains constant value. That's what it is law of conservation of momentum.
More precisely, the momentum conservation law 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 if 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 work.
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, a 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 on one of the coordinate axes is equal to zero (that is, the forces are balanced in the direction of this axis), then the momentum conservation law is fulfilled in this direction.
The law of conservation of momentum is also called law of conservation of momentum .
Most a prime example application of the law of conservation of momentum - jet propulsion.
Jet propulsion
Jet motion is the movement of a body that occurs when a part of it separates from it at a certain speed. The body itself receives an oppositely directed momentum.
The simplest example of jet propulsion is the flight of a balloon from which air escapes. If we inflate the balloon and let it go, 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 ejection of liquid from the fruit of a mad 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.
Reactive 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 ejected matter, is a closed system, the momentum of such a system does not change with time.
The reactive force arises as a result of 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 momentum of the rocket and fuel was equal to zero. Therefore, according to the law of conservation of momentum, after the engines are turned on, the sum of these impulses is also equal to zero.
where is the mass of the rocket
Gas flow rate
Rocket speed change
∆m f - fuel mass consumption
Let's assume the rocket worked for a time t .
Dividing both sides of the equation by ∆ t, we get the expression
According to Newton's second law, the reactive force is
Reactive force, or jet thrust, provides 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.
IN Everyday life in order to characterize a person who commits spontaneous acts, the epithet "impulsive" is sometimes used. At the same time, some people do not even remember, and a significant part do not even know with what physical quantity this word is related. What is hidden under the concept of “body momentum” and what properties does it have? The answers to these questions were sought by such great scientists as Rene Descartes and Isaac Newton.
Like any science, physics operates with clearly formulated concepts. On the this moment the following definition for the quantity called the momentum of the body is adopted: it is a vector quantity, which is a measure (quantity) of the mechanical movement of the body.
Let us assume that the issue is considered within the framework of classical mechanics, i.e. it is considered 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 momentum modulus of the body is calculated by formula 1 (see photo below).
Thus, by definition, this quantity is equal to the product of the mass of the body and its speed, with which its vector is codirected.
As a unit of momentum in SI ( international system units) is taken as 1 kg/m/s.
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 a flying boulder has an impetus directly proportional to its speed, which would be the same if there were no air resistance. At the same time, according to this philosopher, bodies with more weight had the ability to "accommodate" more of this driving force.
The concept, later called impulse, was further developed by Rene Descartes, who designated it with the words “quantity of motion”. 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 fact that the amount of motion must also have a direction was the first to guess the English scientist John Vallis. It happened in 1668. However, it took another couple of years for him to formulate the well-known law of conservation of momentum. The 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, named after him.
Momentum of the system of material points
Let us first consider the case when we are talking about velocities much smaller than the speed of light. Then, according to the laws of classical mechanics, the total momentum of the system of material points is a vector quantity. He is equal to the sum 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 co-directed with the velocity of the particle.
If we are talking about a body of finite size, then first it is mentally divided into small parts. Thus, the system of material points is again considered, however, its momentum is calculated not by the usual summation, but by integration (see formula 4).
As you can see, there is no time dependence, so the momentum of a system that is not affected by external forces (or their influence is mutually compensated) remains unchanged in time.
Proof of the conservation law
Let us 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.
Note that the system is closed. Then, summing over all points and applying Newton's Third Law, we obtain expression 6.
Thus, the momentum of a closed system is a constant.
The conservation law is also valid in those cases when the total sum of the forces that act on the system from the outside is equal to zero. From this follows one important particular assertion. It states that the momentum of a 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 a hit with a club, the puck must maintain its momentum. This situation will be observed even despite the fact that this body is affected by the force of gravity and the reactions of the support (ice), since, although they are equal in absolute value, 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 momentum of the mechanical system of material points is the sum of the impulses of all the material points included in the system.
The second property of this quantity is that it remains unchanged under interactions that change only mechanical characteristics systems.
In addition, momentum is invariant with respect to any rotation of the frame of reference.
Relativistic case
Let us assume that we are talking about non-interacting material points having velocities of the order of 10 to the 8th power or slightly less in the SI system. The three-dimensional momentum is calculated by formula 7, where c is understood as the speed of light in 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 reference frame. There is also a 4D version. For one material point, it is determined by formula 8.
Momentum and energy
These quantities, as well as the mass, are closely related to each other. In practical problems, relations (9) and (10) are usually used.
Definition via de Broglie waves
In 1924, a hypothesis was put forward that not only photons, but also any other particles (protons, electrons, atoms) have wave-particle duality. Its author was the French scientist Louis de Broglie. If we translate this hypothesis into the language of mathematics, then it can be argued that any particle with energy and momentum is associated with a wave with a frequency and length expressed by formulas 11 and 12, respectively (h is Planck's constant).
From the last relation, we obtain 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 incommensurable with the speed of light, then the momentum modulus is calculated in the same way as in classical mechanics (see formula 1). Consequently, 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, i.e., its momentum.
Now you know that the momentum 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 who are far from physics observe it in everyday life. For example, everyone knows that firearms and artillery pieces recoil when fired. The law of conservation of momentum is also clearly demonstrated by playing billiards. It can be used to predict the direction of expansion of the balls after the impact.
The law has found application in the calculations necessary to study the consequences of possible explosions, in the field of creating jet vehicles, in the design of firearms, and in many other areas of life.
