Let's do some simple transformations with formulas. According to Newton's second law, the force can be found: F = m * a. Acceleration is found as follows: a = v⁄t. Thus, we get: F = m * v/ t.
Determination of body impulse: formula
It turns out that force is characterized by a change in the product of mass and speed in time. If we designate this product by a certain value, then we will receive the change in this value over time as a characteristic of force. This value was called the momentum of the body. The body impulse is expressed by the formula:
where p is the momentum of the body, m is the mass, v is the speed.
Momentum is a vector quantity, while its direction always coincides with the direction of the velocity. The unit of impulse is kilogram per meter per second (1 kg * m / s).
What is body impulse: how to understand?
Let's try in a simple way, "on the fingers" to figure out what a body impulse is. If the body is at rest, then its momentum is zero. It is logical. If the speed of the body changes, then a certain impulse appears in the body, which characterizes the magnitude of the force applied to it.
If there is no effect on the body, but it moves at a certain speed, that is, it has a certain impulse, then its impulse means what effect this body can have when interacting with another body.
The impulse formula includes the mass of the body and its speed. That is, the more mass and / or speed a body has, the more impact it can have. This is also understandable from life experience.
A small force is needed to move a body of small mass. The more body weight, the more effort will have to be made. The same goes for the speed that is imparted to the body. In the case of the impact of the body itself on another, the impulse also shows the amount with which the body is capable of acting on other bodies. This value directly depends on the speed and mass of the original body.
Impulse in the interaction of bodies
Another question arises: what will happen to the momentum of a body when it interacts with another body? The mass of a body cannot change if it remains intact, but the speed can change easily. In this case, the speed of the body will change depending on its mass.
Indeed, it is clear that when bodies with very different masses collide, their speed will change in different ways. If a soccer ball flying at high speed crashes into an unprepared person, for example, a spectator, then the viewer may fall, that is, gain some low speed, but will definitely not fly like a ball.
This is because the mass of the spectator is much greater than the mass of the ball. But at the same time, the total impulse of these two bodies will remain unchanged.
The law of conservation of momentum: formula
This is the law of conservation of momentum: when two bodies interact, their total momentum remains unchanged. The law of conservation of momentum operates only in a closed system, that is, in such a system in which there is no external force or their total effect is zero.
In reality, there is almost always an outside influence on the system of bodies, but the general impulse, like energy, does not disappear into anywhere and does not arise out of nowhere, it is distributed among all participants in the interaction.
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 impossible to determine these values. Let's consider a few of these situations.When two billiard balls or cars collide, we can assert about the acting forces that this is their nature, elastic forces act here. However, we will not be able to establish precisely their modules or their directions, especially since these forces have an extremely short duration of action.When rockets and jet planes move, we also have little to say about the forces that set these bodies in motion.In such cases, methods are used that make it possible to get away from solving the equations of motion, and immediately use the consequences of these equations. At the same time, new physical quantities... Consider one of these quantities, called the momentum of the body
An arrow shot from a bow. The longer the contact of the bowstring with the arrow (∆t) lasts, the greater the change in the momentum of the arrow (∆), and, consequently, 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 impulses 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 during the same period of time cause the same changes in momentum in different bodies, regardless of the mass of the latter.
2. One and the same change in the momentum of a body can be achieved either by acting with a small force for a long period of time, or by acting with a short-term 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 the body to the period of time during which this change occurred is equal to the sum of the forces acting on the body.
After analyzing 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, when moving, burn fuel, and the products of this burned are thrown into the surrounding space. Naturally, the mass of an aircraft or rocket decreases as fuel is consumed.
Despite the fact that Newton's second law in the form "the resultant force is equal to the product of the mass of a body and its acceleration" allows solving 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, linking 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 the solution of 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 the force and the momentum of the body, we can derive the second and third laws of Newton.
Newton's second law is derived from the ratio of the momentum of the force and the momentum of the body.
The impulse of force is equal to the change in the impulse of the body:
Having made the appropriate transfers, we get the dependence of the force on the acceleration, because the acceleration is defined as the ratio of the change in speed to the time during which this change took place:
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 vectoriality of speed, that is, the fact that the speed can change in direction. After transformations we get:
Since the time interval in a closed system was a constant value for both bodies, we can write:
We got 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., Yavorskiy B.M. Physics (basic level) - M .: Mnemosina, 2012.
- Gendenshtein L.E., Dick Yu.I. Physics grade 10. - M .: Mnemosina, 2014.
- Kikoin I.K., Kikoin A.K. Physics - 9, Moscow, Education, 1990.
Homework
- Give a definition of the impulse of the body, the impulse of power.
- How are the impulse of the body connected with 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 ().
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 body's momentum vector is directed, and also provides an example of solving the problem.
Movement amount
To find out where the momentum vector of the body is directed, one should, first of all, understand its physical meaning. The term was first explained by Isaac Newton, but it is important to note that the Italian scientist Galileo Galilei has already used a similar concept in his works. To characterize a moving object, he introduced a quantity called urge, pressure, 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 understand by the impulse of the body, to call the amount of movement. Indeed, the mathematical formula for the value under consideration is written in the form:
Here m is the mass of the body, v¯ is its velocity. 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 momentum.
It is important to note that this formula does not follow from mathematical proofs or expressions. Its appearance 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.
Where the concept of a vector of momentum of a body came from is considered below.
Impulse of force - the reason for the change in momentum
Intuitively introduced characteristic Newton was able to connect with the second law bearing his name.
The impulse of force is a known physical quantity, which is equal to the product of the applied external force to a certain body by the time of its action. Using the well-known Newton's law and assuming that the force does not depend on time, one 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, the multiplication of the acceleration of a body by the period of time that it acts, gives an increase in speed. This fact makes it possible to rewrite the formula above in a slightly different form:
F¯ * Δt = m * Δv¯, where Δv¯ = a¯ * Δt.
The right side of the equality 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 impulse of a force, one can come to an important conclusion: the effect of an external force on an object for some time leads to a change in its momentum.
Now it becomes clear why the momentum is usually called impulse, because its change coincides with the impulse of force (the word "force" is usually omitted).
The vector quantity p¯
Some quantities (F¯, v¯, a¯, p¯) have a bar. This means that we are talking about a vector characteristic. That is, the momentum, as well as the speed, force and acceleration, in addition to the absolute value (modulus), is also described by the direction.
Since each vector can be decomposed into separate components, using a Cartesian rectangular coordinate system, you can write the following equalities:
1) p¯ = m * v¯;
2) p x = 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 a vector form of representing the momentum, the 2nd set of formulas allows calculating each of the components of the momentum p¯, knowing the corresponding components of the velocity (the indices x, y, z speak about the projection of the vector onto the corresponding coordinate axis). Finally, the 3rd formula allows you to calculate the length of the impulse vector (the absolute value of the quantity) in terms of 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 of linear velocity. Indeed, it is known from mathematics that multiplying 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, that is, | 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 codirection of the body's momentum vector p¯, it should be said that it is codirectional with 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 reasoning in the previous paragraph, we can say that the direction of the change in the momentum Δp¯ coincides with the direction of the force vector F¯ (Δt> 0) or with the direction of the vector of the velocity change Δv¯ (m> 0).
It is important not to be confused here that we are talking about changing values. In the general case, the vectors p¯ and Δp¯ do not coincide, since they are in no way related to each other. For example, if the force F¯ will act against the velocity v¯ of the displacement 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 momentum?
The questions discussed above: where the vector of the momentum of the body and the vector of its change are directed, are not due to simple curiosity. The point is that the law of conservation of momentum p¯ is satisfied 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, which are not acted upon by external forces (Δp¯ = 0).
How to use this law and vector representations of the quantity p¯ to solve problems of interaction (collision) of bodies?
Problem with two balls
The picture below shows two balls of different masses that fly at different angles to the horizontal line. Let the masses of the balls be equal to m 1 = 1 kg, m 2 = 0.5 kg, their velocities v 1 = 2 m / s, v 2 = 3 m / s. It is necessary to determine the direction of the impulse after the impact of the balls, assuming the latter is absolutely inelastic.
Starting to solve the problem, one should write down the law of invariability of the momentum in vector form, that is:
p 1 ¯ + p 2 ¯ = const.
Since each component of the momentum must be conserved, it is necessary to rewrite this expression, taking into account also that after the collision, two balls will begin to move as a single object (absolutely 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 selected vector of the ordinate axis (see Fig.).
Now we need to express the unknown components of the velocity u, and then substitute the known values into the expressions (the corresponding projections of the velocities are determined by multiplying the moduli 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 (30 o);
u x = (1 * 2 * 0.7071 + 0.5 * 3 * 0.866) / (1 + 0.5) = 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 (30 o);
u y = (-1 * 2 * 0.7071 + 0.5 * 3 * 0.5) / (1 + 0.5) = -0.4428 m / s.
These are two components of the body's velocity after impact and "sticking" of the balls. Since the direction of the velocity coincides with the momentum vector p¯, the answer to the question of the problem is possible if we determine u¯. Its angle relative to the horizontal axis will be equal to the arctangent of the ratio of the components u y and u x:
α = arctan (-0.4428 / 1.8088) = -13.756 o.
The minus sign indicates that the impulse (velocity) after the impact will be directed downward from the x-axis.
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 impossible to determine these values. Let's consider a few of these situations.When two billiard balls or cars collide, we can assert about the acting forces that this is their nature, elastic forces act here. However, we will not be able to establish precisely their modules or their directions, especially since these forces have an extremely short duration of action.When rockets and jet planes move, we also have little to say about the forces that set these bodies in motion.In such cases, methods are used that make it possible to get away from solving the equations of motion, and immediately use the consequences of these equations. At the same time, new physical quantities are introduced. Consider one of these quantities, called the momentum of the body
An arrow shot from a bow. The longer the contact of the bowstring with the arrow (∆t) lasts, the greater the change in the momentum of the arrow (∆), and, consequently, 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 impulses 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 during the same period of time cause the same changes in momentum in different bodies, regardless of the mass of the latter.
2. One and the same change in the momentum of a body can be achieved either by acting with a small force for a long period of time, or by acting with a short-term 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 the body to the period of time during which this change occurred is equal to the sum of the forces acting on the body.
After analyzing 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, when moving, burn fuel, and the products of this burned are thrown into the surrounding space. Naturally, the mass of an aircraft or rocket decreases as fuel is consumed.
Despite the fact that Newton's second law in the form "the resultant force is equal to the product of the mass of a body and its acceleration" allows solving 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, linking 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 the solution of 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 the force and the momentum of the body, we can derive the second and third laws of Newton.
Newton's second law is derived from the ratio of the momentum of the force and the momentum of the body.
The impulse of force is equal to the change in the impulse of the body:
Having made the appropriate transfers, we get the dependence of the force on the acceleration, because the acceleration is defined as the ratio of the change in speed to the time during which this change took place:
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 vectoriality of speed, that is, the fact that the speed can change in direction. After transformations we get:
Since the time interval in a closed system was a constant value for both bodies, we can write:
We got 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., Yavorskiy B.M. Physics (basic level) - M .: Mnemosina, 2012.
- Gendenshtein L.E., Dick Yu.I. Physics grade 10. - M .: Mnemosina, 2014.
- Kikoin I.K., Kikoin A.K. Physics - 9, Moscow, Education, 1990.
Homework
- Give a definition of the impulse of the body, the impulse of power.
- How are the impulse of the body connected with 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 ().
In classical mechanics, there are two conservation laws: the law of conservation of momentum and the law of conservation of energy.
Body impulse
For the first time the concept of impulse was introduced by a French mathematician, physicist, mechanic and the philosopher Descartes, who called the impulse amount of movement .
From Latin "impulse" is translated as "push, move".
Any body that moves has momentum.
Imagine a cart standing motionless. Its momentum is zero. But as soon as the cart starts to move, its momentum will cease to be zero. It will begin to change as the speed changes.
The momentum of a material point, or amount of movement, Is a vector quantity equal to the product of the point's mass by its velocity. 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 impulse of such a body is called the product of the mass of this body by the speed of the center of mass.
How to calculate body impulse? You can imagine that the body consists of many material points, or a system of material points.
If is the impulse of one material point, then the impulse of the system of material points
That is, material point system momentum 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 by their speed.
The unit of measurement of impulse in the international SI system is kilogram-meter per second (kg m / s).
Impulse of force
In mechanics, there is a close connection between the momentum of a body and force. These two quantities are connected by a quantity called impulse of power .
If a constant force acts on the bodyF for 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.
A quantity equal to the product of the force acting on the body by the time during which it acts is called impulse of power .
As we can see from the equation, the impulse of force is equal to the difference in impulses of the body at the initial and final moment of time, or the change in impulse over time.
Newton's second law in impulse 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 originally formulated his law in this way.
The impulse of force is also a vector quantity.
The momentum conservation law follows from Newton's third law.
It must be remembered that this law operates only in a closed, or isolated, physical system. A closed system is a system in which bodies interact only with each other and do not interact with external bodies.
Let's 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
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 impulse of force is
By simple calculations, we obtain a mathematical expression for the law of conservation of momentum:
where m 1 and m 2 - body masses,
v 1 and v 2 - the velocities of the first and second bodies before interaction,
v 1 " and v 2" – velocities of the first and second bodies after interaction .
p 1 = m 1 · v 1 - the impulse of the first body before interaction;
p 2 = m 2 · v 2 - the impulse of the second body before the interaction;
p 1 "= m 1 · v 1 " - impulse of the first body after interaction;
p 2 "= m 2 · v 2 " - impulse of the second body after interaction;
That is
p 1 + p 2 = p 1 " + p 2 "
In a closed system, bodies only exchange impulses. And the vector sum of the 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.
Recoil occurs when firing a cannon. The projectile flies forward, and the weapon itself rolls back. A projectile and a cannon are a closed system in which the law of conservation of momentum operates.
The impulse of each of the bodies 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, it remains constant. That's what it is momentum conservation law.
More precisely, the law of conservation of momentum is formulated as follows: the vector sum of the impulses of all bodies of a closed system is a constant value if there are no external forces acting on it, or their vector sum is equal to zero.
The momentum of a system of bodies can only change as a result of external forces acting 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, shot, etc., then in this case the effect of external forces on the system is neglected, and the system itself is considered as closed.
In addition, if external forces act on the system, but the sum of their projections onto one of the coordinate axes is zero, (that is, the forces are balanced in the direction of this axis), then the law of conservation of momentum is fulfilled in this direction.
The momentum conservation law is also called momentum conservation law .
The most striking example of the application of the law of conservation of momentum is jet propulsion.
Jet propulsion
Reactive 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 impulse.
The simplest example of jet propulsion is the flight of a balloon, from which air comes out. If we inflate the balloon and release it, it will start flying in the direction opposite to the movement of the air coming out of it.
An example of jet propulsion in nature is the release of liquid from a mad cucumber when it bursts. In this case, the cucumber itself flies in the opposite direction.
Jellyfish, cuttlefish and other inhabitants of the deep sea move by taking in water and then throwing it out.
Jet thrust is based on the law of conservation of momentum. We know that when a rocket with a jet engine moves, as a result of the combustion of fuel, a jet of liquid or gas is ejected from the nozzle ( jet stream ). As a result of the interaction of the engine with the outflowing substance, Reactive force ... Since the rocket together with the ejected substance is a closed system, the momentum of such a system does not change with time.
Reactive force arises from the interaction of only parts of the system. External forces have no effect on its appearance.
Before the rocket started moving, the sum of the rocket and fuel impulses was zero. Consequently, according to the law of conservation of impulse after turning on the motors, the sum of these impulses is also equal to zero.
where is the mass of the rocket
Gas outflow rate
Rocket speed change
∆ m f - fuel mass consumption
Suppose the rocket has been working for a period of time t .
Dividing both sides of the equation by ∆ t, we get the expression
According to Newton's second law, the reactive force is
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.