Conservation laws. Pulse. Law of conservation of momentum. Physics. Grade 10.
Teacher's comments
Newton's laws make it possible to solve various practically important problems concerning the interaction and motion of bodies.
A large number of such problems are associated, for example, with finding the acceleration of a moving body if all the forces acting on this body are known. And then other quantities (instantaneous speed, displacement, etc.) are determined by acceleration. But it is often very difficult to determine the forces acting on a body. Therefore, to solve many problems, another important physical quantity is used - the momentum of the body.
- The momentum of a body p is a vector physical quantity equal to the product of the mass of the body and its speed
p = mv.
Momentum is a vector quantity. The direction of the body's momentum vector always coincides with the direction of the motion velocity vector.
The SI unit of impulse is the impulse of a body weighing 1 kg moving at a speed of 1 m/s. This means that the SI unit of momentum of a body is 1 kg • m/s.
When making calculations, use the equation for projections of vectors: рх = mvx.
Depending on the direction of the velocity vector relative to the selected X-axis, the projection of the momentum vector can be either positive or negative.
The word “impulse” (impulsus) translated from Latin means “push”. Some books use the term "momentum" instead of the term "impulse".
This quantity was introduced into science at approximately the same period of time when Newton discovered the laws that were later named after him (i.e. at the end of the 17th century).
When bodies interact, their impulses can change. This can be verified by simple experience.
Two balls of equal mass are suspended on thread loops from a wooden ruler mounted on a tripod ring, as shown in Figure a.
Rice. Demonstration of the law of conservation of momentum
Ball 2 is deflected from the vertical by an angle a (Fig. b) and released. Returning to his previous position, he hits ball 1 and stops. In this case, ball 1 begins to move and deviates by the same angle a (Fig. c).
In this case, it is obvious that as a result of the interaction of the balls, the momentum of each of them has changed: by how much the momentum of ball 2 decreased, the momentum of ball 1 increased by the same amount.
If two or more bodies interact only with each other (that is, they are not exposed to external forces), then these bodies form a closed system.
The momentum of each of the bodies included in a closed system can change as a result of their interaction with each other. But
- the vector sum of the impulses of the bodies that make up a closed system does not change over time for any movements and interactions of these bodies
This is the law of conservation of momentum.
The law of conservation of momentum is also satisfied if the bodies of the system are acted upon by external forces whose vector sum is equal to zero. Let's show this by using Newton's second and third laws to derive the law of conservation of momentum. For simplicity, let us consider a system consisting of only two bodies - balls of masses m1 and m2, which move rectilinearly towards each other with velocities v1 and v2 (Fig.).
Rice. A system of two bodies - balls moving rectilinearly towards each other
The forces of gravity acting on each of the balls are balanced by the elastic forces of the surface on which they roll. This means that the action of these forces can be ignored. The forces of resistance to movement in this case are small, so we will not take their influence into account either. Thus, we can assume that the balls interact only with each other.
The figure shows that after some time the balls will collide. During a collision lasting for a very short period of time t, interaction forces F1 and F2 will arise, applied respectively to the first and second ball. As a result of the action of forces, the speed of the balls will change. Let us denote the velocities of the balls after the collision by the letters v1 and v2.
In accordance with Newton's third law, the interaction forces between the balls are equal in magnitude and directed in opposite directions:
According to Newton's second law, each of these forces can be replaced by the product of the mass and acceleration received by each of the balls during interaction:
m1a1 = -m2a2.
Accelerations, as you know, are determined from the equalities:
Replacing the acceleration forces in the equation with the corresponding expressions, we obtain:
As a result of reducing both sides of the equality by t, we obtain:
m1(v'1 - v1) = -m2(v'2 - v2).
Let's group the terms of this equation as follows:
m1v1′ + m2v2′ = m1v1 = m2v2. (1)
Considering that mv = p, we write equation (1) in this form:
P'1 + P'2 = P1 + P2.(2)
The left sides of equations (1) and (2) represent the total momentum of the balls after their interaction, and the right sides represent the total momentum before the interaction.
This means that, despite the fact that the momentum of each of the balls changed during the interaction, the vector sum of their momentum after the interaction remained the same as before the interaction.
Equations (1) and (2) are a mathematical representation of the law of conservation of momentum.
Since this course considers only the interactions of bodies moving along one straight line, to write the law of conservation of momentum in scalar form, one equation is sufficient, which includes projections of vector quantities onto the X axis:
m1v'1x + m2v'2x= m1v1x + m2v2x.
Homework.
Task 1. Answer the questions.
- What is the impulse of a body?
- What can be said about the directions of the momentum vectors and the speed of a moving body?
- What does it mean to say that several bodies form a closed system?
- Formulate the law of conservation of momentum.
- For a closed system consisting of two bodies, write the law of conservation of momentum in the form of an equation that would include the masses and velocities of these bodies. Explain what each symbol in this equation means.
Task 2. Solve problems.
1. Car collision.
Explain these situations from the point of view of the law of conservation of momentum.
2.Why does a big fish swim backwards?
How are these situations different?
3. For future defenders.
When shooting, there is a recoil phenomenon and bruises may appear on the shoulder of the soldier to whom he applies the rifle. Why doesn’t a soldier holding a bazooka (hand grenade launcher) on his shoulder experience recoil when firing?
In which case does the gun shoot further: when it is fixed motionless, or when it is suspended?
The file “Rebuses” is attached to the lesson. You can download the file at any time convenient for you.
Used sources:
- https://www.youtube.com/watch?v=xD7bQXCLuM0
- https://www.youtube.com/watch?v=GdoZwHDYmp8
- https://www.youtube.com/watch?v=EH-UsBy3ENU
- https://www.youtube.com/watch?v=1I-A2dQl7pY
- https://www.youtube.com/watch?v=dqijLBR1qZc
- https://www.youtube.com/watch?v=L_KBBaxfAfQ
- https://znaika.ru/catalog/10-klass/physics/
- https://interneturok.ru/ru/school/physics/10-klass/
INFOPHYS is my world...
Lecture 10. The law of conservation of momentum and reactive motion.
Movement in nature does not arise from nothing and does not disappear - it is transmitted from one object to another. Under certain conditions, movement is able to accumulate, but when released, it reveals its ability to be preserved.
Have you ever wondered why:
- A football player can stop a ball flying at high speed with his foot or head, but a person cannot stop a carriage moving on the rails even very slowly (the mass of the carriage is much greater than the mass of the ball).
- A glass of water is placed on a long strip of strong paper. If you pull the strip slowly, the glass moves along with the paper. and if you sharply pull the strip of paper, the glass remains motionless. (the glass will remain motionless due to inertia - the phenomenon of maintaining the speed of a body constant in the absence of the action of other bodies on it)
- A tennis ball, hitting a person, does not cause any harm, but a bullet, which is less in mass, moves at high speed (600-800 m/s), turns out to be deadly (the speed of the bullet is much higher than that of the ball).
This means that the result of the interaction of bodies depends on both the mass of the bodies and their speed at the same time.
Another great French philosopher, mathematician, physicist and physiologist, founder of modern European rationalism and one of the most influential metaphysicians of modern times, René Descartes
introduced such a concept as “quantity of motion”. He also expressed the law of conservation of momentum and gave the concept of impulse of force.
“I accept that in the Universe... there is a certain amount of motion that never increases or decreases, and thus, if one body sets another in motion, it loses as much of its motion as it imparts.” R. Descartes
Descartes, judging by his statements, understood the fundamental significance of the concept of momentum - or momentum of a body - introduced by him in the 17th century - as the product of the mass of a body by the value of its speed. And although he made the mistake of not considering momentum as a vector quantity, the law of conservation of momentum that he formulated has stood the test of time with honor. At the beginning of the 18th century, the error was corrected, and the triumphant march of this law in science and technology continues to this day.
As one of the fundamental laws of physics, it has given scientists an invaluable research tool, prohibiting some processes and opening the way for others. Explosion, jet motion, atomic and nuclear transformations - this law works perfectly everywhere. And in how many everyday situations the concept of impulse helps to understand, today, we hope, you will see for yourself.
Quantity of motion is a measure of mechanical motion, equal for a material point to the product of its mass m and speed v. The amount of motion mv is a vector quantity, directed in the same way as the speed of the point. Sometimes the quantity of motion is also called impulse . The amount of motion, at any moment in time, is characterized by the speed of an object of a certain mass when moving from one point in space to another.
The momentum of a body (or momentum) is a vector quantity equal to the product of the mass of the body and its speed:
The momentum of the body is directed in the same direction as the speed of the body .
SI unit of 1 kg m/s.
A change in the momentum of a body occurs when bodies interact, for example, during impacts. (Video “Billiard balls”). When bodies interact, impulse
one body can be partially or completely transferred to another body.
Types of collisions:
An absolutely inelastic impact is an impact interaction in which bodies connect (stick together) with each other and move on as one body.
The bullet gets stuck in the block and then they move as one piece. A piece of plasticine sticks to the wall.
An absolutely elastic impact is a collision in which the mechanical energy of a system of bodies is conserved.
After a collision, the balls bounce off each other in different directions. The ball bounces off the wall.
Let a body of mass m be acted upon by a force F for some short period of time Δt.
Under the influence of this force, the speed of the body changed by
Consequently, during the time Δt the body moved with acceleration
From the basic law of dynamics (Newton’s second law) it follows:
A physical quantity equal to the product of a force and the time of its action is called force impulse :
Force impulse is also a vector quantity .
The impulse of force is equal to the change in the momentum of the body (Newton’s II law in impulse form):
Denoting the momentum of the body with the letter p, Newton’s second law can be written as:
It was in this general form that Newton himself formulated the second law. The force in this expression represents the resultant of all forces applied to the body.
To determine the change in momentum, it is convenient to use a pulse diagram, which depicts the pulse vectors, as well as the vector of the sum of pulses, constructed according to the parallelogram rule.
When considering any mechanical problem, we are interested in the motion of a certain number of bodies. The set of bodies whose movement we study is called a mechanical system or simply a system.
In mechanics, there are often problems when it is necessary to simultaneously consider several bodies moving in different ways. These are, for example, problems about the movement of celestial bodies, about the collision of bodies, about the recoil of a firearm, where both the projectile and the gun begin to move after the shot, etc. In these cases, we speak about the movement of a system of bodies: the solar system, a system of two colliding bodies , gun-shell systems, etc. Some forces act between the bodies of the system. In the solar system these are forces of universal gravity, in the system of colliding bodies - elastic forces, in the "gun - projectile" system - forces created by powder gases.
The impulse of the system of bodies will be equal to the sum of the impulses of each of the bodies. included in the system.
In addition to the forces acting from some bodies of the system on others (“internal forces”), bodies can also be acted upon by forces from bodies that do not belong to the system (“external” forces); for example, the force of gravity and the elasticity of the table also act on colliding billiard balls, the force of gravity also acts on the cannon and the projectile, etc. However, in a number of cases, all external forces can be neglected. Thus, when studying the collision of rolling balls, the forces of gravity are balanced for each ball separately and therefore do not affect their movement; When fired from a cannon, gravity will have an effect on the flight of the projectile only after it leaves the barrel, which will not affect the magnitude of the recoil. Therefore, one can often consider the movements of a system of bodies, assuming that there are no external forces.
If a system of bodies is not affected by external forces from other bodies, such a system is called closed.
A CLOSED SYSTEM IS A SYSTEM OF BODIES THAT INTERACT ONLY WITH EACH OTHER.
Law of conservation of momentum.
In a closed system, the vector sum of the impulses of all bodies included in the system remains constant for any interactions of the bodies of this system with each other.
The law of conservation of momentum serves as the basis for explaining a wide range of natural phenomena and is used in various sciences:
- The law is strictly observed in the phenomena of recoil when fired, the phenomenon of jet propulsion, explosive phenomena and the phenomena of collision of bodies.
- The law of conservation of momentum is used: when calculating the velocities of bodies during explosions and collisions; when calculating jet vehicles; in the military industry when designing weapons; in technology - when driving piles, forging metals, etc.
Law of conservation of momentum in projection onto the horizontal axis
If before and after the collision the velocities of the bodies are directed along the horizontal axis, then the law of conservation of momentum should be written in projections onto the OX axis. We must not forget that the vector projection sign:
- positive if its direction coincides with the direction of the OX axis;
- negative if it is directed opposite to the direction of the OX axis.
Important!
In an inelastic collision of two bodies moving towards each other, the speed of the joint motion will be directed in the direction where the body with high momentum was moving before the collision.
Special cases of the law of conservation of momentum (in projections onto the horizontal axis)
| Inelastic collision with a stationary body | m1v1 = (m1 + m2)v |
| Inelastic collision of moving bodies | ± m1v1 ± m2v2 = ±(m1 + m2)v |
| At the initial moment the system of bodies is motionless | 0 = m1v'1 – m2v'2 |
| Before the interaction, the bodies moved at the same speed | (m1 + m2)v = ± m1v'1 ± m2v'2 |
Questions
1. What is called the impulse of a body? 2. What can be said about the directions of the momentum vectors and speed of a moving body? 3. Tell us about the course of the experiment depicted in Figure 44. What does it indicate? 4. What does it mean to say that several bodies form a closed system? 5. Formulate the law of conservation of momentum. 6. For a closed system consisting of two bodies, write the law of conservation of momentum in the form of an equation that would include the masses and velocities of these bodies. Explain what each symbol in this equation means.
Lesson objectives:- Continue the formation of concepts about body impulse and force impulse, as well as the ability to apply them to the analysis of the phenomenon of interaction of bodies in the simplest cases;
- To ensure that students understand the formulation of the law of conservation of momentum, to teach schoolchildren to write the equation of the law in vector form for two interacting bodies;
- Require students to analyze the mechanical interaction of bodies; the ability to identify the signs of a phenomenon by which it is detected; indicate the conditions under which the phenomenon in question occurs; explain examples of the use of the phenomenon;
- To repeat Galileo’s principle of relativity, to reveal the meaning of relativity as applied to the law of conservation of momentum;
- To familiarize students with the application of the law of conservation of momentum in military and space technology, to explain the principle of jet propulsion.
Lesson plan:
Content:
- Repetition of the topic: “Body impulse.”
- Learning new material.
- Introduction of the concept of a mechanical system.
- Theoretical derivation of the law of conservation of momentum.
- Conditions for applying the law of conservation of momentum.
- Rationale for the statement: the law of conservation of momentum is valid in all inertial frames of reference.
- The law of conservation of momentum in technology and nature.
- Consolidation.
- Homework assignment.
Methods and techniques:
- Testing. Conversation, discussion of test results. Working with the textbook.
- Abstraction, modeling.
- Conversation. Demonstration of experiments. Working with the textbook.
- Conversation. Working with the textbook. Computer experiment.
- Working with the textbook. Observations. Generalization of observations. Proposing a hypothesis. Theoretical foresight. Experiment.
- Conversation. Observations. Generalization of observations.
- Demonstration. Observation. Computer modelling.
- Repetition of the main points of the lesson. Discussion of qualitative issues.
- Entries in diaries.
Update:
Teacher:
In the previous lesson, we became acquainted with one of the basic concepts of mechanics - impulse: the impulse of force and the impulse of the body. What does the word “impulse” mean when translated into Russian?
Student:
Impulse translated from Latin means “push, blow, urge.” Previously, the term “quantity of motion” was used.
Teacher:
Who first introduced the concept of momentum into physics?
Student:
The concept of momentum was first introduced into physics in the 17th century. French scientist R. Descartes when he studied the laws of mechanical motion.
Next, a portrait of R. Descartes is shown and his brief biography is given.
Teacher:
The effects produced by a blow or a push have always caused surprise:
- Why does a heavy hammer lying on a piece of iron only press it against the support, while the same hammer, striking the metal, changes the shape of the product?
- What is the secret of the circus trick, when a crushing blow of a hammer on a massive anvil does not cause any harm to the person on whose chest this anvil is installed?
- how does a jellyfish, squid, etc. move?
- Why is a rocket used for space flights, what does it push off from when it moves?
You can answer these and other similar questions by learning in class about one of the basic laws of physics - the law of conservation of momentum, which is used not only in mechanics, but also in other areas of physics, and is of great importance for the scientific and practical activities of man. We will return to discuss some of these issues at the end of the lesson.
is announced to the students
: “The Law of Conservation of Momentum”, as well as the objectives of the lesson:
- Let us once again remember what an impulse of force and an impulse of a body are, let us repeat how these physical quantities are related to each other;
- Let's study the law of conservation of momentum and consider the conditions of its applicability;
- We will learn what significance this law has in living nature and how it is applied in aviation and space technology.
Repetition of the topic “Momentum of a material point”
To test knowledge on the topic “Momentum of a material point”, a test consisting of four questions in two versions is used. Each question is shown on the screen in PowerPoint. The time allotted for completing each task is limited, the questions change automatically on the screen. Students provide their answers on two forms provided in advance. One of the forms is handed over to the teacher after finishing the work, the students leave the second to check the result and analyze their work. After finishing the work, the options for correct answers are shown on the screen and, if necessary, the teacher can return to the questions using hyperlinks or comment on the correct answer. The proposed test questions test the following elements of knowledge:
- the concept of “body impulse” and “force impulse”, the direction of the impulse;
- connection between force impulse and body impulse;
- vector character of the impulse, elastic and inelastic impact, direction of change of impulse;
- Galileo's principle and the relativity of body momentum in ISO.
Presentation of new material:
Teacher:
Tell me, why was it necessary to introduce the concept of momentum into physics?
Student:
The main problem of mechanics - determining the position of a body at any moment in time - can be solved using Newton's laws if the initial conditions and forces acting on the body are given as functions of coordinates, velocities and time. To do this, it is necessary to write down Newton's second law: the student writes on the board and explains the entry: <Figure 1>.
Student:
From this record it is clear that the force required to change the speed of a moving body over a certain period of time is directly proportional to both the mass of the body and the magnitude of the change in its speed.
Teacher:
What other conclusion can be drawn from the obtained record of Newton’s II law?
Student:
The momentum of a body changes under the influence of a given force in the same way for all bodies if the time of action of the force is the same.
Teacher:
Right. This is a very important conclusion and this form of writing Newton’s II law is used in solving many practical problems in which it is necessary to determine the final result of the action of a force. And, in addition, this notation allows us to relate the action of a force directly to the initial and final velocities of bodies, without clarifying the intermediate state of the system of interacting bodies, since in practice this, as a rule, is not always possible. Thus, it is clear that it is difficult to overestimate the role of mechanical shock in technology. It is not surprising that the laws (but not the theory) of impact were established empirically long before the discovery of the basic principles of dynamics.
The historical background “Study of elastic and inelastic impacts” is shown in PowerPoint. In the process of reporting historical information, the results of studies of elastic and inelastic impact are demonstrated: <Figure 2>.
In experiment “a” it is proven that when a ball rolls down an inclined chute with a tray, the momentum acquired by the ball at point A is proportional to the range of its flight in the horizontal direction, and therefore the speed in this direction.
In experiment “b” it is shown that during an elastic collision of identical balls located on a horizontal section of the tray at the moment of impact at point A, an exchange of impulses occurs.
In experiment “c” it is shown that during an inelastic central collision of balls of the same mass (a small piece of plasticine is placed between them), both balls travel the same distances, i.e. the total momentum of the balls before and after the impact is the same.
Introduction to the concept of a mechanical system
Teacher:
Since one of our main goals in the lesson is to derive the law of conservation of momentum of interacting bodies and to clarify the limits of its applicability, we will begin our consideration of this issue by analyzing the interaction of two bodies in a closed system. The teacher analyzes Figure 104 from. Additional drawings are made on the board: <Figure 4>.
Teacher:
A physical system is considered closed if external forces do not act on this system.
However, it is impossible to actually create such a system, since, for example, the action of gravitational forces extends to infinity, so we will assume that a closed system is a system of bodies in which the action of external forces is compensated.
But, strictly speaking, even in this case the closed system is an abstraction, because the action of some external forces (for example, friction) is not always possible to compensate. In this case, such forces are usually neglected.
Derivation of the law of conservation of momentum
Teacher:
We explore the physical model of the absolutely elastic interaction of two balls forming a closed system: students work with the textbook, analyzing Figure 104 from the textbook, which is duplicated on the board in PowerPoint: <Figure 3>.
Teacher:
What are the main features of the model of the physical phenomenon under consideration?
Students:
— we consider the balls to be material points (or the impact is central);
— the impact is absolutely elastic, which means that there is no deformation: the total kinetic energy of the bodies before the impact is equal to the total kinetic energy of the bodies after the impact;
— we neglect the action of resistance and gravity forces, as well as other possible external forces.
Teacher:
The action of what forces and at what moment is shown in the drawing?
Student:
When the balls collide, elastic forces F12 and F21 act between them, which, according to Newton’s III law, are equal in magnitude and opposite in direction.
Teacher:
Write it down mathematically.
The student writes down the expression on the board: <Figure 5>
Teacher:
What can be said about the time of action of these forces on bodies?
Student:
The time of action of bodies on each other during interaction is the same.
Teacher:
Applying Newton's second law, rewrite the resulting equality using the initial and final momenta of the interacting bodies.
A student on the board, commenting, derives the law of conservation of momentum: <Figure 6>
Teacher:
What conclusion did you come to?
Student:
The geometric sum of the momenta of bodies after interaction is equal to the geometric sum of the momenta of these bodies before interaction.
Teacher:
Yes, indeed, this statement is the law of conservation of momentum:
The total momentum of a closed system of bodies remains constant for any interaction of the bodies of the system with each other.
Teacher:
Read the formulation of the law of conservation of momentum on page 128 of the textbook and answer the question: Can the internal forces of a system change the total momentum of the system?
Student:
The internal forces of the system cannot change the momentum of the system.
Teacher:
Right. Watch the experience and explain it.
Experiment: Four identical rollers are placed parallel to each other on a smooth horizontal surface of a demonstration table. A strip of thick cardboard about 80 cm long is placed on them. The mechanical toy moves in one direction, and the cardboard in the opposite direction.
The teacher draws students' attention to the fact that in this experiment, when impulses are exchanged between bodies in a closed system, the center of mass of this system does not change its position in space. The moving body and the support form a closed system of interacting bodies. When these bodies interact, internal forces arise, the bodies exchange impulses, but the total impulse of the system does not change, this can be seen from the fact that the center of mass of the system does not change its position in space. Internal forces change the impulses of individual bodies of the system, but they cannot change the impulse of the entire system.
Conditions for the applicability of the law of conservation of momentum
Teacher:
We formulated the law of conservation of momentum taking into account the introduced limitation in the form of a model of interacting bodies of a closed system. But all real systems, strictly speaking, are not closed. However, in many cases the law of conservation of momentum can be applied. In what cases do you think this is acceptable?
Student 1:
If external forces are small compared to the internal forces of the system, and their action can be neglected.
Student 2:
When external forces compensate each other.
Teacher:
It should be added to what has been said that the law of conservation of momentum can also be applied if the initial and final states of the system are separated by a small time interval (for example, a grenade explosion, a gun shot, etc.). During this time, external forces such as gravity and friction will not noticeably change the momentum of the system.
But these are not all possible conditions for the application of the law of conservation of momentum. Tell me, will the system of bodies on the Earth or near the Earth’s surface be closed, for example, two balls and a cart?
Student:
No, because these bodies are affected by gravity, which is an external force.
Teacher:
This statement is correct, let’s remember it and do three experiments: <Figure 7>
In the first experiment, we will observe a ball falling into a cart, rolling down the right chute. Then we repeat the experiment, releasing the ball from the same height along the left chute. Finally, both balls fall from the same height along both chutes into the same cart. Explain why the cart moved in the first two experiments, but remained motionless in the third.
Student:
In the first two experiments, the cart moved in different directions, but the same distance. She received impulses when interacting with each of the balls.
Teacher:
Right. What can you say about the horizontal projections of the balls' impulses? Explain the results of the third experiment.
Student:
Since the balls move from the same height and have equal masses, the horizontal projections of their impulses are equal and oppositely directed. Therefore, their sum is zero, so the cart remains motionless.
Teacher:
This happens because the force of gravity does not act on bodies in the horizontal direction, and the friction force and air resistance force are small. In such cases, the law of conservation of momentum is applied, since the system of bodies is considered closed along a certain direction.
Further on in the textbook (p. 129 example: the “rifle - bullet” system) it is shown that: The law of conservation of momentum can be applied if the projection of the resultant external forces on the chosen direction is zero.
Relativity of the law of conservation of momentum
Teacher:
Let's try to answer the question: is the law of conservation of momentum valid in all inertial frames of reference? Can a reference frame related to the Earth have an advantage over other reference frames?
Next, the experiment on the interaction of bodies on a stationary and moving platform is demonstrated. Uniform movement is ensured by a technical toy with an electric motor. On the screen, the results of the experiment are duplicated in a pre-prepared demonstration presentation.
Teacher:
Are the impulses of bodies in the “Earth” and “platform” reference systems the same?
Student:
No, the speeds of the carts relative to the Earth and the platform are different.
Teacher:
Right. This shows the relativity of momentum. Write down the impulses of the bodies interacting on the platform using the notations introduced in the figure.
Student:
(commenting):
In the “Earth” reference system: <Figure 8>
In the “Platform” reference system: <Figure 9>
Teacher:
What do we know about the momentum of a system of bodies relative to the Earth?
Student:
The momentum of a closed system of bodies relative to the Earth is conserved.
Teacher:
Express the speed of the bodies relative to the platform in terms of the speed of the bodies relative to the Earth and analyze the resulting expression.
Student:
(commenting): <Figure 10>
thus: <Figure 11>
Since: <Figure 12>, (m1 + m2) and v0 also do not change with time, it means that the momentum of bodies in the “Platform” reference frame is also conserved: <Figure 13>
Teacher:
So, we have shown that the law of conservation of momentum is satisfied in all inertial frames of reference. This corresponds to Galileo's principle of relativity.
Law of conservation of momentum in technology and nature
Examples of jet propulsion in technology and nature are shown on the screen in PowerPoint.
Teacher:
What do a squid, a dragonfly larva and the Space Shuttle have in common?
Student:
All considered bodies use the principle of reactive motion in their movement.
Teacher:
Right. Let's take a closer look at the principle of jet propulsion, studied earlier in the 9th grade. Reactive motion is a movement that occurs when any part of it is separated from the body at a certain speed.
Jet motion is demonstrated by the example of the movement of a balloon on a platform: <Figure 14>.
Teacher:
Let's consider a model of jet propulsion.
Next is the student’s report: “The design of a rocket.” During the student's report, the rocket structure is demonstrated using PowerPoint: .
Teacher:
Let's simulate the action of a jet engine.
— neglecting the interaction of the rocket with external bodies, we will consider the “rocket – gases” system to be closed;
— fuel and oxidizer burn out immediately;
— M is the mass of the shell, v is the speed of the shell, m is the mass of gas ejected from the nozzle, u is the speed of gas outflow.
The rocket shell and combustion products form a closed system. Consequently, the shell, together with the second stage, acquires momentum p0 = Mv
, and the gas flowing out of the nozzle acquires momentum
pg = - mu
.
Since before the launch the momentum of the shell and gas was equal to 0, then p0 = - pg
and the remaining part of the rocket will move at a speed
v = mu/M
in the direction opposite to the direction of the outflow of combustion products. After the fuel of the first stage is completely burned and the oxidizer is consumed, the fuel and oxidizer tanks of this stage turn into excess ballast. Therefore, they are automatically discarded, and the smaller remaining mass of the ship accelerates further. Reducing mass allows for significant savings in fuel and oxidizer in the second stage and increases its speed.
After this, “A Brief History of Space Shuttle Launching” is considered.
Law of conservation of momentum in living nature
Teacher:
Note that, in essence, almost any change in the nature of movement is reactive movement and it occurs according to the law of conservation of momentum. In fact, when a person walks or runs, he pushes the Earth back with his feet. Due to this, he himself moves forward. Of course, the speed of the Earth turns out to be the same number of times less than the speed of a person, how many times the mass of the Earth is greater than the mass of a person. That is why we do not notice the movement of the Earth. But if you jump from a boat onto the shore, the boat’s rollback in the opposite direction will be quite noticeable.
The principle of jet propulsion is very often used in living nature; for example, squids, octopuses, and cuttlefish use a similar type of movement.
During its movement, the jellyfish draws water into the body cavity, and then sharply throws it out of itself and moves forward due to the recoil force.
Consolidation, generalization
Reinforcement questions are shown on the screen in PowerPoint:
Conclusion
Concluding the lesson, I would like to say that the laws in physics cannot be considered as the ultimate truth; they should be treated as models that can be applied to solving individual problems and finding solutions that are in good agreement with experience confirmed by specially designed experiments. Today in class we studied one of the most fundamental models: the law of conservation of momentum. We are convinced that the use of this law makes it possible to explain and predict phenomena not only of mechanics, which speaks of the great philosophical meaning of this model. The law of conservation of momentum serves as proof of the unity of the material world: it confirms the indestructibility of the movement of matter.
