body momentum. Jet propulsion
Lesson #14
Topic. body momentum. Law of conservation of momentum. Jet propulsion.
Target: to form students' knowledge about physical quantities - the momentum of the body and the momentum of the force, and the relationship between them; help to understand the law of conservation of momentum; to form knowledge about jet propulsion.
Lesson type: learning lesson.
Equipment: a steel ball, a magnet, a glass of water, a sheet of paper, identical balls (2 or 4) on strings, a balloon, a pallet, a children's car, a glass of water and a tap.
^
Lesson plan
| Lesson stages | Time, min | Methods and forms of working with a class |
| I. Organizational stage | 2 | |
| II. Updating of basic knowledge | 5 | Frontal survey |
| III. Reporting the topic, purpose and objectives of the lesson | 2 | Determining the purpose of the lesson according to the plan for studying the topic |
| IV. Motivation learning activities | 2 | Reasoned explanation |
| V. Perception and initial understanding of new material | 20 | Teacher's explanation with elements of heuristic conversation |
| VI. Fixing new material | 10 | Self test |
| VII. Summing up the lesson and reporting homework | 4 | Teacher explanation, instruction |
^ Lesson progress
Organizational stage
Actualization and correction of basic knowledge
Questions to the class
State Newton's first law of dynamics.
Formulate Newton's second law of dynamics.
Formulate Newton's third law of dynamics.
Which system of bodies is called isolated or closed?
Reporting the topic, purpose and objectives of the lesson
Topic study plan
Force impulse.
body momentum.
Isolated telephone system. Law of conservation of momentum.
Jet propulsion. Rocket movement is like jet propulsion.
Motivation for learning activities
Like Newton's laws, conservation laws are the result of a theoretical generalization of research facts. These are the fundamental laws of physics, which are of exceptional importance, since they apply not only to mechanics,butAnd inother branches of physics.
Perception and initial understanding of new material
Under the term "impulse" (from the Latin. "impulsus "- push) in mechanics understand the impulse of force and the momentum of the body.
Question to the class. Do you think the result of the interaction depends on time or is it determined only by the strength of the interaction?
Demo 1. Place a steel ball on a horizontal surface and quickly pass a magnet over it. The ball will barely budge (Fig. 1,but). Repeat the experiment, passing the magnet slowly. The ball will move behind the magnet (Fig. 1, b).
Demo 2. Put a sheet of paper on the edge of the table and place a glass of water on it. If the sheet is pulled slowly, then the glass moves with it (Fig. 2,but), and if the sheet is pulled, it will pull out from under the glass, and the glass will remain in place (Fig. 2, b).
^ Question to the class. What do these experiences indicate?
The interaction of bodies depends not only on the force, but also on the time of its action, therefore, to characterize the action of the force, a special characteristic was introduced - the impulse of the force.
^ Power Impulse
- physical quantity, which is a measure of the action of a force over a certain time interval and numerically
equal to the product of force and time eyoactions: 
.
The SI unit is the newton second (N∙ c). The impulse of force is a vector quantity: the direction of the impulse of force coincides with the direction of the force acting on the body.
^2. Body Momentum
Imagine that a ball of mass 40 g is thrown at a speed of 5 m/s. Such a ball can be stopped by substituting a sheet of thick cardboard or a thick cloth. But if the ball is fired from a rifle at a speed of 800 m/s, then even with the help ofyox thick boards, it is almost impossible to stop him.
^ Question to the class. What conclusion can be drawn from this example?
To characterize the movement, it is not enough to know only body mass and speed. Therefore, as one of the measures mechanical movement body momentum (or momentum) is introduced.
^ Body Momentum
- a physical quantity, which is a measure of mechanical movement and is numerically determined by the product of the body mass and the speed of its movement: 
.
The SI unit is the kilogram-meter per second (kg∙m/s) . The body's momentum is a vector quantity, its direction coincides with the direction of the body's velocity.
If the body massmmoves with a speed υ, and then during the time it interacts with another body with a force F , then in the process of this interaction the body will move with acceleration a:
, 
.
The last formula demonstrates the relationship between the momentum of a force and the change in the body's momentum.
Thus, the change in the momentum of the body is equal to the momentum of the interaction force.
^ 3. Isolated system of bodies. Law of conservation of momentum
Isolated (orclosed) system of bodies - this is a system of bodies interacting only with each other and not interacting with bodies that are not included in this system.
There are no isolated systems of bodies in the full sense of the word, this is an idealization. All bodies in the world interact. But in some cases real systems can be considered as isolated, excluding from consideration those interactions that are insignificant in this case.
Demo 3. Elastic impact of two balls of the same mass suspended on threads (Fig. 3).
So, when studying the elastic impact of two identical balls, the system of balls can be considered as isolated, since at the moment of impact the gravity forces of the balls are balanced by the reaction forces of the threads, the resistance forces of the air of the balls are small, they can be neglected.
Give examples of other systems that can be considered isolated.
If we again turn to the system of balls with massesT
1
AndT
2
,
which at the initial moment of time in the chosen inertial frame of reference have velocities 
And
, then after a moment of time
t
it can be seen that their velocities as a result of the interaction have changed to 
And
.
According to Newton's second law:
Because according to Newton's third law
It can be seen from the expression obtained that the vector sum of the momenta of the bodies included in the closed system remains constant. This is the law of conservation of momentum.
^ 4. Jet propulsion. Rocket movement like jet propulsion
The law of conservation of momentum explains jet propulsion.
^ Jet propulsion - this is the movement of a body resulting from the separation of a part from it or the ejection of matter by it at a certain speed relative to the body.
Demo 4 . Inflate the balloon and then release. The ball will move due to the gases that "flow" from it.
Demo 5. Put a children's car in the tray and install a glass of water with a tap on it. If you open the tap, water will flow out of the glass, and the machine will go.
^ Assignment to the class. Give examples of jet propulsion. (Jet propulsion is carried out by aircraft flying at speeds of several thousand kilometers per hour, shells of the well-known Katyushas, space rockets. Jet propulsion is inherent, for example, in squid, cuttlefish, octopuses.)
Consider Fig. 4. Any rocket consists of a tubular body 1, closed at one end. At the second end there is a nozzle 2. Every rocket has fuel 3. When the rocket is stationary, its total momentum is zero: the fuel and body are stationary. We will assume that the rocket fuel burns instantly. Rafromred-hot gases 4 burst out under high pressure.
In this case, the rocket body moves in the direction opposite to the movement of hot gases.
Let be mG υ G is the projection of the momentum of gases on the axisOU, but m toυ to- projection of the momentum of the rocket body. According to the law of conservation of momentum, the sum of the impulses of the rocket body and the outflowing gases is equal to the total impulse of the rocket at the start, which, as you know, is equal to zero. Accordingly 0 = m r υ r + m to υ to
m to υ to = - m Gυ G
It follows that the rocket body receives the same modulus of momentum as the gases emitted from the nozzle. Consequently,
Here, the “-” sign indicates that the direction of the velocity of the rocket body is opposite to the direction of the velocity of the outgoing gases. Therefore, to move the rocket in a given direction, the jet of gases emitted by the rocket must be directed opposite to the given direction of movement. As you can see, the rocket moves without interacting with other bodies, and therefore can move in space.
^ Assignment to the class. After analyzing the last formula, answer the question: how can you increase the speed of a rocket?
Rocket speed can be increased in two ways:
increase the speed of gases flowing from the rocket nozzle;
increase the amount of fuel burned.
VI. Fixing new material
^ Self test
Mark the correct answer in your opinion.
The momentum of the body is called:
B the product of a body's mass and its speed
IN the product of the force acting on the body and the speed of the body
G the product of the force acting on the body and the time of its action
Specify the unit of momentum of the body.
Specify the unit of force impulse.
The change in momentum of the body is:
B the difference between the initial and final speed of the body
IN momentum of force
G change in body weight per unit of time
The reactive movement occurs:
B movement various parts body relative to the center of mass of the body
^B dividing the body into parts
G separation from the body of a part of its mass with a certain speed of movement relative to the rest
Determine in which reference systems the law of conservation of momentum is satisfied.
B Non-inertial D Any
Choose an example that demonstrates jet propulsion.
B pendulum swing
IN flight of the moth
G Falling leaves from the trees
The rocket rises uniformly vertically upwards. Determine how and whythe momentum of the rocket changes.
B Does not change as mass decreases and speed movement increases
IN Increasing as the rocket rises higher off the ground
G Doesn't change because the speed is constant
Specifycorrect notation of the law of conservation of momentum.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| B | IN | G | IN | G | IN | BUT | BUT | BUT |
VII. Lesson summary and message homework
The teacher sums up the lesson, evaluates the activities of students.
Homework
Learn the theoretical material from the textbook.
Describe jet propulsion as physical phenomenon according to the generalized planacting of a physical phenomenon.
Think of a demonstration of jet propulsion, describe and explain it.
When bodies interact, the momentum of one body can be partially or completely transferred to another body. If external forces from other bodies do not act on a system of bodies, then such a system is called closed.
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.
This fundamental law of nature is called law of conservation of momentum . It is a consequence of Newton's second and third laws.
Consider any two interacting bodies that are part of a closed system. The forces of interaction between these bodies will be denoted by and According to Newton's third law
If these bodies interact over time t, then the impulses of the interaction forces are identical in absolute value and directed in opposite directions:
Apply to these bodies Newton's second law:
Where and are the momenta of the bodies at the initial moment of time, and are the momenta of the bodies at the end of the interaction. From these relations it follows that as a result of the interaction of two bodies, their total momentum has not changed:
Law of conservation of momentum:
Considering now all possible pair interactions of bodies included in a closed system, we can conclude that internal forces of a closed system cannot change its total impulse, i.e., the vector sum of the impulses of all bodies included in this system.
Rice. 1.17.1 illustrates the law of conservation of momentum with an example off-center impact two balls of different masses, one of which was at rest before the collision.
Shown in fig. 1.17.1 the momentum vectors of the balls before and after the collision can be projected onto the coordinate axes OX And OY. The law of conservation of momentum is also satisfied for the projections of vectors on each axis. In particular, from the momentum diagram (Fig. 1.17.1) it follows that the projections of the vectors and momenta of both balls after the collision on the axis OY must be the same modulo and have different signs so that their sum is equal to zero.
Law of conservation of momentum in many cases makes it possible to find the velocities of interacting bodies even when the values active forces unknown. An example would be jet propulsion .
When firing from a gun, there is return- the projectile moves forward, and the gun rolls back. A projectile and a gun are two interacting bodies. The speed that the gun acquires during recoil depends only on the speed of the projectile and the mass ratio (Fig. 1.17.2). If the velocities of the gun and projectile are denoted by and and their masses by M And m, then, based on the law of conservation of momentum, it can be written in projections onto the axis OX
Based on the principle of bestowal jet propulsion. IN rocket during fuel combustion, gases heated to a high temperature are ejected from the nozzle at high speed relative to the rocket. Let us denote the mass of ejected gases through m, and the mass of the rocket after the outflow of gases through M. Then for the closed system “rocket + gases”, based on the law of conservation of momentum (by analogy with the problem of firing a gun), we can write:
where V is the speed of the rocket after the outflow of gases. In this case, it is assumed that the initial velocity of the rocket was zero.
The resulting formula for rocket speed is valid only if the entire mass of burnt fuel is ejected from the rocket simultaneously. In fact, the outflow occurs gradually during the entire time of the accelerated movement of the rocket. Each subsequent portion of gas is ejected from the rocket, which has already acquired a certain speed.
To obtain an exact formula, the process of gas outflow from a rocket nozzle must be considered in more detail. Let the rocket in time t has mass M and moves with speed (Fig. 1.17.3 (1)). For a short period of time Δ t a certain portion of gas will be ejected from the rocket with a relative velocity Rocket at the moment t + Δ t will have speed and its mass will be equal to M + Δ M, where ∆ M < 0 (рис. 1.17.3 (2)). Масса выброшенных газов будет, очевидно, равна –ΔM> 0. Velocity of gases in the inertial system OX will be equal to Apply the law of conservation of momentum. At the point in time t + Δ t the momentum of the rocket is , and the momentum of the emitted gases is 
. At the point in time t the momentum of the entire system was equal. Assuming the “rocket + gases” system to be closed, we can write:
The quantity can be neglected, since |Δ M| << M. Dividing both parts of the last relation by Δ t and passing to the limit at Δ t→0, we get:
|
Figure 1.17.3. A rocket moving in free space (without gravity). 1 - at the time t. Rocket mass M, its speed 2 - Rocket at time t + Δ t. Rocket weight M + Δ M, where ∆ M < 0, ее скорость масса выброшенных газов –ΔM> 0, relative velocity of gases velocity of gases in the inertial frame |
Value 
is the fuel consumption per unit of time. The value is called jet thrust The reactive thrust force acts on the rocket from the outgoing gases, it is directed in the direction opposite to the relative velocity. Ratio 
expresses Newton's second law for a body of variable mass. If gases are ejected from the rocket nozzle strictly backward (Fig. 1.17.3), then in scalar form this ratio takes the form:
where u is the relative velocity module. Using the mathematical operation of integration, from this relation, one can obtain formulaTsiolkovskyfor the final velocity υ of the rocket:
where is the ratio of the initial and final masses of the rocket.
It follows from it that the final speed of the rocket can exceed the relative speed of the outflow of gases. Consequently, the rocket can be accelerated to high speeds required for space flights. But this can only be achieved by consuming a significant mass of fuel, which is a large fraction of the initial mass of the rocket. For example, to achieve the first space velocity υ \u003d υ 1 \u003d 7.9 10 3 m / s at u\u003d 3 10 3 m / s (velocities of outflow of gases during fuel combustion are of the order of 2–4 km / s) starting mass single-stage rocket should be about 14 times the final weight. To reach the final speed υ = 4 u ratio should be 50.
A significant reduction in the launch mass of the rocket can be achieved by using multi-stage rockets when the rocket stages separate as the fuel burns out. Masses of containers containing fuel, spent engines, control systems, etc. are excluded from the process of subsequent rocket acceleration. It is along the path of creating economical multi-stage rockets that modern rocket science is developing.
Law of conservation of momentum
In subsection (5.8), the concept of momentum of an arbitrary body was introduced and equation (5.19) was obtained, which describes the change in momentum under the action of external forces. Since the change in momentum is due only to outside forces, then equation (5.19) is convenient to apply to describe the interactions of several bodies. In this case, interacting bodies are considered as one complex body (system of bodies). It can be shown that complex body momentum (system of bodies) is equal to the vector sum of the impulses of its parts:
p \u003d p 1 + p 2 + ... (9.13)
For a system of bodies, an equation of the form (5.13) is written without any changes:
dp = F dt.(9.14)
Change of momentum system of bodies is equal to the impulse of external forces acting on it.
Consider some examples illustrating the operation of this law.
On fig. 9.10, and the athlete is standing, leaning with her right foot on the skateboard, and with her left foot she is pushing off the ground. The speed achieved during the push depends on the force of the push and on the time during which this force acts.
On fig. 9.10, b depicts a javelin thrower. The speed that a javelin of a given mass will acquire depends on the force applied by the athlete's hand and on the time during which it is applied.
Rice. 9.10. a) An athlete on a skateboard; b) javelin thrower
Rice. 9.11.
Shot put
Therefore, before throwing the javelin, the athlete brings his hand far back. A similar process is analyzed in more detail in the example of an athlete pushing the shot, fig. 9.11.
Equation (9.14) implies one important thing for practical application consequence called the law of conservation of momentum. Consider a system of bodies that is not acted upon by external forces. Such a system is called closed.
A system of bodies that interact only with each other and do not interact with other bodies is called closed.
There are no external forces for such a system. (F= 0 and dp= 0). Therefore, there is law of conservation of momentum.
The vector sum of the impulses of the bodies, included in a closed system, remains unchanged (preserved).
In other words, for any two moments of time, the momenta of a closed system are the same:
p1=p2(9.15)
The law of conservation of momentum is a fundamental law of nature that knows no exceptions. It is absolutely strictly observed both in the macrocosm and in the microcosm.
Of course, a closed system is an abstraction, since in almost all cases there are external forces. However, for some types of interactions with a very short duration, the presence of external forces can be neglected, since with a small interval of action, the force impulse can be considered equal to zero:
F dt 0→dp 0.
Short duration processes are
Collisions of moving bodies
Disintegration of the body into parts (explosion, shot, throw).
Examples
In action films, there are often scenes in which, after being hit by a bullet, a person is thrown away in the course of the shot. On the screen, it looks pretty impressive. Let's see if this is possible? Let the mass of people M\u003d 70 kg and at the moment the bullet hits it is at rest. We take the mass of the bullet equal to t = 9 g and her speed v= 750 m/s. If we assume that after hitting a bullet, a person starts moving (in fact, this can be prevented by the friction force between the soles and the floor), then for the man-bullet system, we can write the law of conservation of momentum: p 1 = r 2. Before the bullet hits the person does not move and in accordance with (9.9) the momentum of the system p 1 \u003d m∙v+0. We will assume that the bullet gets stuck in the body. Then the final momentum of the system R 2 = (M + m)∙u, where And- the speed that a person received when a bullet hit. Substituting these expressions into the momentum conservation law, we obtain:
The result obtained shows that there can be no question of any flying off of a person by several meters (by the way, a body thrown upwards at a speed of 0.1 m / s will rise to a height of only 0.5 mm!).
2) Clash of hockey players.
Two hockey players M 1 And M 2 move towards each other with speeds, respectively, v1, v2(Fig. 9.12). Determine the total speed of their movement, counting the collision absolutely inelastic(with absolutely inelastic impact bodies "cling" and move further as a whole).
Rice. 9.12. Absolutely inelastic collision of hockey players
We apply the law of conservation of momentum to a system consisting of two hockey players. Momentum of the system before the collision p 1 \u003d M 1 ∙v 1- M 2 v 2. There is a “-” sign in this formula because the speeds v1 And v2 directed towards each other. Speed direction v1 considered positive, and the direction of velocity v2- negative. After an inelastic collision, the bodies move with a common velocity v and momentum of the system p 2 \u003d (M l + M 2) ∙ v. We write down the law of conservation of momentum and find the speed v:
Speed direction v determined by its sign.
Let us pay attention to one important circumstance: the law of conservation of momentum can only be applied to free bodies. If the motion of one of the bodies is limited by external constraints, then the total momentum will not be conserved.
Jet propulsion
Jet propulsion is based on the use of the law of conservation of momentum. This is the name of the movement of the body that occurs when a part of it separates from the body at some speed. Consider rocket propulsion. Let the rocket and its mass along with the fuel M rests. The initial momentum of the rocket with fuel is zero. During the combustion of a portion of the fuel mass T gases are formed, which are ejected through the nozzle with a speed u. According to the law of conservation of momentum, the total momentum of the rocket and fuel saved: p 2 = p 1→ m∙u +(M - m)∙v = 0, where v- the speed obtained by the rocket. From this equation we find: v = ─t∙u /(M ─ t). We see that the rocket acquires a speed directed in the direction opposite to the direction of the gas ejection. As the fuel burns, the speed of the rocket continuously increases.
An example of jet propulsion is the recoil when fired from a rifle. Let the rifle, the mass of which m 1 = 4.5 kg, shoots a bullet with a mass t 2 = 11g flying at speed v 1 = 800 m/s. From the law of conservation of momentum, the recoil velocity can be calculated:
Such a significant recoil rate will occur if the rifle is not pressed to the shoulder. In this case, the shooter will receive a strong blow with the butt. With the correct shooting technique, the shooter presses the rifle to the shoulder and the recoil is perceived by the entire body of the shooter. With an arrow mass of 70 kg, the recoil velocity in this case will be equal to 11.8 cm / s, which is quite acceptable.
BODY MOMENTUM IS A vector quantity, equal PRODUCT BODY MASS AT ITS SPEED:
The unit of momentum in the SI system is the momentum of a body with a mass of 1 kg moving at a speed of 1 m/s. This unit is called KILOGRAM-METER PER SECOND (kg . m/s).
A SYSTEM OF BODIES THAT DO NOT INTERACT WITH OTHER BODIES NOT INCLUDED IN THIS SYSTEM IS CALLED CLOSED.
In a closed system of bodies, the momentum obeys the conservation law.
IN A CLOSED SYSTEM OF BODIES THE GEOMETRIC SUM OF IMPULSES OF THE BODIES REMAINS CONSTANT FOR ANY INTERACTIONS OF THE BODIES OF THIS SYSTEM BETWEEN THEM.
Reactive motion is based on the law of conservation of momentum. During the combustion of fuel, gases heated to a high temperature are ejected from the rocket nozzle at a certain speed. At the same time, they interact with the rocket. If, before the engine starts, the sum of impulses
|
|
where M is the mass of the rocket; V is the speed of the rocket;
m is the mass of ejected gases; v is the speed of the outflow of gases.
From here we get the expression for the speed of the rocket:
main feature jet engine is that for movement it does not need a medium with which it can interact. Therefore, a rocket is the only vehicle capable of moving in a vacuum.
The great Russian scientist and inventor Konstantin Eduardovich Tsiolkovsky proved the possibility of using rockets for space exploration. He developed a scheme for the rocket device, found the necessary fuel components. The works of Tsiolkovsky served as the basis for the creation of the first spacecraft.
First in the world artificial satellite Earth was launched in our country on October 4, 1957, and on April 12, 1961, Yuri Alekseevich Gagarin became the first cosmonaut of the Earth. Currently spacecraft explore other planets solar system, comets, asteroids. American astronauts have landed on the moon, and a manned flight to Mars is being prepared. Scientific expeditions have been working in orbit for a long time. Developed spaceships reusable "Shuttle" and "Challenger" (USA), "Buran" (Russia), work is underway to create a scientific station "Alpha" in Earth's orbit, where scientists from different countries will work together.
Jet propulsion is also used by some living organisms. For example, squids and octopuses move by throwing a jet of water in the opposite direction of movement.
4/2. Experimental task on the topic "Molecular physics": observation of changes in air pressure with changes in temperature and volume.
Connect the corrugated cylinder to the pressure gauge, measure the pressure inside the cylinder.
Place the cylinder in a container of hot water. What's happening?
Compress the cylinder. What's happening?
