What speed should the moon's satellite have? Lesson on solving problems on the movement of artificial satellites. Three cosmic speeds

Subject: Artificial Earth satellites. First escape velocity.

The purpose of the lesson: Consider the trajectory of a body in a gravitational field, calculate 1, 2 and 3 escape velocity. Define a stationary satellite of the Earth.

Tasks:

Educational: know the conditions under which a body can become an artificial satellite of the Earth; be able to calculate the first, as well as the second and third escape velocity.

Developmental:develop speech, thinking, the ability to identify essential features, and find interdisciplinary connections.

Educational:to develop respect for the work of scientists and pride in Russian inventors of space technology.

Lesson type: lesson on learning new material

Lesson type: lesson lecture

Teaching methods: reproductive, explanatory and illustrative

Equipment : textbooks, blackboard, chalk.

Lesson plan:

Organizing time

Updating of reference knowledge

Presentation of new material

Problem solving

Homework

During the classes

Structural element of the lesson

Activities of teacher and students

Organizing time

Greeting, checking absentees. Getting students in the mood for work.

Updating of reference knowledge

In the last lesson, we formulated the law of universal gravitation.

One of the students is called to the board and writes a formula for the law of universal gravitation. Question for the class: Formulate the law of universal gravitation.

Learning new material

Under the influence of gravitational forceGravity causes the rotation of the Earth around the Sun and the movement of the planets' satellites.

Let's try to figure out what speeds the satellites move at.

Let the body be at some heightH, the force of gravity acts on it from the side of the Earth, directed towards the center of the Earth. If the initial velocity is zero, then the body freely falls to the Earth in a straight line, along the force of gravity.In the presence of a horizontal component, the body moves almost along a parabolic trajectory.

Starting at some speedthe body moves away so quickly that it does not fall to Earth. And it becomes an artificial satellite of the Earth, and moves around it alongcircular orbit - this speed is called the first cosmic speed. |
If a body is launched in a circular orbit from the Earth’s surface (H = 0), then

First cosmicThe actual speed is = 7.9 km/s, if g ≈9.8 m/ With 2 , AR ≈6, 4 * 10 6 m. A body whose speed is 7.9 km/s and is directed horizontally relative to the Earth’s surface becomes an artificial satellite,

moving in a circular orbit at a low altitude above the Earth

If the speed of the body ishigher than the first cosmic one, then the Earth’s gravitational force will hold it, but the satellite will move in an elliptical orbit. With a further increase in launch speed, the body moves further and further from the Earth, while the elliptical orbit becomes significantly elongated.

N
Finally, there will be a speed at which the body is capable of tearing outXiainto outer space, overcoming the gravity of the Earth, i.e. it will move awayFROMEarth to infinity long distance. (The trajectory is parabolic.).

At speed V 0 > V III ≈ 16.7 km/s – the body leaves the limits Solar System. This speed is called the third escape velocity. A stationary artificial Earth satellite is a satellite that is constantly located above the same point on the equator. In order for a satellite to “hover” over a given point on the equator, it must have the same period of revolution as the Earth, i.e. 24 hours

There are two significant dates for our country. October 4, 1957 The first satellite was launched in the Soviet Union. The satellite looked like a ball with a diameter of 58 cm and a mass of 83.6 kg. April 12, 1961, the world's first cosmonaut, our compatriot Yuri Alekseevich Gagarin committed on the satellite ship "Vostok".

Problem solving

Solve consolidation problems.

Task No. 1

Calculate the first escape velocity for the Moon if the radius of the Moon is 1700 km, and the acceleration of free fall of bodies on the Moon is 1.6 m/s 2 .

Problem No. 2

What speed must an artificial satellite have to orbit in a circular orbit at an altitude of 600 km above the Earth's surface?

Problem No. 3

What is the speed of an artificial satellite moving at an altitude of 300 km above the Earth's surface?

Setting homework

New textbook by G.Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky § 32

Old textbook by G.Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky § 34

Lesson Objectives:

educational:

Formation of skills to independently obtain knowledge;

Formation of skills for accurate and error-free calculation of the first and second cosmic velocities of the Earth and other planets, acceleration of free fall.

Formation of skills and abilities to find rational ways to solve problems for calculating the period of revolution of planets, the density of planets;

Formation of skills to apply the necessary formulas;

developing:

Development of independent work skills;

Practicing methods for solving problems;

Develop the ability to think logically;

Develop the ability to draw conclusions when solving problems;

educational:

Formation of critical assessment of results;

Fostering a sense of pride in one’s homeland.

Lesson type: Lesson on applying knowledge, skills and abilities.

Equipment: computer, multimedia console, disk with a physics training program on the topic: “Mechanics”, student presentations, assessment form, assignment sheets.

Lesson plan:

1. Organizational moment.

3. Updating the basic knowledge necessary for the formation of skills.

4. Consolidation of primary skills and abilities

5. Exercises in applying knowledge and skills in changed conditions

6. Creative application of knowledge and skills.

7. Lesson summary.

8. Homework.

During the classes

1. Organizational moment.

2. Statement of the topic of the lesson and its objectives.

On the screen is a video fragment of the launch of the first ARTIFICIAL EARTH SATELLITE

Now he has already become invisible.
Having overcome the force of gravity...
A satellite disappears in a gray haze
And the Earth signals in a singsong voice,
In the midnight starry sky
He will float like a new star,
To get another magical
There is a “golden key” from the Universe.
M. Romanova

3. Updating of basic knowledge.

1) Frontal.

• What needs to be done for the body to become an artificial satellite? (Tell the body the speed with which you can overcome the force of gravity);
• Why do satellites, orbiting around the Earth under the influence of gravity, not fall to Earth? (Because they have a fairly high speed, directed tangentially to the circle along which it moves)
• Can the motion of a satellite around the Earth be considered a free fall? (Yes, it is possible, because the centripetal acceleration when the satellite moves around the Earth is equal to the acceleration of gravity);
• What is the direction of the velocity vector when moving around a circle? (Tangential to the circle);
• What is the direction of acceleration of a body moving in a circle? (Towards the center of the circle);
• Let us arrange the values ​​of the speeds in accordance with the trajectory of the body's movement

7.9 km/s; circle

More than 7.9 km/s; ellipse

11.2 km/s; parabola

More than 11.2 km/s. hyperbola

• Let us repeat the units of measurement of the following physical quantities, building a correspondence between physical quantities and their units of measurement:

Weight; - newton;

Force; - meter;

Acceleration; - meter per second;

Density; - kilogram;

Volume; - meter per second squared;

Speed; - cubic meter;

• Let's remember the mathematical formulas:

2) Checking homework.

Now let's check how you learned output 1 of escape velocity.

If desired, go to the board and write the conclusion of the first cosmic velocity for the Earth (the children write the conclusion of the cosmic velocity on the wings of the boards on the reverse side).

3) Task on the correspondence of formulas and their names.

While the guys are working at the board, we will do work on knowledge of formulas.

1 option

1) F T = m g A) formula for the first cosmic velocity;

2) T = B) formula for centripetal acceleration;

3) F = B) formula for calculating gravity;

4) a c = G) formula for the force of universal gravitation;

5) D) formula for calculating the period when moving in a circle.

Option 2

1) A) Acceleration of free fall;

2) B) formula for the density of matter;

3) B) formula for the volume of a sphere;

4) D) formula for escape velocity at altitude above the Earth;

5) D) formula for linear speed when moving in a circle.

We will check the work mutual verification with your desk neighbor.

4. Formation, consolidation of primary skills and abilities and their application in standard situations - by analogy.

Imagine that your spaceships landed on the planets of the solar system: Mercury, Venus, Mars, Jupiter. What speeds must your ships have to overcome the gravity of the planets?

Your task is to calculate the first escape velocity and the free fall acceleration of the planet on which you are located. The crew of the 1st row starts from Mercury, the second row - from Venus, and the third - from Mars. We take the data for calculating speeds and acceleration from the table, write the answers in the table, and solve the problem in a notebook.

You have 5 minutes to decide. Those interested can work at the board and find the acceleration of gravity and the first escape velocity of Jupiter

	Weight, kg	Radius, km
Mercury

So, we finished the solution and entered the answers into the table. What are we observing?

What determine the acceleration of free fall and the first cosmic velocities? (The greater the mass of the planet, the greater the acceleration of gravity and the first escape velocity)

5. Exercises in applying knowledge and skills in changed conditions.

Now let’s calculate the acceleration of gravity and the first escape velocity at different altitudes.

The first row calculates for a height equal to the radius of the Earth;

The second row is for a height equal to two radii of the Earth;

The third row is for a height equal to three radii of the Earth;

We put the results in a table, solve them in a notebook, and divide the work in pairs yourself.

h height in R z
First escape velocity, km/s
Gravity acceleration, m/s 2

After solving and recording the results, we determine how the acceleration of gravity and the first escape velocity change.

We solve more complex problems.

Let's look at the slide from the multimedia educational disc "Mechanics".

6. Creative application of knowledge and skills.

Differentiated problem solving.

Option #1

First level

1. An artificial satellite moves around the Earth in a circular orbit. Choose the correct statement.

A. The satellite moves with constant acceleration.

B. The satellite's speed is corrected to the center of the Earth.

B. The satellite attracts the Earth with less force than the Earth attracts the satellite.

2. Calculate the acceleration of gravity at a height equal to two Earth radii.

A. 1.1 m/s 2 . B. 5 m/s 2 . V. 4.4 m/s 2 .

3. What keeps the artificial Earth satellite in orbit?

Enough level

1. The Moon moves around the Earth in a circular orbit at a speed of 1 km/s, with an orbital radius of 384,000 km. What is the mass of the Earth?
2. Can a satellite orbit the Earth in a circular orbit at a speed of 1 km/s? Under what conditions is this possible?

High level

1. The spacecraft entered a circular orbit with a radius of 10 million km around the star it discovered. What is the mass of the star if the spacecraft's orbital period is 628,000 s?
2. The satellite orbits in a circular orbit at a low altitude above the planet. Satellite orbital period 6 hours Assuming the planet to be a homogeneous sphere, find its density.

Option No. 2

First level

1. What will happen to an artificial Earth satellite if it is launched into orbit at a speed slightly less than the first escape velocity? Choose the correct statement.

A. Will return to Earth.

B. Will move in a more distant orbit.

B. It will move towards the Sun.

2. What is the acceleration of gravity at a height equal to half the radius of the Earth? The radius of the Earth is taken to be 6400 km.

A. 4.4. m/s 2 V. 9.8 m/s 2 . V. 16.4 m/s 2 .

3. Why are artificial earth satellites launched from the earth in the direction of the east?

Enough level

1. What speed must an artificial satellite of the Moon have in order for it to revolve around it in a circular orbit at an altitude of 40 km? The gravitational acceleration for the Moon at this altitude is 1.6 m/s2, and the radius of the Moon is 1.760 km.
2. Determine the acceleration of free fall of a body at an altitude of 600 km above the Earth's surface. The radius of the Earth is 6400 km.

High level

1. The orbital period of the satellite is 1 hour 40 minutes 47 seconds. At what altitude above the Earth's surface is the satellite moving? The radius of the Earth is R = 6400 km, the mass of the Earth is M = 6 10 24 kg.
2. An artificial satellite orbits the Earth at a speed of 6 km/s. After the maneuver, it moves in another orbit at a speed of 5 km/s. How many times did the orbital radius and orbital period change as a result of the maneuver?

7. Lesson summary.

Summing up the lesson.

Students give grades for their work in the lesson in the following table:

Job title	Grade (average score)
solving a formula matching task
problem solving in pairs
output of the first escape velocity.
solving problems at the board
solving differentiated problems
oral responses

8. Homework.

	Weight, kg	Radius, km	Gravity acceleration, m/s 2	First escape velocity, km/s
Neptune

In space, gravity provides the force that causes satellites (such as the Moon) to orbit larger bodies (such as the Earth). These orbits generally have the shape of an ellipse, but most often this ellipse is not very different from a circle. Therefore, to a first approximation, the orbits of satellites can be considered circular. Knowing the mass of the planet and the height of the satellite’s orbit above the Earth, we can calculate what it should be speed of the satellite around the Earth.

Calculation of the speed of a satellite around the Earth

Rotating in a circular orbit around the Earth, a satellite at any point in its trajectory can only move at a constant absolute speed, although the direction of this speed will constantly change. What is the magnitude of this speed? It can be calculated using Newton's second law and the law of gravity.

To maintain a circular orbit of a mass satellite in accordance with Newton's second law, a centripetal force will be required: , where is the centripetal acceleration.

As is known, centripetal acceleration is determined by the formula:

where is the speed of the satellite, is the radius of the circular orbit along which the satellite moves.

Centripetal force is provided by gravity, therefore, in accordance with the law of gravity:

where kg is the mass of the Earth, m 3 ⋅kg -1 ⋅s -2 is the gravitational constant.

Substituting everything into the original formula, we get:

Expressing the required speed, we find that the speed of the satellite around the Earth is equal to:

This is a formula for the speed that an Earth satellite must have at a given radius (i.e. distance from the center of the planet) to maintain a circular orbit. The speed cannot change in magnitude as long as the satellite maintains a constant orbital radius, that is, as long as it continues to orbit the planet in a circular path.

When using the resulting formula, there are several details to consider:

Artificial satellites of the Earth, as a rule, orbit the planet at an altitude of 500 to 2000 km from the surface of the planet. Let's calculate how fast such a satellite should move at an altitude of 1000 km above the Earth's surface. In this case km. Substituting the numbers, we get:

Material prepared by Sergei Valerievich