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
1. Bodies 1 and 2 move uniformly in circles with radii of 60 and 40 cm, respectively. The acceleration of which body is greater and by how many times, if: a) the velocities of the bodies are the same; b) the circulation periods are the same?
2. A satellite moves in a circular orbit at an altitude of 400 km around a planet with a radius of 5000 km. What is the speed and acceleration of the satellite if its orbital period is 81 minutes?
3. The satellite moves in a circular orbit at an altitude of 600 km, its period of revolution around the Earth is 97.5 minutes. Determine the speed and acceleration of the satellite. Assume that the radius of the Earth is 6400 km.
4. Determine the average orbital speed of the satellite if the average altitude of its orbit above the Earth is 1200 km and the rotation period is 105 minutes. The radius of the Earth is 6400 km.
5. An artificial Earth satellite moves in a circular orbit at a speed of 8 km/s and with a period of 96 minutes. Determine the satellite's flight altitude above the Earth's surface if the Earth's radius is 6400 km.
6. The world's first orbital space station moved at a speed of 7.8 km/s, and its orbital period was 88.85 minutes. Assuming its orbit is circular, find the altitude of the station's orbit above the Earth's surface. Consider the radius of the Earth to be 6400 km.
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 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 guys write the conclusion of the cosmic velocity on the wings of the boards on the back 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 gravity;
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 should 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
- 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?
- 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
- 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 ship's orbital period is 628,000 s?
- 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. It 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
- 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.
- 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
- 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.
- 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 « Physics - 10th grade"
To solve problems, you need to know the law of universal gravitation, Newton's law, as well as the relationship between the linear speed of bodies and the period of their revolution around the planets. Please note that the radius of the satellite's trajectory is always measured from the center of the planet.
Task 1.
Calculate the first escape velocity for the Sun. The mass of the Sun is 2 10 30 kg, the diameter of the Sun is 1.4 10 9 m.
Solution.
The satellite moves around the Sun under the influence of a single force - gravity. According to Newton's second law, we write:
From this equation we determine the first escape velocity, i.e. the minimum speed with which a body must be launched from the surface of the Sun in order for it to become its satellite:
Task 2.
A satellite is moving around a planet at a distance of 200 km from its surface at a speed of 4 km/s. Determine the density of the planet if its radius is equal to two radii of the Earth (Rpl = 2R 3).
Solution.
Planets have the shape of a ball, the volume of which can be calculated using the formula then the density of the planet
Determine the average distance from Saturn to the Sun if the period of Saturn's revolution around the Sun is 29.5 years. The mass of the Sun is 2 10 30 kg.
Solution.
We believe that Saturn moves around the Sun in a circular orbit. Then, according to Newton’s second law, we write:
where m is the mass of Saturn, r is the distance from Saturn to the Sun, M c is the mass of the Sun.
Saturn's orbital period from here
Substituting the expression for speed υ into equation (4), we obtain
From the last equation we determine the required distance from Saturn to the Sun:
Comparing with the tabular data, we will make sure that the found value is correct.
Source: “Physics - 10th grade”, 2014, textbook Myakishev, Bukhovtsev, Sotsky
Dynamics - Physics, textbook for grade 10 - Cool physics
Task No. 1. ⋅ 10 23 kg, its radius is 3300 km.Problem No. 2
Task No. 3 2 ?
Problem No. 4
Problem No. 5
Problem No. 6
An approximate version of the test on the topic “The Law of Universal Gravitation. Movement of a body in a circle. Artificial Earth satellites"
Task No. 1. Calculate the acceleration of free fall of bodies near the surface of Mars. The mass of Mars is 6⋅ 10 23 kg, its radius is 3300 km.Problem No. 2 . Determine the speed of the satellite moving around the Earth in a circular orbit at an altitude equal to two radii of the Earth, if the first escape velocity at the Earth’s surface is 8 km/s.
Task No. 3 . What distance does a body travel along an arc of a circle with a radius of 3 m in 2.5 s if its centripetal acceleration is 12 cm/s 2 ?
Problem No. 4 . In the spacecraft, instruments noted a decrease in the acceleration of free fall by 3 times. How far did the spacecraft move from the surface of the Earth?
Problem No. 5 . Determine the mass of the Sun if the speed of the Earth's revolution in a circular orbit around the Sun is 30 km/s, and the radius of the Earth's orbit is 1.5 million km.
Problem No. 6 . How far does a freely falling body travel in the fifth second of its motion?
An approximate version of the test on the topic “The Law of Universal Gravitation. Movement of a body in a circle. Artificial Earth satellites"
Task No. 1. Calculate the acceleration of free fall of bodies near the surface of Mars. The mass of Mars is 6⋅ 10 23 kg, its radius is 3300 km.Problem No. 2 . Determine the speed of the satellite moving around the Earth in a circular orbit at an altitude equal to two radii of the Earth, if the first escape velocity at the Earth’s surface is 8 km/s.
Task No. 3 . What distance does a body travel along an arc of a circle with a radius of 3 m in 2.5 s if its centripetal acceleration is 12 cm/s 2 ?
Problem No. 4 . In the spacecraft, instruments noted a decrease in the acceleration of free fall by 3 times. How far did the spacecraft move from the surface of the Earth?
Problem No. 5 . Determine the mass of the Sun if the speed of the Earth's revolution in a circular orbit around the Sun is 30 km/s, and the radius of the Earth's orbit is 1.5 million km.
Problem No. 6 . How far does a freely falling body travel in the fifth second of its motion?
An approximate version of the test on the topic “The Law of Universal Gravitation. Movement of a body in a circle. Artificial Earth satellites"
Task No. 1. Calculate the acceleration of free fall of bodies near the surface of Mars. The mass of Mars is 6⋅ 10 23 kg, its radius is 3300 km.Problem No. 2 . Determine the speed of the satellite moving around the Earth in a circular orbit at an altitude equal to two radii of the Earth, if the first escape velocity at the Earth’s surface is 8 km/s.
Task No. 3 . What distance does a body travel along an arc of a circle with a radius of 3 m in 2.5 s if its centripetal acceleration is 12 cm/s 2 ?
Problem No. 4 . In the spacecraft, instruments noted a decrease in the acceleration of free fall by 3 times. How far did the spacecraft move from the surface of the Earth?
Problem No. 5 . Determine the mass of the Sun if the speed of the Earth's revolution in a circular orbit around the Sun is 30 km/s, and the radius of the Earth's orbit is 1.5 million km.
Problem No. 6 . How far does a freely falling body travel in the fifth second of its motion?
