“Uniform and uneven movement” - t 2. Uneven movement. Yablonevka. L 1. Uniform and. L2. t 1. L3. Chistoozernoe. t 3. Uniform movement. =.
“Curvilinear motion” - Centripetal acceleration. UNIFORM MOTION OF A BODY AROUND A CIRCLE There are: - curvilinear motion with a constant velocity; - movement with acceleration, because speed changes direction. Direction of centripetal acceleration and velocity. Motion of a point in a circle. Movement of a body in a circle with a constant absolute speed.
“Motion of bodies on a plane” - Evaluate the obtained values of unknown quantities. Substitute numerical data into a general solution and perform calculations. Make a drawing, depicting interacting bodies on it. Perform an analysis of the interaction of bodies. Ftr. Movement of a body along an inclined plane without friction. Study of the movement of a body on an inclined plane.
“Support and movement” - An ambulance brought a patient to us. Slender, stooped, strong, strong, fat, clumsy, dexterous, pale. Game situation “Concilium of doctors”. Sleep on a hard bed with a low pillow. “Body support and movement. Rules for maintaining correct posture. Correct posture when standing. Children's bones are soft and elastic.
"Space Speed" - V1. THE USSR. That's why. April 12, 1961 Message to extraterrestrial civilizations. Third escape velocity. On board Voyager 2 is a disk with scientific information. Calculation of the first escape velocity at the Earth's surface. The first manned flight into space. Voyager 1 trajectory. The trajectory of bodies moving at low speed.
“Body dynamics” - What underlies dynamics? Dynamics is a branch of mechanics that examines the causes of the movement of bodies (material points). Newton's laws apply only to inertial frames of reference. Frames of reference in which Newton's first law is satisfied are called inertial. Dynamics. In what frames of reference do Newton's laws apply?
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This is the minimum speed at which a body moving horizontally above the surface of the planet will not fall onto it, but will move in a circular orbit.
Useful information about escape velocity:
If at the moment of entering orbit the spacecraft has a speed equal to First cosmic speed, perpendicular to the direction of the center of the Earth, then its orbit (in the absence of any other forces) will be circular. When the speed of the vehicle is equal to less than , its orbit has the shape of an ellipse, and the point of entry into orbit is located at the apogee. If this point is at an altitude of about 160 km, then immediately after entering orbit the satellite enters the underlying dense layers of the atmosphere and burns up. That is, for the specified height first Cosmic speeds is the minimum for a spacecraft to become a satellite of the Earth. At high altitudes, a spacecraft can become a satellite and at a speed somewhat lower First Space Speed, calculated for this height. So, at an altitude of 300 km, it is enough for a spacecraft to have a speed 45 m/sec less than First Space Speed
There is also:
Second escape velocity: 
In the formula we used:
Gravitational constant
Any object, being thrown up, sooner or later ends up on the earth's surface, be it a stone, a sheet of paper or a simple feather. At the same time, a satellite launched into space half a century ago, a space station or the Moon continue to rotate in their orbits, as if they were not affected by our planet at all. Why is this happening? Why is the Moon not in danger of falling to the Earth, and why is the Earth not moving towards the Sun? Are they really not affected by universal gravity?
From the school physics course we know that universal gravity affects any material body. Then it would be logical to assume that there is some force that neutralizes the effect of gravity. This force is usually called centrifugal. Its effect can be easily felt by tying a small weight to one end of the thread and untwisting it in a circle. Moreover, the higher the rotation speed, the stronger the tension of the thread, and the slower we rotate the load, the greater the likelihood that it will fall down.
Thus, we are very close to the concept of “cosmic velocity”. In a nutshell, it can be described as the speed that allows any object to overcome the gravity of a celestial body. The role can be a planet, its or another system. Every object that moves in orbit has escape velocity. By the way, the size and shape of the orbit depend on the magnitude and direction of the speed that the given object received at the time the engines were turned off, and the altitude at which this event occurred.
There are four types of escape velocity. The smallest of them is the first. This is the lowest speed it must have for it to enter a circular orbit. Its value can be determined by the following formula:
V1=õ/r, where
µ - geocentric gravitational constant (µ = 398603 * 10(9) m3/s2);
r is the distance from the launch point to the center of the Earth.
Due to the fact that the shape of our planet is not a perfect sphere (at the poles it seems to be slightly flattened), the distance from the center to the surface is greatest at the equator - 6378.1. 10(3) m, and the least at the poles - 6356.8. 10(3) m. If we take the average value - 6371. 10(3) m, then we get V1 equal to 7.91 km/s.
The more the cosmic velocity exceeds this value, the more elongated the orbit will acquire, moving away from the Earth to an ever greater distance. At some point, this orbit will break, take the shape of a parabola, and the spacecraft will set off to plow the expanses of space. In order to leave the planet, the ship must have a second escape velocity. It can be calculated using the formula V2=√2µ/r. For our planet, this value is 11.2 km/s.
Astronomers have long determined what the escape velocity is, both the first and the second, for each planet of our home system. They can be easily calculated using the above formulas if you replace the constant µ with the product fM, in which M is the mass of the celestial body of interest, and f is the gravitational constant (f = 6.673 x 10(-11) m3/(kg x s2).
The third cosmic speed will allow anyone to overcome the gravity of the Sun and leave their native solar system. If you calculate it relative to the Sun, you get a value of 42.1 km/s. And in order to enter solar orbit from Earth, you will need to accelerate to 16.6 km/s.
And finally, the fourth escape velocity. With its help, you can overcome the gravity of the galaxy itself. Its magnitude varies depending on the coordinates of the galaxy. For ours, this value is approximately 550 km/s (if calculated relative to the Sun).
Details Category: Man and Sky Published 07/11/2014 12:37 Views: 9512Humanity has long been striving for space. But how to break away from the Earth? What prevented man from flying to the stars?
As we already know, this was prevented by gravity, or the gravitational force of the Earth - the main obstacle to space flights.
Earth gravity
All physical bodies located on Earth are subject to the action law of universal gravitation . According to this law, they all attract each other, that is, they act on each other with a force called gravitational force, or gravity .
The magnitude of this force is directly proportional to the product of the masses of the bodies and inversely proportional to the square of the distance between them.
Since the mass of the Earth is very large and significantly exceeds the mass of any material body located on its surface, the gravitational force of the Earth is significantly greater than the gravitational force of all other bodies. We can say that compared to the gravitational force of the Earth they are generally invisible.
The earth attracts absolutely everything to itself. Whatever object we throw upward, under the influence of gravity it will definitely return to Earth. Drops of rain fall down, water flows from the mountains, leaves fall from the trees. Any item we drop also falls to the floor, not the ceiling.
The main obstacle to space flights
Earth's gravity prevents aircraft from leaving the Earth. And it is not easy to overcome it. But man learned to do it.
Let's observe the ball lying on the table. If he rolls off the table, the gravity of the Earth will cause him to fall to the floor. But if we take the ball and forcefully throw it into the distance, it will not fall immediately, but after some time, describing a trajectory in the air. Why was he able to overcome gravity at least for a short time?
And this is what happened. We applied a force to it, thereby imparting acceleration, and the ball began to move. And the more acceleration the ball receives, the higher its speed will be and the further and higher it can fly.
Let us imagine a cannon mounted on the top of a mountain, from which projectile A is fired at high speed. Such a projectile is capable of flying several kilometers. But in the end, the projectile will still fall to the ground. Its trajectory under the influence of gravity has a curved appearance. Projectile B leaves the cannon at higher speed. Its flight path is more elongated, and it will land much further. The more speed a projectile receives, the straighter its trajectory becomes and the greater the distance it travels. And finally, at a certain speed, the trajectory of projectile C takes the shape of a closed circle. The projectile makes one circle around the Earth, another, a third and no longer falls on the Earth. It becomes an artificial satellite of the Earth.
Of course, no one sends cannon shells into space. But spacecraft that have reached a certain speed become Earth satellites.
First escape velocity
What speed must a spacecraft achieve to overcome gravity?
The minimum speed that must be imparted to an object in order to put it into a near-Earth circular (geocentric) orbit is called first escape velocity .
Let's calculate the value of this speed relative to the Earth.
A body in orbit is acted upon by a gravitational force directed toward the center of the Earth. It is also a centripetal force trying to attract this body to the Earth. But the body does not fall to the Earth, since the action of this force is balanced by another force - centrifugal, which tries to push it out. Equating the formulas of these forces, we calculate the first escape velocity.
Where m – mass of the object in orbit;
M – mass of the Earth;
v 1 – first escape velocity;
R – radius of the Earth
G – gravitational constant.
M = 5.97 10 24 kg, R = 6,371 km. Hence, v 1 ≈ 7.9 km/s
The value of the first earth's cosmic velocity depends on the radius and mass of the Earth and does not depend on the mass of the body being launched into orbit.
Using this formula, you can calculate the first cosmic velocities for any other planet. Of course, they differ from the first escape velocity of the Earth, since celestial bodies have different radii and masses. For example, the first escape velocity for the Moon is 1680 km/s.
An artificial Earth satellite is launched into orbit by a space rocket that accelerates to the first cosmic velocity and higher and overcomes gravity.
Beginning of the space age
The first cosmic speed was achieved in the USSR on October 4, 1957. On this day, earthlings heard the call sign of the first artificial Earth satellite. It was launched into orbit using a space rocket created in the USSR. It was a metal ball with antennae, weighing only 83.6 kg. And the rocket itself had enormous power for that time. After all, in order to launch just 1 additional kilogram of weight into orbit, the weight of the rocket itself had to increase by 250-300 kg. But improvements in rocket designs, engines and control systems soon made it possible to send much heavier spacecraft into Earth orbit.
The second space satellite, launched in the USSR on November 3, 1957, already weighed 500 kg. On board there was complex scientific equipment and the first living creature - the dog Laika.
The space age began in human history.
Second escape velocity
Under the influence of gravity, the satellite will move horizontally above the planet in a circular orbit. It will not fall to the surface of the Earth, but it will not move to another, higher orbit. And for him to be able to do this, he needs to be given a different speed, which is called second escape velocity . This speed is called parabolic, escape speed , release speed . Having received such a speed, the body will cease to be a satellite of the Earth, will leave its surroundings and become a satellite of the Sun.
If the speed of a body when starting from the Earth's surface is higher than the first escape velocity, but lower than the second, its near-Earth orbit will have the shape of an ellipse. And the body itself will remain in low-Earth orbit.
A body that has received a speed equal to the second escape velocity when starting from the Earth will move along a trajectory shaped like a parabola. But if this speed even slightly exceeds the value of the second escape velocity, its trajectory will become a hyperbola.
The second escape velocity, like the first, has different meanings for different celestial bodies, since it depends on the mass and radius of this body.
It is calculated by the formula:
The relationship between the first and second escape velocity remains
For the Earth, the second escape velocity is 11.2 km/s.
The first rocket to overcome gravity was launched on January 2, 1959 in the USSR. After 34 hours of flight, she crossed the orbit of the Moon and entered interplanetary space.
The second space rocket towards the Moon was launched on September 12, 1959. Then there were rockets that reached the surface of the Moon and even made a soft landing.
Subsequently, spacecraft went to other planets.
02.12.2014
Lesson 22 (10th grade)
Subject. Artificial Earth satellites. Development of astronautics.
On the motion of thrown bodies
In 1638, Galileo’s book “Conversations and Mathematical Proofs Concerning Two New Branches of Science” was published in Leiden. The fourth chapter of this book was called “On the motion of thrown bodies.” Not without difficulty, he managed to convince people that in airless space “a grain of lead should fall as fast as a cannonball.” But when Galileo told the world that a cannonball fired horizontally from a cannon was in flight for the same amount of time as a cannonball that simply fell from its mouth to the ground, they did not believe him. Meanwhile, this is really true: a body thrown from a certain height in a horizontal direction moves to the ground in the same time as if it had simply fallen vertically down from the same height.
To verify this, we will use a device, the principle of operation of which is illustrated in Figure 104, a. After being hit with a hammer M on an elastic plate P the balls begin to fall and, despite the difference in trajectories, simultaneously reach the ground. Figure 104, b shows a stroboscopic photograph of falling balls. To obtain this photograph, the experiment was carried out in the dark, and the balls were illuminated with a bright flash of light at regular intervals. At the same time, the camera shutter was open until the balls fell to the ground. We see that at the same moments in time when the flashes of light occurred, both balls were at the same height and they reached the ground at the same time.
Free fall time from height h(near the surface of the Earth) can be found using the formula known from mechanics s=аt2/2. Replacing here s on h And A on g, we rewrite this formula in the form
from where, after simple transformations, we get
A body thrown from the same height in a horizontal direction will spend the same time in flight. In this case, according to Galileo, “the uniform unhindered movement is joined by another, caused by the force of gravity, due to which a complex movement arises, composed of uniform horizontal and naturally accelerated movements.”
During the time determined by expression (44.1), moving in the horizontal direction with speed v0(i.e., with the speed with which it was thrown), the body will move horizontally a distance
From this formula it follows that the flight range of a body thrown in a horizontal direction is proportional to the initial speed of the body and increases with increasing height of the throw.
To find out which trajectory the body moves in this case, let us turn to experience. We attach a rubber tube equipped with a tip to the water tap and direct the stream of water in a horizontal direction. The water particles will move in exactly the same way as a body thrown in the same direction. By turning away or, conversely, turning on the tap, you can change the initial speed of the stream and thereby the flight range of water particles (Fig. 105), however, in all cases the stream of water will have the shape parabolas. To verify this, a screen with pre-drawn parabolas should be placed behind the jet. The water jet will exactly follow the lines shown on the screen.
So, a freely falling body whose initial velocity is horizontal moves along a parabolic trajectory.
By parabola The body will also move if it is thrown at a certain acute angle to the horizon. The flight range in this case will depend not only on the initial speed, but also on the angle at which it was directed. By conducting experiments with a jet of water, it can be established that the greatest flight range is achieved when the initial speed makes an angle of 45° with the horizon (Fig. 106).
At high speeds of movement of bodies, air resistance should be taken into account. Therefore, the flight range of bullets and shells in real conditions is not the same as it follows from formulas valid for movement in airless space. So, for example, with an initial bullet speed of 870 m/s and an angle of 45° in the absence of air resistance, the flight range would be approximately 77 km, while in reality it does not exceed 3.5 km.
First escape velocity
Let's calculate the speed that must be imparted to the artificial Earth satellite so that it moves in a circular orbit at an altitude h above the ground.
At high altitudes, the air is very rarefied and offers little resistance to bodies moving in it. Therefore, we can assume that the satellite is only affected by gravitational force directed towards the center of the Earth ( Fig.4.4).
According to Newton's second law.
The centripetal acceleration of the satellite is determined by the formula 
, Where h- the height of the satellite above the Earth's surface. The force acting on the satellite, according to the law of universal gravitation, is determined by the formula 
, Where M- mass of the Earth.
Substituting the values F And a into the equation for Newton's second law, we get
From the resulting formula it follows that the speed of the satellite depends on its distance from the Earth’s surface: the greater this distance, the lower the speed it will move in a circular orbit. It is noteworthy that this speed does not depend on the mass of the satellite. This means that any body can become a satellite of the Earth if it is given a certain speed. In particular, when h=2000 km=2 10 6 m speed v≈ 6900 m/s.
The minimum speed that must be imparted to a body on the surface of the Earth for it to become a satellite of the Earth moving in a circular orbit is called first escape velocity.
The first escape velocity can be found using formula (4.7), if we accept h=0:
Substituting into formula (4.8) the value G and values of quantities M And R for the Earth, you can calculate the first escape velocity for the Earth satellite:
If such a speed is imparted to a body in the horizontal direction at the surface of the Earth, then in the absence of an atmosphere it will become an artificial satellite of the Earth, revolving around it in a circular orbit.
Only sufficiently powerful space rockets can convey such speed to satellites. Currently, thousands of artificial satellites orbit the Earth.
Any body can become an artificial satellite of another body (planet) if it is given the necessary speed.
Movement of artificial satellites
In Newton's works you can find a wonderful drawing showing how you can make the transition from a simple fall of a body along a parabola to the orbital motion of a body around the Earth (Fig. 107). “A stone thrown onto the ground,” Newton wrote, “will deviate under the influence of gravity from a straight path and, having described a curved trajectory, will finally fall to the Earth. If you throw it at a faster speed, it will fall further." Continuing these arguments, it is not difficult to come to the conclusion that if a stone were thrown from a high mountain with a sufficiently high speed, then its trajectory could become such that it would never fall to the Earth at all, turning into its artificial satellite.
The minimum speed that must be imparted to a body at the surface of the Earth in order to turn it into an artificial satellite is called first escape velocity.
To launch artificial satellites, rockets are used that lift the satellite to a given height and impart the required speed to it in the horizontal direction. After this, the satellite is separated from the launch vehicle and continues further movement only under the influence of the Earth's gravitational field. (We neglect the influence of the Moon, Sun and other planets here.) The acceleration imparted by this field to the satellite is the acceleration of gravity g. On the other hand, since the satellite is moving in a circular orbit, this acceleration is centripetal and is therefore equal to the ratio of the square of the satellite's speed to the radius of its orbit. Thus,
Where
Substituting expression (43.1) here, we get
We got the formula circular speed satellite
, i.e., the speed that the satellite has when moving in a circular orbit with a radius r on high h from the surface of the Earth.
To find the first escape velocity v1, it should be taken into account that it is defined as the speed of the satellite near the Earth’s surface, i.e. when h<
Substituting numerical data into this formula leads to the following result:
It was possible to communicate such a huge speed to the body for the first time only in 1957, when the world’s first artificial earth satellite(abbreviated ISZ). The launch of this satellite (Fig. 108) is the result of outstanding achievements in the fields of rocketry, electronics, automatic control, computer technology and celestial mechanics.
In 1958, the first American satellite Explorer 1 was launched into orbit, and a little later, in the 60s, other countries also launched satellites: France, Australia, Japan, China, Great Britain, etc., and many The satellites were launched using American launch vehicles.
Nowadays, the launch of artificial satellites is commonplace, and international cooperation has long been widespread in the practice of space research.
Satellites launched in different countries can be divided according to their purpose into two classes:
1. Research satellites. They are designed to study the Earth as a planet, its upper atmosphere, near-Earth space, the Sun, stars and the interstellar medium.
2. Application satellites. They serve to satisfy the earthly needs of the national economy. This includes communications satellites, satellites for studying the Earth’s natural resources, meteorological satellites, navigation satellites, military satellites, etc.
AES intended for human flight includes manned satellite ships And orbital stations.
In addition to working satellites in near-Earth orbits, so-called auxiliary objects also revolve around the Earth: the last stages of launch vehicles, nose fairings and some other parts that are separated from satellites when they are launched into orbit.
Note that due to the enormous air resistance near the Earth's surface, the satellite cannot be launched too low. For example, at an altitude of 160 km it is capable of making only one revolution, after which it descends and burns up in dense layers of the atmosphere. For this reason, the first artificial Earth satellite, launched into orbit at an altitude of 228 km, lasted only three months.
With increasing altitude, atmospheric resistance decreases and at h>300 km becomes negligible.
The question arises: what will happen if you launch a satellite at a speed greater than the first cosmic speed? Calculations show that if the excess is insignificant, then the body remains an artificial satellite of the Earth, but it no longer moves in a circle, but in a elliptical orbit. With increasing speed, the satellite’s orbit becomes more and more elongated, until it finally “breaks”, turning into an open (parabolic) trajectory (Fig. 109).
The minimum speed that must be imparted to a body at the surface of the Earth in order for it to leave it, moving along an open trajectory, is called second escape velocity.
The second escape velocity is √2 times greater than the first escape velocity:
At this speed, the body leaves the region of gravity and becomes a satellite of the Sun.
To overcome the gravity of the Sun and leave the solar system, you need to develop even greater speed - third space. The third escape velocity is 16.7 km/s. Having approximately the same speed, the automatic interplanetary station Pioneer 10 (USA) in 1983 for the first time in human history went beyond the Solar System and is now flying towards Barnard's star.
Examples of problem solving
Problem 1. A body is thrown vertically upward at a speed of 25 m/s. Determine the altitude and flight time.
Given: Solution:
; 0=0+25 . t-5 . t 2
; 0=25-10. t 1 ; t 1 =2.5c; H=0+25. 2.5-5. 2.5 2 =31.25 (m)
t- ? 5t=25; t=5c
H - ? Answer: t=5c; H=31.25 (m)
Rice. 1. Selection of reference system
First we must choose a frame of reference. Frame of reference we select one connected to the ground, the starting point of movement is designated 0. The Oy axis is directed vertically upward. The speed is directed upward and coincides in direction with the Oy axis. The acceleration of gravity is directed downward along the same axis.
Let's write down the law of body motion. We must not forget that speed and acceleration are vector quantities.
Next step. Note that the final coordinate, at the end when the body has risen to a certain height and then fell back to the ground, will be equal to 0. The initial coordinate is also equal to 0: 0=0+25 . t-5 . t 2.
If we solve this equation, we get the time: 5t=25; t=5 s.
Let us now determine the maximum lift height. First, we determine the time it takes for the body to rise to the top point. To do this we use the velocity equation: .
We wrote the equation in general form: 0=25-10. t 1,t 1 =2.5 s.
When we substitute the values known to us, we find that the time the body rises, time t 1, is 2.5 s.
Here I would like to note that the entire flight time is 5 s, and the rise time to the maximum point is 2.5 s. This means that the body rises exactly as long as it takes to fall back to the ground. Now let's use the equation we've already used, the law of motion. In this case, we put H instead of the final coordinate, i.e. maximum lift height: H=0+25. 2.5-5. 2.5 2 =31.25 (m).
Having made simple calculations, we find that the maximum lifting height of the body will be 31.25 m. Answer: t=5c; H=31.25 (m).
In this case, we used almost all the equations that we studied when studying free fall.
Problem 2. Determine the height above ground level at which acceleration of gravity decreases by half.
Given: Solution:
RZ =6400 km; ; 
.
N -? Answer: H ≈ 2650 km.
To solve this problem we need, perhaps, one single datum. This is the radius of the Earth. It is equal to 6400 km.
Acceleration of gravity is determined on the Earth's surface by the following expression: . This is on the surface of the Earth. But as soon as we move away from the Earth at a great distance, the acceleration will be determined as follows: .
If we now divide these values by each other, we get the following: .
Constant quantities are reduced, i.e. the gravitational constant and the mass of the Earth, and what remains is the radius of the Earth and the height, and this ratio is equal to 2.
Now transforming the resulting equations, we find the height: .
If we substitute the values into the resulting formula, we get the answer: H ≈ 2650 km.
Task 3.A body moves along an arc of radius 20 cm at a speed of 10 m/s. Determine centripetal acceleration.
Given: SI Solution:
R=20 cm 0.2 m
V=10 m/s
and C - ? Answer: a C = .
Formula for calculation centripetal acceleration known. Substituting the values here, we get: . In this case, the centripetal acceleration is huge, look at its value. Answer: a C =.