The words “body moves” do not have a definite meaning, since it is necessary to say in relation to which bodies or in relation to which frame of reference this movement is considered. Let's give some examples.

The passengers of a moving train are motionless relative to the walls of the car. And the same passengers move in the frame of reference connected with the Earth. The elevator goes up. A suitcase standing on its floor rests relative to the walls of the elevator and the person in the elevator. But it moves relative to the Earth and the house.

These examples prove the relativity of motion and, in particular, the relativity of the concept of speed. The speed of the same body is different in different frames of reference.

Imagine a passenger in a wagon moving uniformly relative to the surface of the Earth, releasing a ball from his hands. He sees how the ball falls vertically downward relative to the car with acceleration g. Associate the coordinate system with the car X 1 ABOUT 1 Y 1 (Fig. 1). In this coordinate system, during the fall, the ball will travel the path AD = h, and the passenger will note that the ball fell vertically down and at the moment of impact on the floor its speed is υ 1 .

Rice. 1

Well, what will an observer standing on a fixed platform, with which the coordinate system is connected, see? XOY? He will notice (let's imagine that the walls of the car are transparent) that the trajectory of the ball is a parabola AD, and the ball fell to the floor with a speed υ 2 directed at an angle to the horizon (see Fig. 1).

So we note that observers in coordinate systems X 1 ABOUT 1 Y 1 and XOY detect trajectories of various shapes, speeds and distances traveled during the movement of one body - the ball.

It is necessary to clearly understand that all kinematic concepts: trajectory, coordinates, path, displacement, speed have a certain form or numerical values ​​in one chosen frame of reference. When moving from one reference system to another, these quantities may change. This is the relativity of motion, and in this sense mechanical motion is always relative.

The relationship of point coordinates in reference systems moving relative to each other is described Galilean transformations. The transformations of all other kinematic quantities are their consequences.

Example. A man walks on a raft floating on a river. Both the speed of a person relative to the raft and the speed of the raft relative to the shore are known.

In the example, we are talking about the speed of a person relative to the raft and the speed of the raft relative to the shore. Therefore, one frame of reference K we will connect with the shore - this is fixed frame of reference, second TO 1 we will connect with the raft - this is moving frame of reference. We introduce the notation for speeds:

• 1 option(speed relative to systems)

υ - speed TO

υ 1 - the speed of the same body relative to the moving reference frame K

u- moving system speed TO TO

$\vec(\upsilon )=\vec(u)+\vec(\upsilon )_(1) .\; \; \; (1)$

• "Option 2

υ tone - speed body relatively stationary reference systems TO(human speed relative to the Earth);

υ top - the speed of the same body relatively mobile reference systems K 1 (human speed relative to the raft);

υ With- moving speed systems K 1 relative to the fixed system TO(velocity of the raft relative to the Earth). Then

$\vec(\upsilon )_(tone) =\vec(\upsilon )_(c) +\vec(\upsilon )_(top) .\; \; \; (2)$

• 3 option

υ A (absolute speed) - the speed of the body relative to the fixed frame of reference TO(human speed relative to the Earth);

υ from ( relative speed) - the speed of the same body relative to the moving reference frame K 1 (human speed relative to the raft);

υ p ( portable speed) - speed of the moving system TO 1 relative to the fixed system TO(velocity of the raft relative to the Earth). Then

$\vec(\upsilon )_(a) =\vec(\upsilon )_(from) +\vec(\upsilon )_(n) .\; \; \; (3)$

• 4 option

υ 1 or υ people - speed first body relative to a fixed frame of reference TO(speed human relative to the earth)

υ 2 or υ pl - speed second body relative to a fixed frame of reference TO(speed raft relative to the earth)

υ 1/2 or υ person/pl - speed first body concerning second(speed human relatively raft);

υ 2/1 or υ pl / person - speed second body concerning first(speed raft relatively human). Then

$\left|\begin(array)(c) (\vec(\upsilon )_(1) =\vec(\upsilon )_(2) +\vec(\upsilon )_(1/2) ,\; \; \, \, \vec(\upsilon )_(2) =\vec(\upsilon )_(1) +\vec(\upsilon )_(2/1) ;) \\ () \\ (\ vec(\upsilon )_(person) =\vec(\upsilon )_(pl) +\vec(\upsilon )_(person/pl) ,\; \; \, \, \vec(\upsilon )_( pl) =\vec(\upsilon )_(person) +\vec(\upsilon )_(pl/person) .) \end(array)\right. \; \; \; (4)$

Formulas (1-4) can also be written for displacements Δ r, and for accelerations a:

$\begin(array)(c) (\Delta \vec(r)_(tone) =\Delta \vec(r)_(c) +\Delta \vec(r)_(top) ,\; \; \; \Delta \vec(r)_(a) =\Delta \vec(r)_(from) +\Delta \vec(n)_(?) ,) \\ () \\ (\Delta \vec (r)_(1) =\Delta \vec(r)_(2) +\Delta \vec(r)_(1/2) ,\; \; \, \, \Delta \vec(r)_ (2) =\Delta \vec(r)_(1) +\Delta \vec(r)_(2/1) ;) \\ () \\ (\vec(a)_(tone) =\vec (a)_(c) +\vec(a)_(top) ,\; \; \; \vec(a)_(a) =\vec(a)_(from) +\vec(a)_ (n) ,) \\ () \\ (\vec(a)_(1) =\vec(a)_(2) +\vec(a)_(1/2) ,\; \; \, \, \vec(a)_(2) =\vec(a)_(1) +\vec(a)_(2/1) .) \end(array)$

Plan for solving problems on the relativity of motion

1. Make a drawing: draw the bodies in the form of rectangles, above them indicate the directions of velocities and movements (if necessary). Select the directions of the coordinate axes.

2. Based on the condition of the problem or in the course of the solution, decide on the choice of a moving frame of reference (FR) and with the notation of velocities and displacements.

• Always start by choosing a mobile CO. If there are no special reservations in the problem regarding which SS the velocities and displacements are given (or need to be found), then it does not matter which system to take as a moving SS. A good choice of the moving system greatly simplifies the solution of the problem.
• Pay attention to the fact that the same speed (displacement) is indicated in the same way in the condition, solution and in the figure.

3. Write down the law of addition of velocities and (or) displacements in vector form:

$\vec(\upsilon )_(tone) =\vec(\upsilon )_(c) +\vec(\upsilon )_(top) ,\; \; \, \, \Delta \vec(r)_(tone) =\Delta \vec(r)_(c) +\Delta \vec(r)_(top) .$

• Do not forget about other ways to write the law of addition:

4. Write down the projections of the law of addition on the 0 axis X and 0 Y(and other axes)

0X: υ tone x = υ with x+ υ top x , Δ r tone x = Δ r with x + Δ r top x , (5-6)

0Y: υ tone y = υ with y+ υ top y , Δ r tone y = Δ r with y + Δ r top y , (7-8)

• Other options:

υ 1 x= υ 2 x+ υ 1/2 x , Δ r 1x = Δ r 2x + Δ r 1/2x ,

0Y: υ a y= υ from y+ υ p y , Δ r and y = Δ r from y + Δ r P y ,

υ 1 y= υ 2 y+ υ 1/2 y , Δ r 1y = Δ r 2y + Δ r 1/2y .

5. Find the values ​​of the projections of each quantity:

υ tone x = …, υ with x= …, υ top x = …, Δ r tone x = …, Δ r with x = …, Δ r top x = …,

υ tone y = …, υ with y= …, υ top y = …, Δ r tone y = …, Δ r with y = …, Δ r top y = …

• Likewise for other options.

6. Substitute the obtained values ​​into equations (5) - (8).

7. Solve the resulting system of equations.

• Note. As the skill of solving such problems is developed, points 4 and 5 can be done in the mind, without writing in a notebook.

Add-ons

1. If the speeds of bodies are given relative to bodies that are now motionless, but can move (for example, the speed of a body in a lake (no current) or in windless weather), then such speeds are considered given relative to mobile system(relative to water or wind). This own speeds bodies, relative to a fixed system, they can change. For example, a person's own speed is 5 km/h. But if a person goes against the wind, his speed relative to the ground will become less; if the wind blows in the back, the person's speed will be greater. But relative to the air (wind), its speed remains equal to 5 km / h.
2. In tasks, the phrase "velocity of the body relative to the ground" (or relative to any other stationary body) is usually replaced by "velocity of the body" by default. If the speed of the body is not given relative to the ground, then this should be indicated in the condition of the problem. For example, 1) the speed of the aircraft is 700 km/h, 2) the speed of the aircraft in calm weather is 750 km/h. In example one, the speed of 700 km/h is given relative to the ground, in the second, the speed of 750 km/h is given relative to the air (see appendix 1).
3. In formulas that include values ​​with indices, the conformity principle, i.e. the indices of the corresponding quantities must match. For example, $t=\dfrac(\Delta r_(tone x) )(\upsilon _(tone x)) =\dfrac(\Delta r_(c x))(\upsilon _(c x)) =\dfrac(\Delta r_(top x))(\upsilon _(top x))$.
4. Displacement during rectilinear motion is directed in the same direction as the speed, so the signs of the projections of displacement and speed relative to the same reference frame coincide.

1. The relativity of motion lies in the fact that when studying motion in frames of reference moving uniformly and rectilinearly relative to the fixed frame of reference, all calculations can be carried out using the same formulas and equations, as if there were no movement of the moving frame relative to the fixed one.

2. In the boat example, how do the water and the shore move relative to the boat?

2. Imagine that the observer is located in the boat at the point O'. Let's draw a coordinate system X"O"Y" through this point. Let's direct the X" axis along the coast, the Y axis "perpendicular to the river flow. The observer in the boat sees that the coast is moving relative to its coordinate system

moving in the direction opposite to the positive direction of the axis

and the water moves relative to the boat, making a movement

3. A combine harvesting grain in a field moves relative to the ground at a speed of 2.5 km / h and, without stopping, pours grain into a car. Relative to what reference body is the car moving and relative to what is it at rest?

3. Relative to the harvester, the car is at rest, and relative to the ground it moves at the speed of the harvester.

In the 7th grade physics course, the relativity of mechanical motion was mentioned. Let us consider this issue in more detail using examples and formulate what exactly is the relativity of motion.

A person walks along the car against the movement of the train (Fig. 16). The speed of the train relative to the ground is 20 m/s, and the speed of a person relative to the car is 1 m/s. Let us determine with what speed and in what direction a person moves relative to the surface of the earth.

Rice. 16. The speed of a person relative to the car and relative to the ground is different in magnitude and direction

Let's argue like this. If a person did not walk along the car, then in 1 s he would move along with the train at a distance equal to 20 m. But during the same time he traveled a distance equal to 1 m, against the train. Therefore, in a time equal to 1 s, it has shifted relative to the earth's surface only by 19 m in the direction of the train. This means that the speed of a person relative to the surface of the earth is 19 m/s and is directed in the same direction as the speed of the train. Thus, in the frame of reference associated with the train, a person moves at a speed of 1 m / s, and in the frame of reference associated with any body on the surface of the earth, at a speed of 19 m / s, and these speeds are directed in opposite directions . It follows that the speed is relative, i.e. the speed of the same body in different frames of reference can be different both in numerical value and in direction.

Now let's turn to another example. Imagine a helicopter descending vertically to the ground. Relative to the helicopter, any point of the propeller, for example, point A (Fig. 17), will always move along a circle, which is shown in the figure as a solid line. For an observer on the ground, the same point will move along a helical path (dashed line). From this example, it is clear that the trajectory of motion is also relative, i.e., the trajectory of motion of the same body can be different in different frames of reference.

Rice. 17. Relativity of trajectory and path

Therefore, the path is a relative value, since it is equal to the sum of the lengths of all sections of the trajectory traveled by the body during the considered period of time. This is especially evident in those cases when the physical body moves in one frame of reference and rests in another. For example, a person sitting in a moving train travels a certain path s in the frame associated with the earth, and in the frame of reference associated with the train, his path is zero.

Thus,

• the relativity of motion is manifested in the fact that the speed, trajectory, path and some other characteristics of motion are relative, i.e. they can be different in different frames of reference

The understanding that the motion of the same body can be considered in different frames of reference played a huge role in the development of views on the structure of the Universe.

For a long time, people have noticed that the stars during the night, just like the Sun during the day, move across the sky from east to west, moving in arcs and making a complete revolution around the Earth in a day. Therefore, for many centuries it was believed that the motionless Earth is in the center of the world, and all celestial bodies revolve around it. Such a system of the world was called geocentric (the Greek word "geo" means "earth").

In the II century. the Alexandrian scientist Claudius Ptolemy summarized the available information about the movement of the stars and planets in the geocentric system and managed to compile fairly accurate tables that make it possible to determine the position of celestial bodies in the past and future, predict the onset of eclipses, etc.

However, over time, when the accuracy of astronomical observations increased, discrepancies began to be found between the calculated and observed positions of the planets. The corrections introduced at the same time made Ptolemy's theory very complicated and confusing. There was a need to replace the geocentric system of the world.

New views on the structure of the universe were detailed in the 16th century. Polish scientist Nicolaus Copernicus. He believed that the Earth and other planets move around the Sun, while simultaneously rotating around their axes. Such a system of the world is called heliocentric, since in it the Sun (in Greek "helios") is taken as the center of the Universe.

Thus, in the heliocentric reference system, the motion of celestial bodies is considered relative to the Sun, and in the geocentric reference system, relative to the Earth.

How, then, with the help of the Copernican world system, can we explain the daily revolution of the Sun around the Earth that we see? Figure 18 schematically depicts the globe, illuminated from one side by the sun's rays, and a person (observer) who is in the same place on the Earth during the day. Rotating with the Earth, he observes the movement of the luminaries.

Rice. 18. In the heliocentric system of the world, the apparent movement of the Sun in the sky during the day and stars at night is explained by the rotation of the Earth around its axis

The imaginary axis around which the Earth rotates, as it were, pierces the globe, passing through the North (N) and South (S) geographic poles. The arrow indicates the direction of the Earth's rotation - from west to east.

In Figure 18, a, the globe is depicted at that moment in time when it, as it were, takes the observer from the dark night side to the daylight illuminated by the Sun. But the observer, rotating together with the Earth about its axis from west to east at a speed approximately equal to 200 m/s 1, nevertheless does not feel this movement, just as we do not feel it. Therefore, it seems to him that the Sun revolves around the Earth, rising from the horizon, moves during the day (Fig. 18, b) from east to west, and goes beyond the horizon in the evening (Fig. 18, c). Then the observer sees the movement of stars from east to west during the night (Fig. 18, d).

So, according to the system of the world of Copernicus, the apparent rotation of the Sun and stars, i.e., the change of day and night, is explained by the rotation of the Earth around its axis. The time it takes for the globe to complete one revolution is called a day.

The heliocentric system of the world turned out to be much more successful than the geocentric one in solving many scientific and practical problems.

Thus, the application of knowledge about the relativity of motion made it possible to take a fresh look at the structure of the Universe. And this, in turn, later helped to discover the physical laws that describe the movement of bodies in the solar system and explain the reasons for such movement.

Questions

1. What is the relativity of motion? Illustrate your answer with examples.
2. What is the main difference between the heliocentric system of the world and the geocentric?
3. Explain the change of day and night on Earth in the heliocentric system (see Fig. 18).

Exercise 9

1. The water in the river moves at a speed of 2 m/s relative to the bank. A raft floats on the river. What is the speed of the raft relative to the shore; about the water in the river?
2. In some cases, the speed of the body may be the same in different frames of reference. For example, a train moves at the same speed in the frame of reference associated with the station building and in the frame of reference associated with a tree growing near the road. Doesn't this contradict the statement that speed is relative? Explain the answer.
3. Under what condition will the speed of a moving body be the same with respect to two frames of reference?
4. Due to the daily rotation of the Earth, a person sitting on a chair in his house in Moscow moves relative to the earth's axis at a speed of about 900 km / h. Compare this speed with the muzzle velocity of the bullet relative to the gun, which is 250 m/s.
5. A torpedo boat moves along the sixtieth parallel of south latitude at a speed of 90 km/h relative to land. The speed of the daily rotation of the Earth at this latitude is 223 m/s. What is (in SI) and where is the speed of the boat relative to the earth's axis, if it moves to the east; to the west?

1 The speed of rotation of points on the Earth's surface relative to the axis depends on the latitude of the area: it increases from zero (at the poles) to 465 m/s (at the equator).

Imagine an electric train. She rides quietly along the rails, carrying passengers to their dachas. And suddenly, the hooligan and parasite Sidorov, sitting in the last car, notices that controllers are entering the car at the Sady station. Of course, Sidorov did not buy a ticket, and he wants to pay a fine even less.

Relativity of a free rider in a train

And so, in order not to be caught, he quickly commits to another car. Controllers, having checked the tickets of all passengers, move in the same direction. Sidorov again moves to the next car, and so on.

And so, when he reaches the first carriage and there is nowhere to go further, it turns out that the train has just reached the Ogorody station he needs, and the happy Sidorov gets out, rejoicing that he drove like a hare and didn’t get caught.

What can we learn from this action-packed story? We can, no doubt, rejoice for Sidorov, and we can, in addition, discover one more interesting fact.

While the train traveled five kilometers from the Sady station to the Ogorody station in five minutes, Sidorov the hare covered the same distance in the same time plus a distance equal to the length of the train in which he rode, that is, about five thousand two hundred meters in the same five minutes.

It turns out that Sidorov was moving faster than the train. However, the controllers following on his heels developed the same speed. Considering that the speed of the train was about 60 km / h, it was just right to give them all several Olympic medals.

However, of course, no one will engage in such stupidity, because everyone understands that Sidorov's incredible speed was developed by him only relative to stationary stations, rails and gardens, and this speed was due to the movement of the train, and not at all Sidorov's incredible abilities.

Regarding the train, Sidorov did not move at all quickly and did not reach not only the Olympic medal, but even the ribbon from it. This is where we come across such a concept as the relativity of motion.

The concept of relativity of motion: examples

The relativity of motion has no definition, since it is not a physical quantity. The relativity of mechanical motion is manifested in the fact that some characteristics of motion, such as speed, path, trajectory, and so on, are relative, that is, they depend on the observer. In different reference systems, these characteristics will be different.

In addition to the above example with citizen Sidorov on the train, you can take almost any movement of any body and show how relative it is. When you go to work, you are moving forward relative to your home, and at the same time you are moving backward relative to the bus you missed.

You are standing still in relation to the player in your pocket, and are rushing at great speed relative to a star called the Sun. Each step you take will be a gigantic distance for the asphalt molecule and insignificant for the planet Earth. Any movement, like all its characteristics, always makes sense only in relation to something else.

Mathematically, the movement of a body (or a material point) with respect to a chosen reference system is described by equations that establish how t coordinates that determine the position of the body (points) in this frame of reference. These equations are called the equations of motion. For example, in Cartesian coordinates x, y, z, the movement of a point is determined by the equations x = f 1 (t) (\displaystyle x=f_(1)(t)), y = f 2 (t) (\displaystyle y=f_(2)(t)), z = f 3 (t) (\displaystyle z=f_(3)(t)).

In modern physics, any movement is considered relative, and the movement of a body should be considered only in relation to some other body (reference body) or system of bodies. It is impossible to indicate, for example, how the Moon moves in general, one can only determine its movement, for example, in relation to the Earth, the Sun, stars, etc.

Other definitions

On the other hand, it was previously believed that there is a certain "fundamental" frame of reference, the simplicity of writing in which the laws of nature distinguishes it from all other systems. So, Newton considered absolute space to be a selected reference frame, and physicists of the 19th century believed that the system, relative to which the ether of Maxwell's electrodynamics rests, is privileged, and therefore it was called the absolute reference frame (AFR). Finally, assumptions about the existence of a privileged reference frame were rejected by the theory of relativity. In modern concepts, no absolute reference system exists, since the laws of nature, expressed in tensor form, have the same form in all reference systems - that is, at all points in space and at all points in time. This condition - local space-time invariance - is one of the verifiable foundations of physics.

Sometimes an absolute reference frame is called a CMB-related frame, that is, an inertial frame of reference in which the CMB does not have a dipole anisotropy.

Reference body

In physics, a reference body is a set of bodies that are motionless relative to each other, in relation to which the movement is considered (in the associated