By studying kinematics, we learn to describe mechanical movement- change in the position of the body relative to other bodies over time. To clarify the very important words "relative to other bodies" we will give an example in which you need to use your imagination.

Let's say we got into a car and drove onto a road heading north. Let's look around. With oncoming cars, it's simple: they always approach us from the north, pass us and move south (look at the picture - the blue car on the left).

With passing cars it is more difficult. Those cars that are going faster than us approach us from behind, overtake us and move away to the north (for example, a gray car in the center). But the cars that we are overtaking approach us from the front and move away from us back (red car on the right). That is, passing cars relative to us can move south at the same time when relative to the road going north!

So, from the point of view of the driver and passengers of our car (at the bottom of the picture, its blue hood), the red car being overtaken is moving south, although, from the point of view of the boy on the side of the road, the same car is going north. In addition, a red car will “fly by with a whistle” past the boy, and by our car it will “slowly float away” back.

In this way, the movement of bodies may look different from the point of view of different observers. This phenomenon is relativity of mechanical motion . It manifests itself in the fact that the speed, direction and trajectory of the same movement are different for different observers. The first two differences (in speed and direction of movement) we have just illustrated by the example of cars. Next, we will show the differences in the form of the trajectory of the same body for different observers (see the figure with yachts).

Recall: kinematics creates a mathematical description of the motion of bodies. But how to do this if the movement looks different from the point of view of different observers? To be certain, in physics always choose a frame of reference.

Reference system call the clock and the coordinate system associated with the reference body (observer). Let's explain this with examples.

Let's imagine that we are on a train and we drop an object. It will fall at our feet, although even at a speed of 36 km / h, the train moves 10 meters every second. Imagine now that a sailor has climbed onto the mast of the yacht and drops the shot (see figure). We also should not be embarrassed that it will fall to the bottom of the mast, despite the fact that the yacht is sailing forward. That is at each moment of time, the nucleus moves both down and forward along with the yacht.

So, in the frame of reference associated with the yacht(let's call it "deck"), the core moves only vertically and travels a path equal to the length of the mast; the trajectory of the nucleus is a straight line segment. But in the reference frame associated with the shore(let's call it "pier"), the core moves both vertically and forward; the trajectory of the core is a branch of a parabola, and the path is clearly greater than the length of the mast. Conclusion: the trajectories and paths of the same nucleus are different in different reference systems: “deck” and “pier”.

What about core speed? Since this is the same body, we consider the time of its fall to be the same in both frames of reference. But since the paths traversed by the nucleus are different, then the speeds of the same movement in different frames of reference are different.

Is it possible to be stationary and still move faster than a Formula 1 car? It turns out you can. Any movement depends on the choice of reference system, that is, any movement is relative. The topic of today's lesson: “Relativity of motion. The law of addition of displacements and velocities. We will learn how to choose a frame of reference in a particular case, how to find the displacement and speed of the body.

Mechanical motion is a change in the position of a body in space relative to other bodies over time. In this definition, the key phrase is "relative to other bodies." Each of us is motionless relative to any surface, but relative to the Sun, together with the entire Earth, we make orbital motion at a speed of 30 km / s, that is, the motion depends on the frame of reference.

Reference system - a set of coordinate systems and clocks associated with the body, relative to which the movement is being studied. For example, when describing the movements of passengers in a car, the frame of reference can be associated with a roadside cafe, or with a car interior or with a moving oncoming car if we estimate the overtaking time (Fig. 1).

Rice. 1. Choice of reference system

What physical quantities and concepts depend on the choice of reference system?

1. Position or coordinates of the body

Consider an arbitrary point . In different systems, it has different coordinates (Fig. 2).

Rice. 2. Point coordinates in different coordinate systems

2. Trajectory

Consider the trajectory of a point located on the propeller of an aircraft in two reference systems: the reference system associated with the pilot, and the reference system associated with the observer on Earth. For the pilot, this point will make a circular rotation (Fig. 3).

Rice. 3. Circular rotation

While for an observer on Earth, the trajectory of this point will be a helix (Fig. 4). It is obvious that the trajectory depends on the choice of the frame of reference.

Rice. 4. Helical trajectory

Relativity of the trajectory. Body motion trajectories in different frames of reference

Let us consider how the trajectory of motion changes depending on the choice of the reference system using the problem as an example.

A task

What will be the trajectory of the point at the end of the propeller in different COs?

1. In the CO associated with the pilot of the aircraft.

2. In CO associated with an observer on Earth.

Solution:

1. Neither the pilot nor the propeller move relative to the aircraft. For the pilot, the trajectory of the point will appear as a circle (Fig. 5).

Rice. 5. Trajectory of the point relative to the pilot

2. For an observer on Earth, a point moves in two ways: rotating and moving forward. The trajectory will be helical (Fig. 6).

Rice. 6. Trajectory of a point relative to an observer on Earth

Answer : 1) circle; 2) helix.

Using the example of this problem, we have seen that the trajectory is a relative concept.

As an independent check, we suggest that you solve the following problem:

What will be the trajectory of the point at the end of the wheel relative to the center of the wheel, if this wheel is moving forward, and relative to points on the ground (stationary observer)?

3. Movement and path

Consider a situation where a raft is floating and at some point a swimmer jumps off it and seeks to cross to the opposite shore. The movement of the swimmer relative to the fisherman sitting on the shore and relative to the raft will be different (Fig. 7).

Movement relative to the earth is called absolute, and relative to a moving body - relative. The movement of a moving body (raft) relative to a fixed body (fisherman) is called portable.

Rice. 7. Move the swimmer

It follows from the example that displacement and path are relative values.

4. Speed

Using the previous example, you can easily show that speed is also a relative value. After all, speed is the ratio of displacement to time. We have the same time, but the movement is different. Therefore, the speed will be different.

The dependence of motion characteristics on the choice of reference system is called relativity of motion.

There have been dramatic cases in the history of mankind, connected precisely with the choice of a reference system. The execution of Giordano Bruno, the abdication of Galileo Galilei - all these are the consequences of the struggle between the supporters of the geocentric reference system and the heliocentric reference system. It was very difficult for mankind to get used to the idea that the Earth is not at all the center of the universe, but a completely ordinary planet. And the motion can be considered not only relative to the Earth, this motion will be absolute and relative to the Sun, stars or any other bodies. It is much more convenient and simpler to describe the motion of celestial bodies in a reference frame associated with the Sun, this was convincingly shown first by Kepler, and then by Newton, who, based on the consideration of the motion of the Moon around the Earth, derived his famous law of universal gravitation.

If we say that the trajectory, path, displacement and speed are relative, that is, they depend on the choice of a reference frame, then we do not say this about time. Within the framework of classical, or Newtonian, mechanics, time is an absolute value, that is, it flows the same in all frames of reference.

Let's consider how to find displacement and speed in one frame of reference, if they are known to us in another frame of reference.

Consider the previous situation, when a raft is floating and at some point a swimmer jumps off it and tries to cross to the opposite shore.

How is the movement of the swimmer relative to the fixed CO (associated with the fisherman) related to the movement of the relatively mobile CO (associated with the raft) (Fig. 8)?

Rice. 8. Illustration for the problem

We called the movement in a fixed frame of reference . From the triangle of vectors it follows that . Now let's move on to finding the relationship between the speeds. Recall that in the framework of Newtonian mechanics, time is an absolute value (time flows in the same way in all frames of reference). This means that each term from the previous equality can be divided by time. We get:

This is the speed at which the swimmer is moving for the fisherman;

This is the swimmer's own speed;

This is the speed of the raft (the speed of the river).

Problem on the law of addition of velocities

Consider the law of addition of velocities using the problem as an example.

A task

Two cars are moving towards each other: the first car at speed , the second - at speed . How fast are the cars approaching (Fig. 9)?

Rice. 9. Illustration for the problem

Solution

Let's apply the law of addition of speeds. To do this, let's move from the usual CO associated with the Earth to the CO associated with the first car. Thus, the first car becomes stationary, and the second moves towards it at a speed (relative speed). With what speed, if the first car is stationary, does the Earth rotate around the first car? It rotates at speed and the speed is in the direction of the speed of the second vehicle (carrying speed). Two vectors that are directed along the same straight line are summed. .

Answer: .

Limits of applicability of the law of addition of velocities. The law of addition of velocities in the theory of relativity

For a long time it was believed that the classical law of velocity addition is always valid and applicable to all frames of reference. However, about a year ago it turned out that in some situations this law does not work. Let's consider such a case on the example of a problem.

Imagine that you are on a space rocket that is moving at a speed of . And the captain of the space rocket turns on the flashlight in the direction of the rocket movement (Fig. 10). The speed of light propagation in vacuum is . What will be the speed of light for a stationary observer on Earth? Will it be equal to the sum of the speeds of light and rocket?

Rice. 10. Illustration for the problem

The fact is that here physics is faced with two contradictory concepts. On the one hand, according to Maxwell's electrodynamics, the maximum speed is the speed of light, and it is equal to . On the other hand, according to Newtonian mechanics, time is an absolute value. The problem was solved when Einstein proposed the special theory of relativity, or rather its postulates. He was the first to suggest that time is not absolute. That is, somewhere it flows faster, and somewhere slower. Of course, in our world of low speeds, we do not notice this effect. In order to feel this difference, we need to move at speeds close to the speed of light. On the basis of Einstein's conclusions, the law of addition of velocities was obtained in the special theory of relativity. It looks like this:

This is the speed relative to the stationary CO;

This is the speed relative to the mobile CO;

This is the speed of the moving CO relative to the stationary CO.

If we substitute the values ​​from our problem, we get that the speed of light for a stationary observer on Earth will be .

The controversy has been resolved. You can also see that if the velocities are very small compared to the speed of light, then the formula for the theory of relativity turns into the classical formula for adding velocities.

In most cases, we will use the classical law.

Today we found out that the movement depends on the frame of reference, that speed, path, displacement and trajectory are relative concepts. And time within the framework of classical mechanics is an absolute concept. We learned how to apply the acquired knowledge by analyzing some typical examples.

Bibliography

1. Tikhomirova S.A., Yavorsky B.M. Physics (basic level) - M.: Mnemozina, 2012.
2. Gendenstein L.E., Dick Yu.I. Physics grade 10. - M.: Mnemosyne, 2014.
3. Kikoin I.K., Kikoin A.K. Physics - 9, Moscow, Education, 1990.

1. Internet portal Class-fizika.narod.ru ().
2. Internet portal Nado5.ru ().
3. Internet portal Fizika.ayp.ru ().

Homework

1. Define the relativity of motion.
2. What physical quantities depend on the choice of reference system?

Associated with the body, in relation to which the movement (or balance) of some other material points or bodies is being studied. Any movement is 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 in relation to the Earth or the Sun and stars, etc.

Mathematically, the movement of a body (or a material point) with respect to the 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. For example, in Cartesian coordinates x, y, z, the movement of a point is determined by the equations X = f1(t), y = f2(t), Z = f3(t), called the equations of motion.

Reference body- the body relative to which the reference system is set.

reference system- juxtaposed with a continuum spanned by real or imagined basic reference bodies. It is natural to present the following two requirements to the basic (generating) bodies of the reference system:

1. Base bodies must be motionless relative to each other. This is checked, for example, by the absence of a Doppler effect during the exchange of radio signals between them.

2. The base bodies must move with the same acceleration, that is, they must have the same indicators of the accelerometers installed on them.

see also

Relativity of motion

Moving bodies change their position relative to other bodies. The position of a car speeding on a highway changes with respect to the mileposts, the position of a ship sailing in the sea near the coast changes with respect to the stars and the coastline, and the movement of an aircraft flying above the earth can be judged by its change in its position relative to the surface of the Earth. Mechanical motion is the process of changing the position of bodies in space over time. It can be shown that the same body can move differently relative to other bodies.

Thus, it is possible to say that some body is moving only when it is clear relative to which other body - the reference body - its position has changed.

Notes

Links

Wikimedia Foundation. 2010 .

See what "Relativity of Motion" is in other dictionaries:

Events is a key effect of SRT, which manifests itself, in particular, in the “twin paradox”. Consider several synchronized clocks located along the axis in each of the frames of reference. Lorentz transformations assume that at the moment ... Wikipedia

Theories of relativity form an essential part of the theoretical basis of modern physics. There are two main theories: private (special) and general. Both were created by A. Einstein, private in 1905, general in 1915. In modern physics, private ... ... Collier Encyclopedia

RELATIVITY- the nature of that which depends on another thing. The scientific theory of relativity has nothing in common with the philosophical theory of the relativity of human knowledge; it is an interpretation of the phenomena of the universe (and not of human knowledge), ... ... Philosophical Dictionary

The angular momentum (kinetic momentum, angular momentum, orbital momentum, angular momentum) characterizes the amount of rotational motion. A value depending on how much mass rotates, how it is distributed relative to the axis ... ... Wikipedia

Einstein, a physical theory that considers the spatio-temporal properties of physical processes. Since the laws established by the theory of relativity are common to all physical processes, they are usually referred to simply as ... ... encyclopedic Dictionary

In a broad sense, any change, in a narrow sense, a change in the position of the body in space. D. became a universal principle in the philosophy of Heraclitus (“everything flows”). The possibility of D. was denied by Parmenides and Zeno of Elea. Aristotle subdivided D. into ... ... Philosophical Encyclopedia

Image of the solar system from the book by Andreas Cellarius Harmonia Macrocosmica (1708) The heliocentric system of the world is the idea that the Sun is the central celestial body around which the Earth and others revolve ... Wikipedia

ZENON OF ELEA- [Greek. Ζήνων ὁ ᾿Ελεάτης] (5th century BC), ancient Greek. philosopher, representative of the philosophical Eleatic school, student of Parmenides, creator of the famous Zeno's aporias. Life and writings The exact date of birth of ZE is unknown. According to Diogenes... Orthodox Encyclopedia

The mechanical motion of a body is the change in its position in space relative to other bodies over time. In this case, the bodies interact according to the laws of mechanics. A section of mechanics that describes the geometric properties of motion without taking into account ... ... Wikipedia

A reference system is a set of a reference body, a coordinate system associated with it and a time reference system, in relation to which the movement (or equilibrium) of any material points or bodies is considered. Mathematically movement ... Wikipedia

Books

• A set of tables. Physics. Statics. Special Relativity (8 tables), . Art. 5-8664-008. Educational album of 8 sheets. Article - 5-8625-008. Equilibrium conditions for translational motion. Equilibrium conditions for rotational motion. Center of gravity. Center of mass...

Questions.

1. What do the following statements mean: speed is relative, trajectory is relative, path is relative?

This means that these quantities (velocity, trajectory and path) for motion differ depending on which reference frame the observation is made from.

2. Show with examples that speed, trajectory and distance traveled are relative values.

For example, a person stands motionless on the surface of the Earth (there is no speed, no trajectory, no path), but at this time the Earth rotates around its axis, and therefore a person, relative to, for example, the center of the Earth, moves along a certain trajectory (in a circle), moves and has a certain speed.

3. Formulate briefly what the relativity of motion is.

The movement of the body (speed, path, trajectory) is different in different frames of reference.

4. What is the main difference between the heliocentric and geocentric systems?

In the heliocentric system, the reference body is the Sun, and in the geocentric system, the Earth.

5. Explain the change of day and night on Earth in the heliocentric system (see Fig. 18).

In the heliocentric system, the change of day and night is explained by the rotation of the Earth.

Exercises.

1. Water in a 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?

The speed of the raft relative to the shore is 2 m/s, relative to the water in the river - 0 m/s.

2. In some cases, the speed of a body can 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.

If both bodies, with which the frames of reference of these bodies are connected, remain motionless relative to each other, then they are connected with the third frame of reference - the Earth, relative to which the measurements take place.

3. Under what condition will the speed of a moving body be the same with respect to two frames of reference?

If these frames of reference are fixed relative to each other.

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 is moving 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 equal to in (SI) and where is the speed of the boat relative to the earth's axis directed if it moves to the east? to the west?

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 O 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. one

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 O 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 speeds 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:

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

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:

0X: υ a x= υ from x+ υ p x , Δ r a x = Δ r from x + Δ r P x ,

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 velocities 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). it 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.