mechanical movement. Uniform and uneven movement. KS. Relativity of motion

I propose a game: choose an object in the room and describe its location. Do this so that the guesser cannot make a mistake. Out? And what will come out of the description if other bodies are not used? The expressions will remain: "to the left of ...", "above ..." and the like. Body position can only be set relative to some other body.

Location of the treasure: "Stand at the eastern corner of the outermost house of the village, facing north, and after walking 120 steps, turn to face east and walk 200 steps. In this place, dig a hole of 10 cubits and you will find 100 ingots of gold." It is impossible to find the treasure, otherwise it would have been dug up long ago. Why? The body in relation to which the description is made is not defined, it is not known in which village that house is located. It is necessary to accurately determine the body, which will be taken as the basis of our future description. Such a body in physics is called reference body. It can be chosen arbitrarily. For example, try choosing two different reference bodies and, relative to them, describe the location of the computer in the room. There will be two dissimilar descriptions.

Coordinate system

Let's look at the picture. Where is the tree, relative to cyclist I, cyclist II, and us looking at the monitor?

Relative to the reference body - cyclist I - the tree is on the right, relative to the reference body - cyclist II - the tree is on the left, relative to us it is in front. One and the same body - a tree, constantly in the same place, at the same time "to the left", and "to the right" and "in front". The problem is not only that different reference bodies are chosen. Consider its location relative to cyclist I.

In this picture, the tree on right from cyclist I

In this picture, the tree left from cyclist I

The tree and the cyclist did not change their location in space, but the tree can be "left" and "right" at the same time. In order to get rid of the ambiguity of the description of the direction itself, we choose a certain direction as positive, the opposite of the chosen one will be negative. The selected direction is indicated by an axis with an arrow, the arrow indicates the positive direction. In our example, we choose and designate two directions. From left to right (the axis on which the cyclist moves), and from us inside the monitor to the tree, this is the second positive direction. If we denote the first direction we have chosen as X, the second as Y, we get a two-dimensional coordinate system.

Relative to us, the cyclist is moving in the negative direction on the x-axis, the tree is in the positive direction on the y-axis

Relative to us, the cyclist is moving in the positive direction on the x-axis, the tree is in the positive direction on the y-axis

Now determine which object in the room is 2 meters in the positive X direction (to your right), and 3 meters in the negative Y direction (behind you). (2;-3) - coordinates this body. The first digit "2" is used to indicate the location along the X axis, the second digit "-3" indicates the location along the Y axis. It is negative, because the Y axis is not on the side of the tree, but on the opposite side. After the body of reference and direction is chosen, the location of any object will be described unambiguously. If you turn your back to the monitor, there will be another object to the right and behind you, but it will also have different coordinates (-2; 3). Thus, the coordinates accurately and unambiguously determine the location of the object.

The space in which we live is a space of three dimensions, as they say, a three-dimensional space. In addition to the fact that the body can be "right" ("left"), "in front" ("behind"), it can be even "above" or "below" you. This is the third direction - it is customary to designate it as the Z axis.

Is it possible to choose different axis directions? Can. But you can not change their direction during the solution of, for example, one problem. Is it possible to choose other axis names? It is possible, but you risk that others will not understand you, it is better not to do so. Is it possible to swap the x-axis with the y-axis? It is possible, but do not get confused in the coordinates: (x;y).

With a rectilinear motion of a body, one coordinate axis is sufficient to determine its position.

To describe motion on a plane, a rectangular coordinate system is used, consisting of two mutually perpendicular axes (Cartesian coordinate system).

Using a three-dimensional coordinate system, you can determine the position of the body in space.

Reference system

Each body at any moment of time occupies a certain position in space relative to other bodies. We already know how to determine its position. If over time the position of the body does not change, then it is at rest. If, over time, the position of the body changes, then this means that the body is moving. Everything in the world happens somewhere and sometime: in space (where?) and in time (when?). If we add to the body of reference, the coordinate system that determines the position of the body, a method of measuring time - hours, we get reference system. With which you can evaluate the movement or rest of the body.

Relativity of motion

The astronaut went into outer space. Is it at rest or in motion? If we consider it relative to the friend of the astronaut, who is nearby, he will rest. And if relative to an observer on Earth, the astronaut moves at great speed. Same with train travel. In relation to the people on the train, you sit still and read a book. But relative to the people who stayed at home, you are moving at the speed of a train.

Examples of choosing a reference body, relative to which in figure a) the train is moving (relative to trees), in figure b) the train is at rest relative to the boy.

Sitting in the car, waiting for departure. In the window we observe the train on a parallel track. When it starts to move, it is difficult to determine who is moving - our car or the train outside the window. In order to decide, it is necessary to assess whether we are moving relative to other stationary objects outside the window. We evaluate the state of our car in relation to different reference systems.

Changing displacement and speed in different reference systems

Displacement and speed change when moving from one frame of reference to another.

The speed of a person relative to the ground (fixed frame of reference) is different in the first and second cases.

Velocity addition rule: The speed of a body relative to a fixed frame of reference is the vector sum of the speed of a body relative to a moving frame of reference and the speed of a moving frame of reference relative to a fixed one.

Similar to the displacement vector. Movement addition rule: The movement of a body relative to a fixed frame of reference is the vector sum of the movement of a body relative to a moving frame of reference and the movement of a moving frame of reference relative to a fixed one.

Let a person walk along the car in the direction (or against) the movement of the train. Man is a body. The earth is a fixed frame of reference. The car is a moving frame of reference.

Changing the trajectory in different frames of reference

The trajectory of a body is relative. For example, consider the propeller of a helicopter descending to Earth. A point on the propeller describes a circle in the frame of reference associated with the helicopter. The trajectory of this point in the reference frame associated with the Earth is a helix.

translational movement

The movement of a body is a change in its position in space relative to other bodies over time. Each body has a certain size, sometimes different points of the body are in different places in space. How to determine the position of all points of the body?

BUT! Sometimes it is not necessary to specify the position of each point of the body. Let's consider such cases. For example, this does not need to be done when all points of the body move in the same way.

All the currents of the suitcase and the machine move in the same way.

The movement of a body in which all its points move in the same way is called progressive

Material point

It is not necessary to describe the movement of each point of the body even when its dimensions are very small compared to the distance it travels. For example, a ship crossing the ocean. Astronomers, when describing the motion of planets and celestial bodies relative to each other, do not take into account their size and their own motion. Despite the fact that, for example, the Earth is huge, relative to the distance from the Sun, it is negligible.

There is no need to consider the movement of each point of the body when they do not affect the movement of the entire body. Such a body can be represented by a point. All the substance of the body, as it were, is concentrated into a point. We get a body model, without dimensions, but it has a mass. That's what it is material point.

One and the same body with some of its movements can be considered a material point, with others it cannot. For example, when a boy goes from home to school and at the same time travels a distance of 1 km, then in this movement he can be considered a material point. But when the same boy does exercises, then he can no longer be considered a point.

Consider moving athletes

In this case, the athlete can be modeled by a material point

In the case of an athlete jumping into the water (figure on the right), it is impossible to model it to the point, since the movement of the whole body depends on any position of the arms and legs

The main thing to remember

1) The position of the body in space is determined relative to the reference body;
2) It is necessary to set the axes (their directions), i.e. a coordinate system that defines the coordinates of the body;
3) The movement of the body is determined relative to the reference system;
4) In different reference systems, the speed of a body can be different;
5) What is a material point

A more complicated situation of adding velocities. Have a person take a boat across a river. The boat is the investigated body. The fixed frame of reference is the earth. The moving frame of reference is a river.

The speed of the boat relative to the ground is the vector sum

What is the displacement of any point located on the edge of the disk with radius R when it is rotated by 600 relative to the stand? at 1800? Solve in reference systems associated with the stand and disk.

In the frame of reference associated with the stand, the displacements are equal to R and 2R. In the frame of reference associated with the disk, the displacement is zero all the time.

Why do raindrops in calm weather leave oblique straight stripes on the windows of a uniformly moving train?

In the reference frame associated with the Earth, the trajectory of the drop is a vertical line. In the frame of reference associated with the train, the movement of the drop on the glass is the result of the addition of two rectilinear and uniform movements: the train and the uniform fall of the drop in the air. Therefore, the trace of a drop on the glass is inclined.

How can you determine your running speed if you train on a treadmill with a broken automatic speed detection? After all, you can’t run a single meter relative to the walls of the hall.

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 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). 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.

95. Give examples of uniform motion.
It is very rare, for example, the movement of the Earth around the Sun.

96. Give examples of uneven movement.
The movement of the car, aircraft.

97. A boy slides down a mountain on a sleigh. Can this movement be considered uniform?
No.

98. Sitting in the car of a moving passenger train and watching the movement of an oncoming freight train, it seems to us that the freight train is going much faster than our passenger train was going before the meeting. Why is this happening?
Relative to the passenger train, the freight train moves with the total speed of the passenger and freight trains.

99. The driver of a moving car is in motion or at rest in relation to:
a) roads
b) car seats;
c) gas stations;
d) the sun;
e) trees along the road?
In motion: a, c, d, e
At rest: b

100. Sitting in the car of a moving train, we watch in the window a car that goes forward, then seems to be stationary, and finally moves back. How can we explain what we see?
Initially, the speed of the car is higher than the speed of the train. Then the speed of the car becomes equal to the speed of the train. After that, the speed of the car decreases compared to the speed of the train.

101. The plane performs a "dead loop". What is the trajectory of movement seen by observers from the ground?
ring trajectory.

102. Give examples of the movement of bodies along curved paths relative to the earth.
The movement of the planets around the sun; the movement of the boat on the river; Flight of bird.

103. Give examples of the movement of bodies that have a rectilinear trajectory relative to the earth.
moving train; person walking straight.

104. What types of movement do we observe when writing with a ballpoint pen? Chalk?
Equal and uneven.

105. Which parts of the bicycle, during its rectilinear movement, describe rectilinear trajectories relative to the ground, and which ones are curvilinear?
Rectilinear: handlebar, saddle, frame.
Curvilinear: pedals, wheels.

106. Why is it said that the Sun rises and sets? What is the reference body in this case?
The reference body is the Earth.

107. Two cars are moving along the highway so that some distance between them does not change. Indicate with respect to which bodies each of them is at rest and with respect to which bodies they move during this period of time.
Relative to each other, the cars are at rest. Vehicles move relative to surrounding objects.

108. Sledges roll down the mountain; the ball rolls down the inclined chute; the stone released from the hand falls. Which of these bodies move forward?
The sled is moving forward from the mountain and the stone released from the hands.

109. A book placed on a table in a vertical position (Fig. 11, position I) falls from the shock and takes position II. Two points A and B on the cover of the book described the trajectories AA1 and BB1. Can we say that the book moved forward? Why?

Give examples of bodies moving relative to the Earth and stationary?

Bodies that move relative to the Earth: meteorites, the Sun, the Moon, satellites, a walking person, a driving car (tram / trolleybus / bus).

And motionless bodies: trees, buildings, mountains. In general, everything that stands on Earth.

I would separate the concepts of the Earth as a planet and the earth as the surface of the planet. The Moon, meteorites, spaceships and stations, satellites, comets, and planets move relative to the Earth-planet. Previously, it was believed that the Sun moves relative to the Earth, although it is rather the opposite, depending on which reference point to take.

Moving relative to the surface of the earth - people, cars, planes, birds, clouds, animals, waves, and much more.

It is unlikely that anything can be considered motionless relative to the planet, because everything in space is in motion, but buildings, trees, rocks, stones, and other objects of inanimate nature are motionless relative to the earth's surface.

But this immobility is precisely relative to the surface, because the continents themselves are not immobile and drift.

well, then everything that stands on earth can be called relatively motionless, the entire structure of mankind and all natural objects, but all space objects in relation to the earth will definitely definitely be mobile.

There are many such examples, as I understand it.

As for bodies moving relative to the earth, they include:

• Moon;
• Mars;
• all planets;
• comets;
• meteorites;
• satellites of planets;
• asteroids;
• space satellites;
• spaceships;
• space debris;
• birds;
• clouds;
• hail;
• aircraft;
• gliders;
• aeronautic vehicles;
• parachutes;
• balloons;
• boomerangs;
• soccer balls in the state of half to the goal;
• trains traveling by rail;
• cars driving on the roads;
• ships and vessels sailing on the seas;
• water in rivers;
• water in the currents of oceans and seas;
• star systems;
• black holes in space;
• the whole universe;
• people going to work
• moving parts and mechanisms of engines;
• underwater rivers and springs.

As for bodies that are motionless relative to the Earth, in my opinion, they include:

• at home;
• pipes;
• stones;
• pyramids of the pharaohs;
• bridges;
• freeways;
• people sleeping peacefully at home;
• factories and enterprises.

Also, in my opinion, it is necessary to mention that our planet, together with the solar system, is not stationary in relation to other bodies and objects in space. We are flying in space, and therefore if we assume that there is a body in space that stands motionless in space in relation to us, then most likely this cannot actually be real. For we also move in space, which means that this combination cannot be called motionless. For example, there are space satellites in geostationary orbit, and it is precisely they that hang above the Earth in the same place almost all the time. The immobility of such satellites is provided by special satellite engines, with which it stabilizes the position, orbit and height, as well as speed.

DEFINITION

Relativity of motion manifests itself in the fact that the behavior of any moving body can only be determined in relation to some other body, which is called the body of reference.

Reference body and coordinate system

The reference body is chosen arbitrarily. It should be noted that the moving body and the reference body are equal in rights. Each of them, when calculating the movement, if necessary, can be considered either as a reference body, or as a moving body. For example, a person stands on the ground and watches a car drive along the road. A person is motionless relative to the Earth and considers the Earth a reference body, the plane and the car in this case are moving bodies. However, the passenger of the car, who says that the road runs away from under the wheels, is also right. He considers the car as the reference body (it is motionless relative to the car), while the Earth is a moving body.

To fix a change in the position of the body in space, a coordinate system must be associated with the reference body. A coordinate system is a way of specifying the position of an object in space.

When solving physical problems, the most common is the Cartesian rectangular coordinate system with three mutually perpendicular rectilinear axes - the abscissa (), ordinate () and applicate (). The SI unit for measuring length is the meter.

When orienting on the ground, the polar coordinate system is used. The map determines the distance to the desired settlement. The direction of movement is determined by azimuth, i.e. the corner that makes up the zero direction with the line connecting the person to the desired point. Thus, in the polar coordinate system, the coordinates are distance and angle.

In geography, astronomy, and when calculating the movements of satellites and spacecraft, the position of all bodies is determined relative to the center of the Earth in a spherical coordinate system. To determine the position of a point in space in a spherical coordinate system, the distance to the origin and the angles and are the angles that the radius vector makes with the plane of the zero Greenwich meridian (longitude) and the equatorial plane (latitude).

Reference system

The coordinate system, the body of reference with which it is associated, and the device for measuring time form a reference system, relative to which the movement of the body is considered.

When solving any problem of motion, first of all, the frame of reference in which the motion will be considered must be indicated.

When considering motion relative to a moving frame of reference, the classical law of addition of velocities is valid: the speed of a body relative to a fixed frame of reference is equal to the vector sum of the speed of a body relative to a moving frame of reference and the speed of a moving frame of reference relative to a fixed one:

Examples of solving problems on the topic "Relativity of Motion"

EXAMPLE