Is it possible to be stationary and still move faster than a Formula 1 car? It turns out that it is possible. 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 reference system in a given case, and how to find the displacement and velocity of a body.

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

A reference system is a set of coordinate systems and clocks associated with the body relative to which motion is being studied. For example, when describing the movements of passengers inside a car, the reference system can be associated with a roadside cafe, or with the inside of a car, or with a moving oncoming car if we are estimating the overtaking time (Fig. 1).

Rice. 1. Selection of reference system

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

1. Body position or coordinates

Let's consider an arbitrary point. IN various systems it has different coordinates (Fig. 2).

Rice. 2. Coordinates of a point in different coordinate systems

2. Trajectory

Consider the trajectory of a point on an airplane propeller in two reference frames: the reference frame associated with the pilot, and the reference frame associated with the observer on Earth. For the pilot, this point will perform a circular rotation (Fig. 3).

Rice. 3. Circular rotation

While for an observer on Earth the trajectory of this point will be a helical line (Fig. 4). Obviously, the trajectory depends on the choice of reference system.

Rice. 4. Helical path

Relativity of trajectory. Trajectories of body motion in various reference systems

Let's consider how the trajectory of movement changes depending on the choice of reference system using the example of a problem.

Task

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

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

2. In the CO associated with the observer on Earth.

Solution:

1. Neither the pilot nor the propeller moves relative to the airplane. For the pilot, the trajectory of the point will appear to be 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 this problem as an example, we were convinced that trajectory is a relative concept.

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

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

3. Movement and path

Let's consider a situation when a raft is floating and at some point a swimmer jumps off it and tries 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 ground is called absolute, and relative to a moving body - relative. The movement of a moving body (raft) relative to a stationary body (fisherman) is called portable.

Rice. 7. Swimmer's movement

From the example it follows that displacement and path are relative quantities.

4. Speed

Using the previous example, you can easily show that speed is also a relative quantity. After all, speed is the ratio of movement to time. Our time is the same, but our travel is different. Therefore, the speed will be different.

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

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

Let's consider how to find displacement and velocity in one reference system if they are known to us in another reference system.

Let's 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 a swimmer relative to a stationary SO (associated with the fisherman) connected with the movement of a relatively mobile SO (associated with the raft) (Fig. 8)?

Rice. 8. Illustration for the problem

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

This is the speed at which a swimmer moves for a 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

Let's consider the law of adding velocities using an example problem.

Task

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

Rice. 9. Illustration for the problem

Solution

Let us apply the law of addition of velocities. To do this, let's move from the usual CO associated with the Earth to CO associated with the first car. Thus, the first car becomes stationary, and the second one moves towards it with speed (relative speed). At what speed, if the first car is stationary, does the Earth rotate around the first car? It rotates at a speed and the speed is directed in the direction of the speed of the second car (transfer 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 addition of velocities is always valid and applies to all reference systems. However, about years ago it turned out that in some situations this law does not work. Let's consider this case using an example problem.

Imagine that you are on a space rocket moving at a speed of . And the captain of the space rocket turns on the flashlight in the direction of the rocket's 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 the 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 we don’t notice low speeds this effect. In order to feel this difference, we need to move at speeds close to the speed of light. Based on Einstein's conclusions, the law of addition of velocities in the special theory of relativity was obtained. It looks like this:

This is the speed relative to a stationary CO;

This is the speed of relatively mobile CO;

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

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

The controversy has been resolved. You can also make sure 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 movement depends on the reference system, that speed, path, movement and trajectory are relative concepts. And time, within the framework of classical mechanics, is an absolute concept. We learned to apply the acquired knowledge by analyzing some typical examples.

Bibliography

1. Tikhomirova S.A., Yavorsky B.M. Physics (basic level) - M.: Mnemosyne, 2012.
2. Gendenshtein L.E., Dick Yu.I. Physics 10th grade. - 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?

Equal to the vector difference of velocities specified relative to a fixed reference frame.

When studying mechanical movement, it is primarily emphasized relativity. When studying various properties of body motion, it is assumed that absolute motion(i.e. motion related to fixed axes). In many cases it becomes necessary to determine relative motion, referred to a reference system moving with respect to the fixed axes.

Relative motion a point in relation to a moving frame of reference can be considered as absolute motion, and has all the properties of absolute motion.

Movement can be viewed in different ways reference systems. The choice of a reporting system is dictated by convenience: it must be chosen so that the movement being studied and its patterns look as simple as possible. To move from one reference system to another, it is necessary to know which characteristics of motion change and how, and which remain unchanged.

Based on experiments, it can be argued that when considering movements occurring at speeds small compared to the speed of light, time is constant in all reference systems, which means that when measured in any reference system, the time interval between two events is the same.

As for spatial characteristics, the position of the body changes when moving to another frame of reference, but the spatial location of these two events does not change.

Now let's consider change in the speed of movement of bodies when moving from one reference system to another, which moves relative to the first.

Let's consider an example of crossing on a ferry moving forward relative to the shores (relative to the ground). The vector of movement of the passenger relative to the shores will be denoted by Δr, and relative to the ferry - through Δr´. Displacement of the ferry relative to the ground in the same time Δt denote by ΔR. In this case

Δr = ΔR + Δr´.

Let's divide the equality term by term into a period of time Δt during which these movements occurred. Going to the limit Δt>0, we get a similar relationship for speeds:

υ = V + υ ´

Where υ - passenger speed relative to the ground, V- speed of the ferry relative to the ground, υ ´ is the speed of the passenger relative to the ferry. This equality expresses speed addition rule, which, with the simultaneous participation of the body in two movements, can be interpreted as speed conversion law bodies when moving from one reference system to another. In fact, υ And υ ´ - passenger speed in two different reference systems, and V- the speed of one system (ferry) relative to another (ground).

From formula (2) it follows that relative speed two bodies is the same in all reference systems. When moving to a new reference system, the same vector is added to the speed of each body V speed of the reference system. Therefore, the difference between the velocity vectors of the bodies υ - υ ´ does not change. Relative speed tel is absolute.

Mathematically, the motion of a body (or a material point) in relation to a chosen frame of reference is described by equations that establish how it changes over time t coordinates that determine the position of the body (point) in this reference system. These equations are called equations of motion. For example, in Cartesian coordinates x, y, z, the motion 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 (body of reference) or system of bodies. It is impossible to indicate, for example, how the Moon moves in general, you 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 was a certain “fundamental” reference system, the simplicity of recording the laws of nature in which distinguishes it from all other systems. Thus, Newton considered absolute space to be a distinguished reference system, 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, the assumptions about the existence of a privileged frame of reference 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 times. This condition - local space-time invariance - is one of the verifiable foundations of physics.

Sometimes an absolute reference frame is called a system associated with the cosmic microwave background radiation, that is, an inertial reference frame in which the cosmic microwave background radiation does not have dipole anisotropy.

Reference body

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

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

Relativity of motion: basic principles

Frame of reference- this is a set of reference body, coordinate system and time associated with the body in relation to which the movement (or equilibrium) of some other material points or bodies is studied. Any movement is relative, and the movement of a body should be considered only in relation to some other body (body of reference) or system of bodies. It is impossible to indicate, for example, how the Moon moves in general, you 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) in relation to a chosen reference system is described by equations that establish how the coordinates that determine the position of the body (point) in this reference system change over time t. 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 specified.

Frame of reference- compared with a continuum stretched over real or imaginary basic bodies of reference. It is natural to present the following two requirements to the basic (generating) bodies of the reference system:

1. The base bodies must be motionless relative to each other. This is checked, for example, by the absence of a Doppler effect when radio signals are exchanged between them.

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

Moving bodies change their position relative to other bodies. The position of a car speeding along a highway changes relative to the markers on kilometer posts, the position of a ship sailing in the sea near the shore changes relative to the stars and the coastline, and the movement of an airplane flying over the earth can be judged by the 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 what other body - the body of reference - its position has changed.

Relativity of motion: a real-life example

Imagine an electric train. She travels quietly along the rails, transporting passengers to their dachas. And suddenly, sitting in the last carriage, the hooligan and parasite Sidorov notices that at the Sady station controllers are entering the carriage. Naturally, Sidorov did not buy a ticket, and he wants to pay the fine even less. And so, so as not to be caught, he quickly moves in a straight, uniform motion to another car. The controllers, having checked the tickets of all passengers, move in the same direction. Sidorov again moves to the next carriage 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 happy Sidorov gets out, rejoicing that he rode like a hare and did not get caught.

What can we learn from this action-packed story? We can, without a doubt, rejoice for Sidorov, and we can, in addition, discover another interesting fact. While the train traveled five kilometers from the Sady station to the Ogorody station in five minutes, the Sidorov hare covered the same distance plus a distance equal to the length of the train in which it was traveling, that is, about five thousand two hundred meters in the same five minutes.

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 vegetable gardens, and this speed was determined by the movement of the train, and not at all by Sidorov’s incredible abilities. In relation to the train, Sidorov was not moving fast at all and did not even reach the Olympic medal, but even the ribbon from it. This is where we come across such a concept as the relativity of motion.

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

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

2. Let’s imagine that the observer is located in a boat at point O’. Let's draw a coordinate system X"O"Y through this point. The X" axis will be directed along the shore, the Y" axis - perpendicular to the flow of the river. The observer in the boat sees that the shore is moving relative to his coordinate system

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

and the water moves relative to the boat making a movement

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

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