Mechanics mechanical motion and its relativity. mechanical movement
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 last 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 bars 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 will 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" indicates 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 system coordinates).
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 to 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 systems reference
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 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 rain drops 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.
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 frames of reference: the frame of reference associated with the pilot, and the frame of reference associated with the observer on Earth. For the pilot, this point will make 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 makes forward movement 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 classical law addition of velocities 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 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, maximum speed is the speed of light, and it is equal to . On the other hand, according to Newtonian mechanics, time is an absolute quantity. 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
- Tikhomirova S.A., Yavorsky B.M. Physics ( a basic level of) - M.: Mnemozina, 2012.
- Gendenstein L.E., Dick Yu.I. Physics grade 10. - M.: Mnemosyne, 2014.
- Kikoin I.K., Kikoin A.K. Physics - 9, Moscow, Education, 1990.
- Internet portal Class-fizika.narod.ru ().
- Internet portal Nado5.ru ().
- Internet portal Fizika.ayp.ru ().
Homework
- Define the relativity of motion.
- What physical quantities depend on the choice of reference system?
Lecture 2. Relativity of mechanical motion. Reference systems. Characteristics of mechanical movement: movement, speed, acceleration.
Mechanics - branch of physics that deals with mechanical movement.
Mechanics is divided into kinematics, dynamics and statics.
Kinematics is a branch of mechanics in which the movement of bodies is considered without clarifying the causes of this movement.Kinematics studies ways of describing movement and the relationship between the quantities that characterize these movements.
The task of kinematics: determination of the kinematic characteristics of movement (trajectory of movement, displacement, distance traveled, coordinates, speed and acceleration of the body), as well as obtaining equations for the dependence of these characteristics on time.
mechanical movement of the body called the change in its position in space relative to other bodies over time.
mechanical movement relatively , the expression "the body moves" is meaningless until it is determined in relation to what the movement is considered. The motion of the same body relative to different bodies turns out to be different. To describe the movement of a body, it is necessary to indicate in relation to which body the movement is considered. This body is calledreference body . Rest is also relative (examples: a passenger in a train at rest looks at a train passing by)
the main task mechanics – be able to calculate the coordinates of body points at any time.
To solve this, you need to have a body from which the coordinates are counted, associate a coordinate system with it and have a device for measuring time intervals.
The coordinate system, the body of reference with which it is associated, and the instrument for measuring time form reference system , relative to which the motion of the body is considered.
Coordinate systems there are:
1. one-dimensional – the position of the body on the straight line is determined by one coordinate x.
2. two-dimensional – the position of a point on the plane is determined by two coordinates x and y.
3. three-dimensional – the position of a point in space is determined by three coordinates x, y and z.
Every body has a certain size. Different parts of the body are in different places in space. However, in many problems of mechanics there is no need to indicate the positions of individual parts of the body. If the dimensions of the body are small compared to the distances to other bodies, then this body can be considered its material point. This can be done, for example, when studying the motion of planets around the Sun.
If all parts of the body move in the same way, then such a movement is called translational.
Progressively moving, for example, cabins in the attraction "Giant Wheel", a car on straight section paths, etc. With the translational motion of the body, it can also be considered as a material point.
material pointa body is called, the dimensions of which, under given conditions, can be neglected .
concept material point plays an important role in mechanics. A body can be considered as a material point if its dimensions are small compared to the distance it travels, or compared to the distance from it to other bodies.
Example . Dimensions orbital station, which is in orbit near the Earth, can be ignored, and when calculating the trajectory of the spacecraft during docking with the station, one cannot do without taking into account its size.
Characteristics of mechanical movement: movement, speed, acceleration.
Mechanical movement is characterized by three physical quantities: displacement, speed and acceleration.
Moving over time from one point to another, the body (material point) describes a certain line, which is called the trajectory of the body.
The line along which the point of the body moves is called trajectory of movement.
The length of the trajectory is called traveled way.
Denotedl, measured inmeters . (trajectory - trace, path - distance)
Distance traveled l equal to length arcs of the trajectory traversed by the body in some time t.Path – scalar .
By moving the body called a directed segment of a straight line connecting the initial position of the body with its subsequent position. Displacement is a vector quantity.
The vector connecting the start and end points of the trajectory is called movement.
DenotedS , measured in meters. (displacement is a vector, displacement modulus is a scalar)
Speed - a vector physical quantity that characterizes the speed of movement of a body, numerically equal to the ratio of movement in a small period of time to the value of this period.
Denoted v
Speed formula:
or
Unit of measurement in SI -m/s .
In practice, the speed unit used is km/h (36 km/h = 10 m/s).
Measure the speedspeedometer .
Acceleration - vector physical quantity characterizing the rate of change of speed, numerically equal to the ratio of the change in speed to the period of time during which this change occurred.
If the speed changes the same throughout the entire time of movement, then the acceleration can be calculated by the formula:
Acceleration is measuredaccelerometer
SI unitm/s 2
Thus, the main physical quantities in the kinematics of a material point are the distance traveledl, displacement, speed and acceleration. Pathl is a scalar value. Displacement, speed and acceleration are vector quantities. To specify a vector quantity, you need to specify its modulus and specify the direction. Vector quantities obey certain mathematical rules. Vectors can be projected onto coordinate axes, they can be added, subtracted, etc.
Relativity of mechanical motion.
Mechanical movement is relative. The motion of the same body relative to different bodies turns out to be different.
For example, a car is moving on a road. There are people in the car. People move along with the car on the road. That is, people move in space relative to the road. But relative to the car itself, people do not move. This manifests itself.
To describe the movement of a body, it is necessary to indicate in relation to which body the movement is considered. This body is called the reference body. Peace is also relative. For example, a passenger on a train at rest looks at a passing train and does not realize which train is moving until they look at the sky or the ground.
All bodies in the universe are moving, so there are no bodies that are in absolute rest. For the same reason, it is possible to determine whether a body is moving or not only relative to some other body.
For example, a car is moving on a road. The road is on the planet Earth. The road is motionless. Therefore, it is possible to measure the speed of a vehicle relative to a stationary road. But the road is stationary relative to the Earth. However, the Earth itself revolves around the Sun. Therefore, the road, along with the car, also revolves around the Sun. Consequently, the car performs not only translational motion, but also rotational (relative to the Sun). But relative to the Earth, the car makes only translational motion. This manifests itselfrelativity of mechanical motion .
The motion of the same body may look different from the point of view of different observers. The speed, direction of movement and the type of body trajectory will be different for different observers. Without specifying the reference body, talking about motion is meaningless. For example, a seated passenger in a train is at rest relative to the carriage, but moves with the carriage relative to the station platform.
Let us now illustrate for different observers the difference in the form of the trajectory of a moving body. Being on Earth, in the night sky you can easily see bright fast-flying dots - satellites. They move in circular orbits around the Earth, that is, around us. Let's sit down now spaceship flying towards the sun. We will see that now each satellite moves not in a circle around the Earth, but in a spiral around the Sun:
Relativity of mechanical motion – this is the dependence of the trajectory of the body, the distance traveled, displacement and speed on the choice reference systems .
The motion of bodies can be described in different frames of reference. From the point of view of kinematics, all frames of reference are equal. However, the kinematic characteristics of motion, such as trajectory, displacement, speed, turn out to be different in different systems. The quantities that depend on the choice of the reference frame in which they are measured are called relative.
Galileo showed that under the conditions of the Earth it is practically validlaw of inertia. According to this law, the action of forces on a body is manifested in changes in speed; to maintain the same movement with a constant magnitude and direction of speed does not require the presence of forces.Frames of reference in which the law of inertia is satisfied, began to be called inertial reference systems (ISO) .
Systems that rotate or accelerate are non-inertial.
The earth cannot be considered completely ISO: it rotates, but for most of our purposesreference systems associated with the Earth, in a fairly good approximation, can be taken as inertial. A reference frame moving uniformly and rectilinearly relative to the IFR is also inertial.
G. Galileo and I. Newton were deeply aware of what we call todaythe principle of relativity , Whereby mechanical laws physicists must be the same in all ISOs under the same initial conditions.
From this it follows: no ISO is no different from another frame of reference. All ISOs are equivalent in terms of mechanical phenomena.
Galileo's principle of relativity comes from some assumptions that are based on our daily experience. In classical mechanicsspace andtime consideredabsolute . It is assumed that the length of the bodies is the same in any frame of reference and that time flows in the same way in different frames of reference. It is assumed thatweight body and alsoall forces remain unchanged when moving from one ISO to another.
We are convinced of the validity of the principle of relativity by everyday experience, for example, in a uniformly moving train or plane, bodies move in the same way as on Earth.
There is no experiment that can be used to establish which frame of reference is really at rest and which is moving. There are no frames of reference in a state of absolute rest.
If a coin is tossed vertically upwards on a moving cart, then only the coordinate of the OS will change in the frame of reference associated with the cart.
In the reference system associated with the Earth, the coordinates of the OU and OX change.
Consequently, the position of bodies and their velocities in different frames of reference are different.
Consider the motion of the same body with respect to two different frames of reference: stationary and moving.
A boat crosses a river perpendicular to the flow of the river, moving at a certain speed relative to the water. The movement of the boat is monitored by 2 observers: one motionless on the shore, the other on a raft floating downstream. Relative to the water, the raft is motionless, and relative to the shore, it moves at the speed of the current.
Associate a coordinate system with each observer.
X0Y is a fixed coordinate system.
X'0'Y' – moving coordinate system.
S is the displacement of the boat relative to the fixed CO.
S 1 – movement of the boat relative to the mobile CO
S 2 – movement of the moving frame of reference relative to the fixed reference frame.
According to the law of vector addition
We get the speed by dividing S by t:
v is the speed of the body relative to the stationary CO
v 1 - the speed of the body relative to the mobile CO
v 2 is the speed of the moving reference frame relative to the fixed reference frame
This formula expressesclassical law of addition of velocities: the speed of the body relative to the stationary CO is equal to geometric sum velocity of the body relative to the mobile CO and the speed of the mobile CO relative to the stationary CO.
In scalar form, the formula will look like:
This formula was first obtained by Galileo.
Galileo's principle of relativity : all inertial frames of reference are equal; the course of time, mass, acceleration and force are written in the same way .
At the very beginning of the study of mechanical motion, it was emphasized relative nature. Motion can be considered in different frames of reference. The specific choice of reference system is dictated by considerations of convenience: it should be chosen so that the studied movement and its laws look as simple as possible.
Movement in different frames of reference. To move from one frame of reference to another, it is necessary to know which characteristics of motion remain unchanged, and which ones change during such a transition, and in what way.
Let's start with time. Experience shows that, as long as we are talking about movements occurring at speeds that are small compared to the speed of light, time "flows" the same way in all frames of reference and in this sense can be considered absolute. This means that the time interval between two events is the same when measured in any frame of reference.
Let's move on to spatial characteristics. The position of the particle, determined by its radius-vector, changes when moving to another frame of reference. However, the relative spatial arrangement of the two events does not change and in this sense is absolute. For example, the relative position of two particles at any one moment of time, given by the difference in their radius-vectors, the spatial dimensions do not depend on the choice of the reference system solids etc.
Thus, according to the classical concepts of non-relativistic physics, time intervals and spatial distances between simultaneous events are absolute. These ideas, as it turned out after the creation of the theory of relativity, are valid only for relatively slow motions of reference systems. In the theory of relativity, ideas about space and time have undergone significant changes. However, the new relativistic concepts, which have replaced the classical ones, pass into them in the limiting case of slow motions.
Let us now consider the change in the speed of the particle in the transition from one frame of reference to another, moving relative to the first one. This question is closely related to the principle of independence of displacements discussed in § 5. Let us return to the example with
ferry crossing the fiord, when the ferry moves forward relative to the coast. Let's denote the vector of the passenger's movement relative to the coast (i.e., in the reference frame associated with the earth) through and its movement relative to the ferry (i.e., in the reference frame associated with the ferry) - through through Then
Dividing this equality term by term by the time during which these movements occurred, and passing to the limit at , we obtain a ratio similar to (1) for velocities:
where is the speed of the passenger relative to the ground, V is the speed of the ferry relative to the ground, the speed of the passenger relative to the ferry. Expressed by equality (2), the rule for adding velocities with the simultaneous participation of the body in two movements can be interpreted as the law of the transformation of the body's speed in the transition from one reporting system to another. Indeed, and are the speeds of the passenger in two different frames of reference, the speed of one of these frames (the ferry) relative to the other (the ground).
Thus, the speed of a body in any frame of reference is equal to the vector sum of the speed of this body in another frame of reference and the speed V of this second frame of reference relative to the first. We note that the velocity transformation law expressed by formula (2) is valid only for relatively slow (nonrelativistic) motions, since its derivation was based on the idea of the absolute character of time intervals (the value was considered to be the same in two frames of reference).
Relative speed and acceleration. It follows from formula (2) that the relative velocity of two particles is the same in all frames of reference. Indeed, in the transition to a new reference frame, the same reference frame velocity vector V is added to the velocity of each of the particles. Therefore, the difference in the particle velocity vectors does not change in this case. The relative velocity of particles is absolute!
The acceleration of a particle generally depends on the frame of reference in which its motion is considered. However, the acceleration in two frames of reference is the same when one of them moves uniformly and rectilinearly relative to the other. This immediately follows from formula (2) for
When studying specific movements or solving problems, you can use any frame of reference. A reasonable choice of reference system can greatly facilitate obtaining the necessary
result. In the examples of the study of motions considered so far, this issue has not been sharpened - the choice of the reference system was, as it were, imposed by the very condition of the problem. However, in all cases, even when the choice of a frame of reference is obvious at first glance, it is useful to think about which frame of reference will actually turn out to be optimal. We illustrate this in the following tasks.
Tasks
1. Downstream and upstream. A boat is moving downstream at a constant speed. At some point, a spare oar falls into the water from the boat. After a time of mines, the loss is detected and the boat turns back. What is the speed of the river if the oar was picked up at a distance of km downstream from the place of loss?
Solution. Let us choose a frame of reference associated with moving water. In this frame of reference, the water is stationary and the oar lies all the time in the place where it fell. The boat first moves away from this place for a period of time then turns back. The return trip to the oar will take the same time, since the speed of the boat through the water does not depend on the direction of movement. During all this time, the current carries the oar a distance relative to the coast. Therefore, the flow rate of mines
To see how a good choice of reference system makes it easier to get an answer to the question here, solve this problem in the reference system associated with the earth.
Let us pay attention to the fact that the above solution does not change if the boat floats along a wide river not downstream, but at a certain angle to it: in the frame of reference associated with moving water, everything happens like in a lake where the water is still. It is easy to figure out that on the way back, the bow of the boat should be pointed directly at the floating oar, and not at the place where it was dropped into the water.
Rice. 58. Car traffic on intersecting roads
2. Crossroads. Two car roads intersect at right angles (Fig. 58). Car A moving along one of them at a speed is at a distance from the intersection at the moment when it is crossed by car B moving at a speed along another road. At what point in time will the distance between the cars in a straight line be the minimum? What does it equal? Where are the cars at this moment?
Solution. In this problem, it is convenient to associate the frame of reference with one of the cars, for example, with the second one. In such a reference frame, the second car is stationary, and the speed of the first is equal to its speed relative to the second, i.e., the difference (Fig. 59):
The movement of the first car relative to the second occurs in a straight line directed along the vector V,. Therefore, the required shortest distance between cars is equal to the length of the perpendicular dropped from point B to the straight line. Considering similar triangles in Fig. 59, we have
The time it takes the cars to reach this distance can be found by dividing the leg length by the speed of the first car relative to the second:
Rice. 59. Velocities in the reference frame associated with one of the cars
The positions of the cars at this moment of time can be found by realizing that in the initial reference frame associated with the earth, the second car will leave the intersection at a distance equal to
The first car in this time will approach the intersection at a distance
3. Oncoming trains. Two trains of the same length move towards each other along parallel tracks with the same speed. At the moment when the cabs of the locomotives are level with each other, one of the trains starts to slow down and moves on with constant acceleration. He stops after a while just at the moment when the tails of the trains caught up. Find the length of the train.
Solution. Associate a reference frame with a uniformly moving train. In this system, it is stationary, and the oncoming train at the initial moment has a speed. The movement of the second train will be uniformly slowed down in this frame of reference. That's why average speed movement of the braking train is equal to The distance traveled during the braking time (relative to the first train) is equal to the total length of the two trains, i.e. 21. Therefore
where do we find
Let us pay attention to the fact that in this problem the transition to a moving frame of reference was used to consider the non-uniform motion of the body, but the motion of the frame of reference itself was uniform. Next tasks
show that it is sometimes convenient to switch to an accelerated frame of reference.
4. "The hunter and the monkey." When shooting at a horizontally moving target, an experienced hunter aims with some “lead”, because during the flight of the shot the target has time to move a certain distance. Where should he aim when shooting at a freely falling target, if the shot is fired simultaneously with the start of its fall?
Solution. Let us choose a frame of reference associated with a freely falling target. In this reference frame, the target is stationary, and the pellets fly uniformly and in a straight line with the speed acquired at the time of the shot. This happens because the free fall of all bodies in the reference frame associated with the earth occurs with the same acceleration.
In a frame of reference freely falling with acceleration where the target is stationary and the pellets fly in a straight line, it becomes obvious that you need to aim exactly at the target. This fact does not depend on the value initial speed shots - it can be any. But if the initial speed is too low, the pellets may simply not have time to reach the target while it is in free fall. If the target falls from a height , and the initial distance to it in a straight line is equal, then, as is easy to see, the inequality must be satisfied
from which the restriction on the initial speed of the pellets is obtained:
With a lower initial velocity, the pellets will fall to the ground before the target.
5. Limit of achievable goals. In the previous paragraph, the boundary of the area to be penetrated was found for set value initial velocity All reasoning was carried out in the frame of reference associated with the Earth. Find this boundary by considering motion in a freely falling frame of reference. which "falls" with the acceleration of free fall Its equation has the form
In fact, this is the equation of a whole family of circles: by giving different meanings, we obtain the circles on which the particles are located at different times. The desired boundary is the envelope of such a family of circles (Fig. 60). It is obvious that its highest point lies above the particle emission point.
We will look for the boundary in the following way. Note that the particles emitted at the same instant of time reach the boundary at different instants of time: the boundary touches different circles.
Rice. 60. The boundary of achievable goals as an envelope of a family of circles
Drawing a horizontal line at a certain level y, we find on it the point farthest from the y-axis, which the particles still reach, without thinking about which circle this point belongs to. The abscissa x of this point obviously satisfies the equation (3) of the family of circles. Rewriting it in the form
Which of the kinematic quantities change when moving from one frame of reference to another, and which ones remain unchanged?
Explain why the relative speed of two particles is the same in all frames of reference.
Give arguments indicating that the classical law of speed transformation in the transition from one frame of reference to another is based on the idea of the absolute nature of time.
What should be relative motion two frames of reference, so that when passing from one of them to the other, the acceleration of the particle changes?
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