Lecture. Relativity of mechanical motion

DEFINITION

Mechanical movement call the change in the position of a body in space over time relative to other bodies.

Based on the definition, the fact of motion of a body can be established by comparing its positions at successive moments of time with the position of another body, which is called the body of reference.

Thus, observing clouds floating across the sky, we can say that they change their position relative to the Earth. A ball that rolls on a table changes its position relative to the table. In a moving tank, the tracks move both relative to the Earth and relative to the tank body. A residential building is at rest relative to the Earth, but changes its position relative to the Sun.

The considered examples allow us to draw an important conclusion that the same body can simultaneously perform different movements relative to other bodies.

Types of mechanical movement

The simplest types of mechanical motion of a body of finite dimensions are translational and rotational motions.

The movement is called translational if the straight line connecting two points of the body moves while remaining parallel to itself (Fig. 1, a). During translational motion, all points of the body move equally.

During rotational motion, all points of the body describe circles located in parallel planes. The centers of all circles lie on the same straight line, which is called the axis of rotation. Points of the body lying on the axis of the circle remain motionless. The axis of rotation can be located both inside the body (rotational rotation) (Fig. 1, b) and outside it (orbital rotation) (Fig. 1, c).

Examples of mechanical motion of bodies

A car moves progressively on a straight section of the road, while the wheels of the car perform a rotational rotational motion. The Earth, revolving around the Sun, performs a rotational orbital motion, and rotating around its axis - a rotational rotational motion. In nature we usually encounter complex combinations various types movements. Thus, a soccer ball flying into a goal simultaneously undergoes translational and rotational motion. Complex movements are performed by parts of various mechanisms, celestial bodies, etc.

TICKET No. 1

Mechanical movement. Relativity of motion. Reference system. Material point. Trajectory. Path and movement. Instant speed. Acceleration. Uniform and uniformly accelerated movement.

The mechanical movement of a body is the change in its position in space relative to other bodies over time.

The trajectory of the body, the distance traveled and the displacement depend on the choice of the reference system. In other words, mechanical motion is relative. The coordinate system, the reference body with which it is associated, and the indication of the origin of time form a reference system.

A body whose dimensions can be neglected under given motion conditions is called material point.

The line along which a point of the body moves is called the trajectory of movement. The length of the trajectory is called the distance traveled.

The vector connecting the starting and ending points of the trajectory is called displacement.

The instantaneous speed of translational motion of a body at time t is the ratio of a very small movement S to the small period of time during which this movement occurred:

υ=S/t υ =1 m/1 s=1 m/s

Movement with a constant speed in magnitude and direction is called uniform rectilinear movement.

When the speed of a body changes, the concept of acceleration of the body is introduced.

Acceleration is a vector quantity equal to the ratio of a very small change in the velocity vector to the small period of time during which this change occurred:

a= υ /t a=1 m/s 2

Motion with acceleration that is constant in magnitude and direction is called uniformly accelerated:

With what force does a magnetic field with B=1.5 T act on a conductor with a length of l=0.03 m, located perpendicular to the magnetic field? Current I=2 A

	=90 0 Sin90 0 =1
	F=2*1.5*3*10 -2 =9*10 -2 H

TICKET No. 2

Interaction of bodies. Strength. Newton's second law.

The reason for a change in the speed of movement of a body is always its interaction with other bodies. After turning off the engine, the car gradually slows down and stops. The main reason for changes in vehicle speed is the interaction of its wheels with the road surface. In physics, the concept of “force” is introduced to quantitatively express the action of one body on another. Examples of forces:
forces of elasticity, gravity, gravity, etc.

Force is a vector quantity, it is denoted by the symbol F. The direction of the force vector is taken to be the direction of the acceleration vector of the body on which the force acts. In the SI system:

F=1 H=1 kg*m/s 2

Newton's 2nd law:

The force acting on a body is equal to the product of the mass of the body and the acceleration imparted by this force:

The meaning of the law is that the force acting on a body determines the change in the speed of the body, and not the speed of movement of the body.

Laboratory work “Measuring the refractive index of glass”

TICKET No. 3

Body impulse. Law of conservation of momentum. Manifestation of the law of conservation of momentum in nature and its use in technology.

There is a physical quantity that changes equally for all bodies under the action of the same forces, if the time of action of the force is the same.

Magnitude, equal to the product the mass of a body on the speed of its movement is called the momentum of the body or momentum.

The change in the momentum of the body is equal to the impulse of the force causing this change.

A physical quantity equal to the product of force F by the time t of its action is called impulse of force.

The momentum of a body is a quantitative characteristic of the translational motion of bodies. The unit of measurement of body impulse is: kg*m/s.

Law of conservation of momentum:

In a closed system, the geometric sum of the momenta of the bodies remains constant for any interaction of the bodies of this system with each other:

m 1 υ 1 +m 2 υ 2 =m 1 υ 1 I + m 2 υ 2 I

where υ 12, υ 12 I are the velocities of the first and second bodies before and after interaction.

A system of bodies that do not interact with other bodies not included in this system is called a closed system.

The law of conservation of momentum manifests itself in inertial reference systems (i.e., in those in which the body, in the absence of external influences, moves rectilinearly and uniformly). This law is used in technology: jet engine. When fuel burns, gases heated to a high temperature are ejected from the rocket nozzle at speed. The rocket begins to move as a result of this interaction and in accordance with this law.

	M – rocket mass υ – rocket speed m – fuel mass U is the speed of burned and ejected fuel.

A battery with an emf of 6 V and an internal resistance of r = 0.1 Ohm powers an external circuit with R = 11.9 Ohm. How much heat will be released in 10 minutes in the entire circuit?

	Q=I 2 *Z*t, where Z is the total resistance
	Q= 2 *(R+r)*t / (R+r) 2
	Q= 2 *t / (R+r)
	Q=36*600 / 12=1800 J

TICKET No. 4

The law of universal gravitation. Gravity. Body weight. Weightlessness.

Newton proved that the movement and interaction of the planets of the solar system occurs under the influence of a gravitational force directed towards the Sun and decreasing in inverse proportion to the square of the distance from it. All bodies in the Universe mutually attract each other.

Newton called the force of mutual attraction between bodies in the Universe the force of universal gravitation. In 1682, Newton discovered the law of universal gravitation:

All bodies attract each other. The force of universal gravitation is directly proportional to the product of the masses of bodies and inversely proportional to the square of the distance between them:

F=G*m 1 *m 2 / R 2

G is the gravitational constant.

The force of attraction exerted by the Earth on all bodies is called gravity:

This force decreases in inverse proportion to the square of the distance from the center of the Earth.

In technology and everyday life, the concept of body weight is widely used - P

The weight of a body is the force with which the body, due to its attraction to the Earth, acts on a horizontal support or suspension.

Body weight on a stationary or uniformly moving horizontal support equal to force gravity, but they are applied to different bodies.

During accelerated motion, the weight of a body, the direction of acceleration of which coincides with the direction of acceleration of free fall, less weight body at rest.

If a body, together with a support, falls freely and the acceleration of the body is equal to the acceleration of free fall, and their directions coincide, then the weight of the body disappears. This phenomenon is called weightlessness:

A=g P=0 weightlessness

At what temperature is the internal energy 20 kg. Argon will be 1.25*10 6 J?

TICKET No. 5

Energy conversion during mechanical vibrations. Free and forced vibrations. Resonance.

In nature and technology, a type of mechanical movement occurs - oscillation.

Mechanical vibration is the movement of a body that is repeated exactly or approximately at equal intervals of time.

The forces acting between bodies within a system are called internal. Forces acting from outside the system on the bodies of this system are called external.

Free vibrations are vibrations that occur under the influence of internal forces. Oscillations under the influence of external periodically changing forces are called forced.

When the pendulum deviates from the equilibrium position, its potential energy increases, because the distance from the Earth's surface increases. When moving towards the equilibrium position, the speed of the pendulum increases, its kinetic energy increases due to a decrease in the potential reserve, as a result of a decrease in the distance from the Earth's surface. At equilibrium, kinetic energy is at its maximum and potential energy is at its minimum. After passing the equilibrium position, the kinetic energy is converted into potential energy, the speed of the pendulum decreases and at maximum deviation becomes equal to zero. In this way, a periodic transformation of energy occurs. But because When moving, bodies interact with other bodies, so part of the mechanical energy is converted into internal energy of thermal motion of atoms and molecules. The amplitude of the oscillations will decrease and after some time the pendulum will stop. Free oscillations are always damped.

In a system, when oscillations are excited under the influence of a periodically changing external force, the amplitude, at first, gradually increases. After some time, oscillations are established with a constant amplitude and a period equal to the period of the external force.

The amplitude also depends on the frequency of force changes. Provided that the frequency of the external force ν coincides with the natural frequency of the system ν 0, the amplitude has a maximum value.

Resonance is a sharp increase in the amplitude of forced oscillations as the frequency of change of the external force acting on the system approaches the frequency of free oscillations. The less friction in the system, the more pronounced the resonance (in Fig. Curve No. 1).

Laboratory work “Determination of the focal length of a collecting lens.”

TICKET No. 6

Experimental substantiation of the main provisions of the molecular kinetic theory of the structure of matter. Mass and size of molecules. Avogadro's constant.

At the beginning of the 19th century, the English scientist D. Dalton showed that many natural phenomena can be explained using the molecular structure of matter. By the beginning of the 20th century, the molecular kinetic theory of matter was finally created and confirmed by experiments. Main provisions of the ICT:

substances consist of molecules between which there are intermolecular intervals.

Molecules move continuously and chaotically.

At short distances between molecules and atoms, both attractive and repulsive forces act. The nature of these forces is electromagnetic.

Chaotic motion is also called thermal, because. it depends on temperature.

Experimental justification:

The fact that substances consist of molecules was proven by photographs taken using an electron microscope. The photographs show the arrangement of the molecules.

The fact that molecules are constantly moving is proven by Brown's experiment. In 1827, he observed how grains of clay moved in water. I couldn't explain. Brownian motion is the movement of clay grains caused by the impacts of chaotically moving water molecules. And another natural phenomenon - diffusion, proves the continuous movement of molecules. Diffusion is the phenomenon of penetration of molecules of one substance into the molecules of another substance. Even in solids, where this penetration process occurs most slowly, diffusion is still observed. For example: a gold plate lies on a lead plate. They are under load. After some time, a molecule of each substance will be discovered in the adjacent contacting body.

3. The fact that molecules are attracted to each other is proven by experience with lead cylinders. They can withstand weight up to 5 kg. Diffusion also proves that molecules interact in solids.

Both repulsive and interaction forces act simultaneously between molecules. They are magnetic in nature. During deformations in solid bodies, forces manifest themselves in the form of elastic forces and determine the strength of the bodies. These forces act over very short distances - within the size of molecules. But the effect will be observed if the molecules are brought closer to a distance greater than their stable equilibrium (when the two types of forces are equal in value), then the repulsive forces will increase and the attraction will decrease.

Experimental studies have shown that the molecules are very small. For example: the mass of an olive oil molecule m 0 = 2.5 * 10 -26 kg, and the size of the molecule d = 3 * 10 -10 m.

Avogadro's number is the number of atoms contained in 0.012 kg of the carbon isotope 12 C. Named after the Italian scientist of the 19th century.

N A =6.02*10 23 mol -1

During the electrolysis of a solution of copper sulfate, work was done

A=1.4*10 7 J. Determine the amount of copper released if the voltage between the electrodes of the bath is U=6 V.

K=3.29*10 -7 J
	m=k*A / U m=3.29*10 -7 *1.4*10 7 / 6=4.6 / 6=0.76 kg

TICKET No. 7

Ideal gas. The main MCT equation for an ideal gas. Temperature and its measurement. Absolute temperature.

In real life, when studying phenomena in nature and technology, it is impossible to take into account all the factors influencing it. For this reason, one can take into account most important factor, for example, the movement of molecules, while others (interactions) are not taken into account. On this basis, a model of the phenomenon is introduced.

Gas molecules hitting the surface of a body or the wall of a vessel exert pressure –P. Pressure depends on the following factors:

from kinetic energy molecular movements. The larger it is, the greater the pressure;

number of molecules per unit volume. The more there are, the greater the pressure.

Basic equation ideal gas can be written as a formula:

P=n*m 0 *υ 2 /3 or P=2*n*E/3

Where n is the concentration of molecules per unit volume (n=N/V), m 0 is the mass of one molecule, E is the average value of the kinetic energy of movement of molecules, υ 2 is the average value of the square of the speed of kinetic movement of molecules.

The pressure of an ideal gas is directly proportional to the average kinetic energy of the translational motion of its molecules and the number of molecules per unit volume. Pressure is measured in Pascals P=Pa. Conditions close to an ideal gas are created in vacuum tubes and devices. A vacuum is created there, because gas molecules are a hindrance - the lamp filament will oxidize and burn out instantly.

Temperature is a quantity characterizing the degree of heating of a body. In order to measure body temperature, a device was created - a thermometer. A hydrogen thermometer was chosen as a reference, in which discharged hydrogen was used as a substance. It expands when heated in the same way as oxygen, nitrogen, etc. A closed vessel with discharged hydrogen was connected to a manometer (a device for measuring pressure) and by increasing the temperature, the gas expanded, thereby changing its pressure. Pressure and temperature are related linearly, so the temperature could be determined from the pressure gauge reading. The temperature scale established by a hydrogen thermometer is called the Celsius scale. The melting temperature of ice at normal temperatures is taken as 0 0 C atmospheric pressure, and beyond 100 0 C is the boiling point of water, also at normal pressure 1. Another construction of the temperature scale is also possible. For a deeper understanding of the physical meaning of phenomena, Kelvin proposed another scale - the thermodynamic one. Now it is called the Kelvin scale. It takes –273 0 C as the starting point. This value is called absolute zero - the temperature at which the translational movement of molecules stops. It does not occur in nature below temperatures. Temperature on this scale is called absolute temperature and is measured in Kelvin - TK.

The speed of molecular movement depends on temperature, so temperature is said to be a measure of the kinetic energy of molecular movement. With increasing temperature, the average speed translational movement of molecules.

E=3*k*T/2 P=nkT Where k is Boltzmann’s constant =1.38*10 -23 J/K

An electrical diagram is given. Determine the resistance of four conductors with the same resistance R 1-4 = 4 Ohms, connected to each other according to the diagram:

Conductors 1,4 are connected in series, and 2,3 in parallel.

Let's find the total resistance of conductors 2.3:

R 23 =R / n R 23 = 4 / 2 = 2 Ohm.

Find the total resistance of the entire circuit:

R=R 1 +R 23 +R 4 R=4+2+4=10 Ohm.

TICKET No. 8

Equation of state of an ideal gas (Mendeleev-Clapeyron equation). Isoprocesses.

In real life, when studying phenomena in nature and technology, it is impossible to take into account all the factors influencing it. For this reason, it is possible to take into account the most important factor, such as the movement of molecules, while others (interaction) are not taken into account. On this basis, a model of the phenomenon is introduced.

An ideal gas is a model of a real gas. This is a gas whose molecular sizes are small compared to the volume of the container and they practically do not interact.

Physical quantities, the value of which is determined by the joint action of a huge number of molecules, are called thermodynamic parameters: P, V, T.

An ideal gas is described by the following parameters that are included in the Mendeleev-Clapeyron equation: PV = m*R*T/ M

where M is the molar mass of the substance, R is the universal gas constant, does not depend on the nature of the gas = 8.31 N*m/Kmol*K, m is the mass of the gas.

An isoprocess is a process in which the mass of a gas and one of its parameters remain constant.

Determine the red limit of the photoelectric effect for a metal with work function A = 3.2 * 10 -19 J.

TICKET No. 9

Evaporation and condensation. Saturated and unsaturated pairs. Air humidity. Air humidity measurement.

Substances pass from one state to another. During chaotic movement, some water molecules with high kinetic energy leave it. At the same time, they overcome the forces of attraction from other molecules. This process is called evaporation. (see poster). But another process can also be observed when the vapor molecules return to the liquid, this process is called condensation. If there is an air flow above the vessel, it carries away vapor molecules and the evaporation process occurs faster. The evaporation process also accelerates when the temperature of the liquid increases.

If the vessel is covered with a lid, then after some time a dynamic equilibrium will be established - the number of molecules leaving the liquid = the number of molecules returning to the liquid.

Vapor that is in dynamic equilibrium with its liquid is called saturated. Even if we begin to compress saturated steam at a constant temperature, initially the equilibrium will be disrupted, but then the concentration of steam molecules will level out again, as in dynamic equilibrium.

Saturated vapor pressure P 0 does not depend on volume at constant temperature.

On Earth there is a continuous formation of water vapor: evaporation from water bodies, vegetation, vapor exhaled by animals. But this water vapor is not saturated, because movement occurs air masses in the atmosphere.

Humidity is the amount of water vapor in the Earth's atmosphere.

Water vapor - humidity - is characterized by parameters. (further see the office posters and tell us about them).

Relative humidity can be measured with several instruments, but let's consider one - a psychrometer. (Further about the device and method of measurement, refer to the posters).

Laboratory work “Measuring the wavelength of light using a diffraction grating.”

TICKET No. 10

Crystal ai amorphous bodies. Elastic and plastic deformations of solids.

Crystals surround us everywhere. Solids are all classified as crystals. But because Since single crystals are not found in nature, we do not see them. Most often, substances consist of many interlocking crystalline grains - polycrystals. U crystalline bodies atoms are arranged in a strict order and form a spatial crystal lattice. As a result, they have a regular external shape. Examples of crystalline bodies: table salt, snowflake, mica, graphite, etc. These bodies have certain properties - graphite writes well in layers, salt breaks with flat edges, mica exfoliates in the longitudinal direction. T. ob. they have the same physical properties in one direction - called anisotropy. In reality, most often anisotropy is not observed, because the body consists of a large number of chaotically fused crystals, the total effect of anisotropy leads to the elimination of this phenomenon. But there are other bodies that do not consist of crystals, i.e. they do not have a crystal lattice, they are called amorphous. They have the properties of elastic and liquid bodies. When hit, they prick, and at high temperatures they flow. Examples of amorphous bodies: glass, plastics, resin, rosin, sugar candy. They have the same physical properties in all directions - called. isotropy.

An external mechanical effect on a body causes a displacement of atoms from equilibrium positions and leads to a change in the shape and volume of the body, i.e. to its deformation. The most simple types deformations are stretching and compression. Cables of cranes, cable cars, towing cables, and strings of musical instruments experience tension. The walls and foundations of buildings are subject to compression. Deformation can be characterized by absolute elongation ∆l = l 2 -l 1, where l 1 is before stretching, l 2 is after it. And the ratio of absolute elongation to the length of the sample is called relative elongation: ε=∆l / l 1. When a body deforms, elastic forces arise. Physical quantity, equal to the ratio of the modulus of the elastic force to the cross-sectional area of ​​the body, is called stress σ=F/S. At small deformations, Hooke's law is satisfied, when the deformation increases proportionally with increasing force on the body. But only up to a certain strength limit. If the stress is increased and after its removal the dimensions of the body are still fully restored, then such deformation is called elastic, otherwise it is called residual or plastic.

...); does he read? mechanically"or consciously. Errors, ... requirements) is divided into relatively complete in semantic terms... ; strength movements; volume movements: accuracy movements; smoothness movements; symmetry movements; presence of synkinesis...

In physics, there is such a thing as mechanical motion, the definition of which is interpreted as a change in the coordinates of a body in three-dimensional space relative to other bodies with the loss of time. Oddly enough, you can, for example, exceed the speed of a bus without moving anywhere. This value is relative and dependent on a given point. The main thing is to fix the frame of reference in order to observe the point in relation to the object.

Description

Physics concepts:

1. A material point is a part of a body or an object with small parameters and mass that are not taken into account when studying the process. This is a quantity that is neglected in physics.
2. Displacement is the distance traveled by a material point from one coordinate to another. The concept should not be confused with motion, since in physics it is the definition of a path.
3. The distance traveled is the distance that an object has passed. What is the distance traveled is considered by the section of physics under called "Kinematics".
4. A trajectory in space is a straight or broken line along which an object travels. You can imagine what a trajectory is, according to the definition from the field of physics, by mentally drawing a line.
5. Mechanical is movement along a given path.

Attention! The interaction of bodies is carried out according to the laws of mechanics, and this section is called kinematics.

Understand what a coordinate system is and what a trajectory is in practice?

It is enough to mentally find a point in space and draw coordinate axes from it, the object will move relative to it along a broken or straight line, and the types of movement will also be different, including translational, carried out when oscillating and rotating.

For example, a cat is in a room, moves to any object or changes its location in space, moving along different trajectories.

The distance between objects may vary because the selected paths are not the same.

Types

Known types of movement:

1. Progressive. Characterized by the parallelism of two interconnected points moving equally in space. An object moves forward when it passes along one line. It is enough to imagine replacing the refill in a ballpoint pen, that is, the refill moves forward along a given path, with each part moving parallel and equally. This happens quite often in mechanisms.
2. Rotational. An object describes a circle in all planes that are parallel to each other. The axes of rotation are the centers of the described ones, and the points located on the axis are motionless. The rotating axis itself can be located inside the body (rotational), and also connected to its external points (orbital). To understand what it is, you can take a regular needle and thread. Hold the latter between your fingers and gradually unwind the needle. The needle will trace a circle and similar species movements should be classified as orbital. An example of a rotational view: spinning an object on a hard surface.
3. Oscillatory. All points of a body moving along a given trajectory are repeated with accuracy or approximately through same time. A good example- a puck suspended on a cord, oscillating left and right.

Attention! Features of forward motion. An object moves in a straight line, and at any time interval all its points move in the same direction - this is forward motion. If a bicycle is riding, then at any time you can separately consider the trajectory of its any point, it will be the same. It doesn’t matter whether the surface is flat or not.

These types of movements occur every day in practice, so it won’t be difficult to play them out mentally.

What is relativity

According to the laws of mechanics, an object moves relative to some point.

For example, if a person stands still and a bus moves, this is called the relativity of the movement of the person in question. vehicle to the object.

At what speed does the object move relative to a certain body in space is also taken into account relative to this body and, accordingly, acceleration also has a relative characteristic.

Relativity is a direct dependence of the trajectory specified during the movement of the body, traversable path, speed characteristics, as well as movement in relation to reference systems.

How is the countdown carried out?

What is a reference system and how is it characterized? The reference in relation to the spatial coordinate system, the primary reference to the time of movement - this is the reference system. In different systems, one body may have different locations.

The point is located in the coordinate system; when it begins to move, its movement time is taken into account.

Reference body - is an abstract object located in given point space. When focusing on its position, the coordinates of other bodies are considered. For example, a car stands still and a person moves; in this case, the reference body is a car.

Uniform movement

The concept of uniform motion - this definition in physics is interpreted as follows.

At the very beginning of the study of mechanical motion, it was emphasized relative character. Movement can be considered in different reference systems. The specific choice of a reference system is dictated by considerations of convenience: it should be chosen so that the movement being studied and its patterns look as simple as possible.

Motion in different reference systems. To transition from one reference system to another, it is necessary to know which characteristics of motion remain unchanged, and which ones change during such a transition and how.

Let's start with time. Experience shows that while we're talking about Regarding movements occurring at speeds small compared to the speed of light, time “flows” equally in all reference systems 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 reference system. However, the relative spatial location 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 in time, determined by the difference in their radius vectors and the spatial dimensions, does not depend on the choice of 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 movements of reference systems. In the theory of relativity, ideas about space and time have undergone significant changes. However, the new relativistic concepts that replaced the classical ones transform into them in the limiting case of slow movements.

Let us now consider the change in the speed of motion of a particle when moving from one frame of reference to another, moving relative to the first. This question is closely related to the principle of independence of movements discussed in § 5. Let us return to the example with

crossing a ferry across a fiord, when the ferry moves progressively relative to the shores. Let's denote the vector of the passenger's movement relative to the shores (i.e., in the reference frame associated with the earth) by and its movement relative to the ferry (i.e., in the reference frame associated with the ferry) by via Then

Dividing this equality term by term by the time during which these movements occurred, and passing to the limit at we obtain a similar (1) relationship for speeds:

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. The rule for adding velocities with the simultaneous participation of a body in two movements, expressed by equality (2), can be interpreted as the law for converting the speed of a body when moving from one reporting system to another. In fact, and are the speeds of the passenger in two different reference systems, the speed of one of these systems (the ferry) relative to the other (the earth).

Thus, the speed of a body in any reference system is equal to the vector sum of the speed of this body in another reference system and the speed V of this second reference system relative to the first. Note that the law of velocity transformation expressed by formula (2) is valid only for relatively slow (non-relativistic) movements, since its derivation was based on the idea of ​​the absolute nature of time intervals (the value was considered the same in two reference systems).

Relative speed and acceleration. From formula (2) it follows that relative speed two particles is the same in all reference systems. In fact, when moving to new system reference, the same vector V of the velocity of the reference system is added to the speed of each particle. Therefore, the difference between the particle velocity vectors does not change. The relative speed of particles is absolute!

The acceleration of a particle in the general case 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 motion studies considered so far, this issue was not emphasized - the choice of a reference system was, as it were, imposed by the very conditions of the problem. However, in all cases, even when the choice of reference system is obvious at first glance, it is useful to think about which reference system will actually be optimal. Let us illustrate this with the following problems.

Tasks

1. Downstream and upstream. A motorboat floats downstream at a constant speed. At some point, a spare oar falls from the boat into the water. After a time of minutes, the loss is discovered and the boat turns back. What is the speed of the river flow if the oar was picked up at a distance of km downstream from the place where it was lost?

Solution. Let us choose a reference system associated with moving water. In this frame of reference, the water is motionless and the oar always lies in the place where it fell. The boat first moves away from this place for a while and then turns back. The return journey to the oar will take the same time since the speed of the boat relative to the water does not depend on the direction of movement. During all this time, the current carries the oar a distance relative to the banks. Therefore, the flow speed min

To make sure how much good choice reference system makes it easier to obtain an answer to the question posed here, solve this problem in a reference frame associated with the earth.

Let us note that the above solution does not change if the boat floats along a wide river not downstream, but at some angle to it: in the reference frame associated with moving water, everything happens as in a lake where the water is motionless. 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. Movement of cars on intersecting roads

2. Crossroads. Two highways intersect at right angles (Fig. 58). Car A, moving along one of them at speed, is at a distance from the intersection at the moment when car B, moving at speed along another road, crosses it. At what point in time will the distance between the cars in a straight line be minimal? What is it equal to? Where are the cars at this moment?

Solution. In this problem, it is convenient to associate the reference frame with one of the cars, for example with the second. 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 along the vector V,. Therefore, the desired 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 for cars to approach 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 in time can be found by realizing that in the original reference frame associated with the ground, the second car will move away from the intersection at a distance equal to

During this time, the first car will approach the intersection at a distance

3. Oncoming trains. Two trains of the same length are moving towards each other along parallel tracks at the same speed. At the moment when the cabins of the diesel locomotives are level with each other, one of the trains begins to brake and moves on with constant acceleration. He stops after a while, just at the moment when the tails of the trains reach each other. Find the length of the train.

Solution. Let us associate the reference frame with a uniformly moving train. In this frame it is motionless, and the oncoming train at the initial moment has the speed The movement of the second train and in this frame of reference will be equally slow. Therefore, the average speed 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 it from?

Let us note that in this problem the transition to a moving reference frame was used to consider the non-uniform motion of the body, but the motion of the reference frame itself was uniform. Next tasks

show that sometimes it is convenient to switch to an accelerated moving frame of reference.

4. “The Hunter and the Monkey.” When shooting at a horizontally moving target, an experienced hunter takes aim with some “lead”, since during the flight of the shot the target manages to move a certain distance. Where should he aim when shooting at a free-falling target, if the shot is fired simultaneously with the beginning of its fall?

Solution. Let us choose a reference system associated with a freely falling target. In this frame of reference, the target is stationary, and the pellets fly uniformly and rectilinearly with the speed acquired at the moment 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 that is freely falling with acceleration, where the target is motionless and the pellets fly straight, it becomes obvious that you need to aim exactly at the target. This fact does not depend on the value initial speed pellets - it can be anything. 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 then, as is easy to see, the inequality must be satisfied

hence the limitation on the initial speed of the pellets:

At a lower initial speed, the pellets will fall to the ground before the target.

5. Boundary of achievable goals. In the previous paragraph, the boundary of the area under fire was found for a given value of the initial velocity. All reasoning was carried out in a reference frame associated with the Earth. Find this boundary by considering motion in a freely falling frame. 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: giving different meanings, we obtain the circles on which the particles are located in various moments time. The required boundary is the envelope of such a family of circles (Fig. 60). Obviously, its highest point lies above the point of departure of the particles.

We will look for the boundary as follows. Note that particles emitted at the same moment of time reach the boundary at different times: the boundary touches different circles.

Rice. 60. The boundary of achievable goals as the envelope of a family of circles

Having drawn a horizontal line at a certain level y, we find on it the point that is most distant from the ordinate axis, which the particles still reach, without thinking about which circle this point belongs to. The abscissa x of this point obviously satisfies equation (3) of the family of circles. Rewriting it in the form

Which of the kinematic quantities change when moving from one reference system to another, and which 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 velocity transformation when moving from one reference system to another is based on the idea of ​​the absolute nature of time.

What should be the relative motion of the two reference systems so that when moving from one of them to the other, the acceleration of the particle changes?

I suggest a game: choose an object in the room and describe its location. Do this in such a way that the guesser cannot make a mistake. Did it work out? What will come of the description if other bodies are not used? The following 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, face north and, having walked 120 steps, turn to face east and walk 200 steps. In this place, dig a hole 10 cubits in size and you will find 100 gold bars.” 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 being made is not defined, it is unknown in which village that very house is located. It is necessary to accurately determine the body that will serve as the basis for our future description. In physics such a body is called reference body. It can be selected arbitrarily. For example, try choosing two different reference bodies and describing the location of a computer in a room relative to them. There will be two descriptions that are different from each other.

Coordinate system

Let's look at the picture. Where is the tree in relation 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 located 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 bodies of reference are chosen. Let's consider its location relative to cyclist I.

In this picture there is a tree right from cyclist I

In this picture there is a tree left from cyclist I

The tree and the cyclist did not change their location in space, but the tree can be “on the left” and “on the right” at the same time. In order to get rid of the ambiguity in 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 indicating the positive direction. In our example, we will select and designate two directions. From left to right (the axis along which the cyclist moves), and from us inside the monitor to the tree - this is the second positive direction. If the first direction we have chosen is designated as X, the second - as Y, we obtain a two-dimensional coordinate system.

Relative to us, the cyclist is moving in a negative direction along the X axis, the tree is in a positive direction along the Y axis

Relative to us, the cyclist is moving in the positive direction along the X axis, the tree is in the positive direction along 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 number “2” usually indicates the location along the X axis, the second number “-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 selected, 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 its coordinates will be different (-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, three-dimensional space. In addition to the fact that the body can be “to the right” (“left”), “in front” (“behind”), it can also be “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 cannot change their directions while solving, for example, one problem. Can I choose other axis names? It is possible, but you risk that others will not understand you; it is better not to do this. Is it possible to swap the X axis with the Y axis? You can, but don't get confused about the coordinates: (x;y).

At straight motion of a body, one coordinate axis is enough to determine its position.

To describe movement 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 a 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 the position of a body does not change over time, then it is at rest. If the position of the body changes over time, 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 a method of measuring time - a clock - to the body of reference, the coordinate system that determines the position of the body, we get reference system. With the help of which you can evaluate whether a body is moving or at rest.

Relativity of motion

The astronaut went out into open space. Is it in a state of rest or movement? If we consider it relative to the cosmonaut's friend who is nearby, he will be at rest. And if relative to an observer on Earth, the astronaut is moving at enormous speed. Same with traveling on a train. Regarding the people on the train, you sit motionless 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 the trees), in figure b) the train is at rest relative to the boy.

Sitting in the carriage, we await departure. In the window we watch the train on a parallel track. When it starts to move, it is difficult to determine who is moving - our carriage or the train outside the window. In order to decide, it is necessary to evaluate whether we are moving relative to other stationary objects outside the window. We evaluate the condition of our carriage relative to various systems countdown.

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 (a fixed frame of reference) is different in the first and second cases.

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

Similar to the displacement vector. Rule for adding movements: The displacement of a body relative to a fixed reference system is the vector sum of the displacement of the body relative to a moving reference system and the displacement of a moving reference system relative to a stationary one.

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

Changing trajectory in different reference systems

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

Forward movement

The movement of a body is a change in its position in space relative to other bodies over time. Each body has certain dimensions, 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 indicate the position of every point on the body. Let's consider similar cases. For example, this does not need to be done when all points of the body move the same way.

All the currents of the suitcase and car move the same way.

The movement of a body in which all its points move equally is called progressive

Material point

There is no need 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, their sizes and their own movement are not taken into account. Despite the fact that, for example, the Earth is huge, relative to the distance to 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. It’s as if we concentrate all the substance of the body into a point. We get a model of the body, without dimensions, but it has mass. This is it material point.

The same body with some of its movements can be considered a material point, but with others it cannot. For example, when a boy walks from home to school and at the same time covers a distance of 1 km, then in this movement he can be considered a material point. But when the same boy performs exercises, 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 water (picture on the right), it is impossible to model it to a point, since the movement of the entire 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 specify 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

More difficult situation addition of speeds. Let a man cross a river in a boat. The boat is the body under study. The fixed frame of reference is the earth. The moving frame of reference is the river.

The boat's speed relative to the ground is a vector sum

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

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

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

In the reference frame associated with the Earth, the trajectory of the drop is vertical line. In the frame of reference associated with the train, the movement of a 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 trail of a drop on 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 move a single meter relative to the walls of the hall.