Many students, studying mathematics, come across averages: arithmetic average, geometric average, etc. In physics, the concept of average is quite common. For example, the concept of average ground speed. Let's take a closer look at this value and learn how to solve problems.

Imagine two treadmills each long. There are two athletes at the start. On command, the athletes begin to run along the tracks. But they run differently. Athlete number 1 runs all the time at a constant speed, and he covers this distance in time

.

Now consider the movement of another athlete. Athlete No. 2 started at the same time as Athlete No. 1. After running some distance at a speed of , he suddenly stumbled and fell. For a while, this athlete stood up (and athlete #1 runs at speed ), and then athlete #2 continued to run at speed. Having overcome some distance, athlete No. 2 noticed that his lace was untied. He stopped and began to lace up his shoes (and athlete number 1 is still running at speed ). After a forced stop, athlete No. 2 ran at a speed of , and both athletes crossed the finish line at the same time. And in this case, we will assume that athlete No. 2 moved uniformly in all sections of the treadmill, i.e., the acceleration and deceleration time is negligible compared to the movement time.

And now we come to the most important thing. If you are asked to find the average ground speed of athlete No. 2, then you will have to divide the distance traveled by athlete No. 2 by the time of movement of this athlete (the time of movement of both athletes is the same, since they started and finished at the same time). And get the speed

i.e. it turns out that the average ground speed of athlete No. 2 is equal to the speed of movement of athlete No. 1. Therefore, if you need to find the average ground speed of a moving body, you just need to

divide the distance traveled by the body by the time of movement (including the time of stops) for which this distance was covered! And nothing more!!! And it doesn't matter to you how this body moved: evenly, or accelerated and slowed down, or was motionless for some time, and then it started moving. You just divide the distance by the time,

We will agree to denote the average speed with angle brackets.

Let's go back to the example of athletes. When you need to find the average ground speed of athlete #2, this means that you need to find the speed of uniform movement at which athlete #2 would run the distance in time . And this is the speed of movement of athlete No. 1.

Let's make one more addition: even if the athletes ran along a curvilinear trajectory, the formula for finding the average ground speed would remain the same.

It remains to find out the following: why in physics they came up with such a physical quantity, which, at first glance, has little to do with reality. The fact is that when a body moves at each moment of time (or at each point of the trajectory) it has a specific speed, this speed is called instantaneous. And in order to give a definition of instantaneous speed, it is precisely what we first need to define what we mean by average speed (to be more precise, we are talking about the average speed of movement). But the conversation will continue about this, but for now our story about the average ground speed of the body is over.

Mechanical movement body is called the change in its position in space relative to other bodies over time. In this case, the bodies interact according to the laws of mechanics.

The section of mechanics that describes the geometric properties of motion without taking into account the causes that cause it is called kinematics.

More generally, motion is any spatial or temporal change in the state of a physical system. For example, we can talk about the motion of a wave in a medium.

Relativity of motion

Relativity - the dependence of the mechanical motion of a body on the frame of reference Without specifying the frame of reference, it makes no sense to talk about motion.

Material point trajectory- a line in three-dimensional space, which is a set of points where a material point was, is or will be when it moves in space. It is essential that the concept of a trajectory has a physical meaning even in the absence of any movement along it. In addition, even in the presence of an object moving along it, the trajectory itself cannot give anything in relation to the causes of movement, that is, about the acting forces.

Way- the length of the section of the trajectory of a material point, passed by it in a certain time.

Speed(often denoted, from English velocity or French vitesse) - a vector physical quantity that characterizes the speed of movement and the direction of movement of a material point in space relative to the selected reference system (for example, angular velocity). The same word can be used to refer to a scalar quantity, more precisely, the modulus of the derivative of the radius vector.

In science, speed is also used in a broad sense, as the rate of change of some quantity (not necessarily the radius vector) depending on another (more often changes in time, but also in space or any other). So, for example, they talk about the rate of temperature change, the rate of a chemical reaction, group velocity, connection rate, angular velocity, etc. The derivative of a function is mathematically characterized.

Speed ​​units

Meter per second, (m/s), SI derived unit

Kilometer per hour, (km/h)

knot (nautical mile per hour)

The Mach number, Mach 1 is equal to the speed of sound in a given medium; Max n is n times faster.

As a unit, depending on the specific environmental conditions, should be additionally determined.

The speed of light in vacuum (denoted c)

In modern mechanics, the movement of a body is divided into types, and there is the following classification of types of body movement:

Translational motion, in which any straight line associated with the body remains parallel to itself when moving

Rotational movement or rotation of a body around its axis, which is considered fixed.

A complex movement of the body, consisting of translational and rotational movements.

Each of these types can be uneven and uniform (with non-constant and constant speed, respectively).

Average speed of uneven movement

Average ground speed is the ratio of the length of the path traveled by the body to the time during which this path was traveled:

Average ground speed, unlike instantaneous speed, is not a vector quantity.

The average speed is equal to the arithmetic mean of the speeds of the body during the movement only if the body moved with these speeds for equal periods of time.

At the same time, if, for example, the car moved half the way at a speed of 180 km/h, and the second half at a speed of 20 km/h, then the average speed would be 36 km/h. In examples like this, the average speed is equal to the harmonic mean of all speeds on separate, equal sections of the path.

Average travel speed

You can also enter the average speed over the movement, which will be a vector equal to the ratio of the movement to the time it took:

The average speed determined in this way can be equal to zero even if the point (body) actually moved (but returned to its original position at the end of the time interval).

If the movement took place in a straight line (and in one direction), then the average ground speed is equal to the modulus of the average speed for movement.

Rectilinear uniform motion- this is a movement in which a body (point) makes the same movements for any equal intervals of time. The point's velocity vector remains unchanged, and its displacement is the product of the velocity vector and time:

If you direct the coordinate axis along the straight line along which the point moves, then the dependence of the point coordinate on time is linear: , where is the initial coordinate of the point, is the projection of the velocity vector onto the x coordinate axis.

A point considered in an inertial frame of reference is in a state of uniform rectilinear motion if the resultant of all forces applied to the point is zero.

rotational movement- a type of mechanical movement. During the rotational motion of an absolutely rigid body, its points describe circles located in parallel planes. The centers of all circles lie in this case on one straight line, perpendicular to the planes of the circles and called the axis of rotation. The axis of rotation can be located inside the body and outside it. The axis of rotation in a given reference system can be either movable or fixed. For example, in the reference frame associated with the Earth, the axis of rotation of the generator rotor at the power plant is stationary.

Characteristics of body rotation

With uniform rotation (N revolutions per second),

Rotation frequency- the number of revolutions of the body per unit time,

Rotation period- the time of one complete revolution. The rotation period T and its frequency v are related by the relation T = 1 / v.

Line speed a point located at a distance R from the axis of rotation

,
Angular velocity body rotation.

Kinetic energy rotary motion

Where Iz- the moment of inertia of the body about the axis of rotation. w is the angular velocity.

Harmonic oscillator(in classical mechanics) is a system that, when displaced from an equilibrium position, experiences a restoring force proportional to the displacement.

If the restoring force is the only force acting on the system, then the system is called a simple or conservative harmonic oscillator. Free oscillations of such a system represent a periodic movement around the equilibrium position (harmonic oscillations). The frequency and amplitude are constant, and the frequency does not depend on the amplitude.

If there is also a friction force (damping) proportional to the speed of movement (viscous friction), then such a system is called a damped or dissipative oscillator. If the friction is not too great, then the system performs an almost periodic motion - sinusoidal oscillations with a constant frequency and an exponentially decreasing amplitude. The frequency of free oscillations of a damped oscillator turns out to be somewhat lower than that of a similar oscillator without friction.

If the oscillator is left to itself, then it is said that it performs free oscillations. If there is an external force (depending on time), then we say that the oscillator experiences forced oscillations.

Mechanical examples of a harmonic oscillator are a mathematical pendulum (with small displacement angles), a weight on a spring, a torsion pendulum, and acoustic systems. Among other analogues of the harmonic oscillator, it is worth highlighting the electric harmonic oscillator (see LC circuit).

Sound, in a broad sense - elastic waves propagating longitudinally in a medium and creating mechanical vibrations in it; in a narrow sense - the subjective perception of these vibrations by special sense organs of animals or humans.

Like any wave, sound is characterized by amplitude and frequency spectrum. Usually, a person hears sounds transmitted through the air in the frequency range from 16 Hz to 20 kHz. Sound below the human hearing range is called infrasound; higher: up to 1 GHz - by ultrasound, more than 1 GHz - by hypersound. Among the audible sounds, phonetic, speech sounds and phonemes (of which oral speech consists) and musical sounds (of which music consists) should also be highlighted.

Physical parameters of sound

Oscillatory speed- a value equal to the product of the oscillation amplitude BUT particles of the medium through which a periodic sound wave passes, by the angular frequency w:

where B is the adiabatic compressibility of the medium; p is the density.

Like light waves, sound waves can also be reflected, refracted, and so on.

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The position of a body (material point) in space can be determined only in relation to other bodies.

The system of fixed bodies (their number must match the dimension of space), with which the coordinate system is rigidly connected, equipped with a clock and used to determine the position in space of bodies and particles at different points in time, is called reference system (CO)

The most common coordinate system is a rectangular Cartesian coordinate system.

The position of an arbitrary point M is characterized by a radius vector drawn from the origin 0 to the point M.

kinematic law or the kinematic equation of motion is the dependence:

.

Vector can be expanded in the basis , ,Cartesian coordinate system:

.

Vector , ,-unit orthogonal vectors (orts): , ,=1

The movement of a point will be completely determined if three continuous and single-valued functions of time are given:

x = x(t); y = y(t); z = z(t).

These equations of motion are also called kinematic equations of motion .

1. 1. 2. Trajectory. Way. Move. Number of degrees of freedom.

A material point in its motion describes a certain line called trajectory . Depending on the shape of the trajectory, rectilinear motion, circular motion and curvilinear motion are distinguished.

	The length of the line section, - the trajectory, between points 1 and 2, is called the path traveled by the particle ( S). The path cannot be negative. Vector drawn from point 1 to point 2 (see Fig. 1.1) is called movement. It is equal to the change in the radius of the point vector over the considered period of time:
Figure 1.1.

When a point moves, its coordinates and radius vector change over time, therefore, to set the law of motion of this point, it is necessary to specify the type of functional dependences on time.

1.1.3. Speed, instantaneous and average speed. Average ground speed.

The speed of movement of the body in space is characterized by speed .

In the case of uniform motion, the value of the speed , which the particle has at each moment of time, can be calculated by dividing the path ( S) for a while ( t).

Consider now the case of non-uniform motion. Let's break the trajectory (see Fig. 1.2) into infinitely small segments of length  S.

Each of the sections is associated with an infinitesimal increment
. Let at the moment t material point M is in a position that is described by the radius vector
.

Some time later  t she will move to M 1 with radius vector .

t get the average speed.

Because
is a function, then by the definition of the derivative

Middle track speed
is called a scalar value equal to the ratio of the length ∆S of the trajectory segment to the duration ∆t of its passage by the point:
.

With curvilinear motion
. Therefore, in general, the average ground speed
not equal to the modulus of average speed
. Here, the equal sign corresponds to the rectilinear section of the trajectory.

The unit of speed measurement is 1 m/s.

Velocity vector decomposition in the basis of a rectangular Cartesian coordinate system has the form:

Example

Example: A material point moves according to the law. Determine the law of change of its speed. Solution: We have

Uneven movement is considered to be a movement with a changing speed. The speed can change direction. It can be concluded that any movement NOT along a straight path is non-uniform. For example, the movement of a body in a circle, the movement of a body thrown into the distance, etc.

The speed can vary by numerical value. This movement will also be uneven. A special case of such motion is uniformly accelerated motion.

Sometimes there is an uneven movement, which consists of alternating different types of movements, for example, at first the bus accelerates (the movement is uniformly accelerated), then it moves evenly for some time, and then stops.

Instant Speed

It is possible to characterize uneven movement only by speed. But the speed is always changing! Therefore, we can only talk about the speed at a given instant of time. When traveling by car, the speedometer shows you the instantaneous speed of movement every second. But in this case, the time should be reduced not to a second, but to consider a much smaller period of time!

average speed

What is average speed? It is wrong to think that it is necessary to add up all the instantaneous speeds and divide by their number. This is the most common misconception about average speed! The average speed is all the way divided by the elapsed time. And it is not defined in any other way. If we consider the movement of the car, we can estimate its average speeds in the first half of the way, in the second, all the way. The average speeds may be the same, or they may be different in these sections.

At average values, a horizontal line is drawn on top.

Average movement speed. Average ground speed

If the movement of the body is not rectilinear, then the path traveled by the body will be greater than its displacement. In this case, the average travel speed is different from the average ground speed. Ground speed is a scalar.

The main thing to remember

1) Definition and types of uneven movement;
2) The difference between the average and instantaneous speeds;
3) The rule for finding the average speed of movement

Often you need to solve a problem where the entire path is divided into equal sections, average speeds are given for each section, it is required to find the average speed for the entire path. The wrong decision will be if you add up the average speeds and divide by their number. Below is a formula that can be used to solve such problems.

The instantaneous speed can be determined using the motion graph. The instantaneous velocity of a body at any point on the graph is determined by the slope of the tangent to the curve at the corresponding point. Instantaneous speed - the tangent of the slope of the tangent to the graph of the function.

Exercises

While driving a car, speedometer readings were taken every minute. Is it possible to determine the average speed of the car from these data?

It is impossible, since in the general case the value of the average speed is not equal to the arithmetic mean value of the instantaneous speeds. But the path and time are not given.

What is the speed of the alternating motion shown by the car's speedometer?

close to instantaneous. Close, since the time interval should be infinitely small, and when taking readings from the speedometer, it is impossible to judge time in this way.

In what case are the instantaneous and average speeds equal to each other? Why?

With uniform motion. Because the speed does not change.

The speed of the hammer on impact is 8m/s. What is the speed: average or instantaneous?

This article is about how to find the average speed. The definition of this concept is given, and two important particular cases of finding the average speed are considered. A detailed analysis of tasks for finding the average speed of a body from a tutor in mathematics and physics is presented.

Determination of average speed

medium speed the movement of the body is called the ratio of the path traveled by the body to the time during which the body moved:

Let's learn how to find it on the example of the following problem:

Please note that in this case this value did not coincide with the arithmetic mean of the speeds and , which is equal to:
m/s.

Special cases of finding the average speed

1. Two identical sections of the path. Let the body move the first half of the way with the speed , and the second half of the way — with the speed . It is required to find the average speed of the body.

2. Two identical movement intervals. Let the body move at a speed for a certain period of time, and then began to move at a speed for the same period of time. It is required to find the average speed of the body.

Here we got the only case when the average speed of movement coincided with the arithmetic average speeds and on two sections of the path.

Finally, let's solve the problem from the All-Russian Olympiad for schoolchildren in physics, which took place last year, which is related to the topic of our today's lesson.

The body moved with, and the average speed of movement was 4 m/s. It is known that for the last few seconds the average velocity of the same body was 10 m/s. Determine the average speed of the body for the first s of movement.

The distance traveled by the body is: m. You can also find the path that the body has traveled for the last since its movement: m. Then for the first since its movement, the body has overcome the path in m. Therefore, the average speed on this section of the path was:
m/s.

They like to offer tasks for finding the average speed of movement at the Unified State Examination and the OGE in physics, entrance exams, and olympiads. Every student should learn how to solve these problems if he plans to continue his education at the university. A knowledgeable friend, a school teacher or a tutor in mathematics and physics can help to cope with this task. Good luck with your physics studies!

Sergey Valerievich