Mechanical movement of a body is 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 reasons that cause it is called kinematics.

In a more general sense, motion is any spatial or temporal change in the state of a physical system. For example, we can talk about the movement of a wave in a medium.

Relativity of motion

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

Trajectory of a material point- a line in three-dimensional space, representing a set of points at which a material point was, is, or will be located when moving in space. It is important that the concept of a trajectory has a physical meaning even in the absence of any movement along it. In addition, even if there is an object moving along it, the trajectory itself cannot give anything regarding the causes of movement, that is, about the acting forces.

Path- the length of the section of the trajectory of a material point traversed by it in a certain time.

Speed(often denoted from the English velocity or French vitesse) is a vector physical quantity that characterizes the speed of movement and 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, or more precisely, the modulus of the derivative of the radius vector.

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

Speed ​​units

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

Kilometer per hour, (km/h)

knot (nautical miles per hour)

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

How the unit depends on specific environmental conditions must be further defined.

The speed of light in a 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 while moving

Rotational motion or rotation of a body around its axis, which is considered stationary.

Complex body movement 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 covered:

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 movement only in the case when the body moved at these speeds for the same 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 will be 36 km/h. In examples like this, the average speed is equal to the harmonic mean of all speeds on individual, equal sections of the path.

Average moving speed

You can also enter the average speed for the movement, which will be a vector equal to the ratio of the movement to the time during which it was completed:

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

If the movement occurred in a straight line (and in one direction), then the average ground speed is equal to the module of the average speed along the movement.

Rectilinear uniform motion- this is a movement in which a body (point) makes identical movements over any equal periods of time. The velocity vector of a point 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’s coordinates 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 reference system is in a state of uniform rectilinear motion if the resultant of all forces applied to the point is equal to zero.

Rotational movement- 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 on the same 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 or outside it. The axis of rotation in a given reference system can be either movable or stationary. For example, in the reference frame associated with the Earth, the axis of rotation of the generator rotor at a power plant is stationary.

Characteristics of body rotation

With uniform rotation (N revolutions per second),

Rotation frequency- number of body revolutions per unit time,

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

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

,
Angular velocity body rotation.

Kinetic energy rotational movement

Where I z- moment of inertia of the body relative to the axis of rotation. w - 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 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 frictional 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 almost periodic motion - sinusoidal oscillations with a constant frequency and 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 its own devices, it is said to oscillate freely. If there is an external force (time-dependent), then the oscillator is said to experience forced oscillations.

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

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

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

Physical parameters of sound

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

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

Like light waves, sound waves can also be reflected, refracted, etc.

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Medium speed tasks (hereinafter referred to as SV). We have already looked at tasks involving linear motion. I recommend looking at the articles "" and "". Typical tasks for average speed are a group of movement problems, they are included in the Unified State Examination in mathematics, and such a task may very likely appear in front of you at the time of the exam itself. The problems are simple and can be solved quickly.

The idea is this: imagine an object of movement, such as a car. He travels certain sections of the path at different speeds. The entire journey takes a certain amount of time. So: average speed is such a constant speed with which a car would cover a given distance in the same time. That is, the formula for average speed is as follows:

If there were two sections of the path, then

If three, then accordingly:

*In the denominator we sum up the time, and in the numerator the distances traveled during the corresponding time intervals.

The car drove the first third of the route at a speed of 90 km/h, the second third at a speed of 60 km/h, and the last third at a speed of 45 km/h. Find the vehicle's IC along the entire route. Give your answer in km/h.

As already said, it is necessary to divide the entire path into the entire time of movement. The condition says about three sections of the path. Formula:

Let us denote the whole by S. Then the car drove the first third of the way:

The car drove the second third of the way:

The car drove the last third of the way:

Thus

Decide for yourself:

The car drove the first third of the route at a speed of 60 km/h, the second third at a speed of 120 km/h, and the last third at a speed of 110 km/h. Find the vehicle's IC along the entire route. Give your answer in km/h.

The car drove for the first hour at a speed of 100 km/h, for the next two hours at a speed of 90 km/h, and then for two hours at a speed of 80 km/h. Find the vehicle's IC along the entire route. Give your answer in km/h.

The condition says about three sections of the path. We will search for the SC using the formula:

The sections of the path are not given to us, but we can easily calculate them:

The first section of the route was 1∙100 = 100 kilometers.

The second section of the route was 2∙90 = 180 kilometers.

The third section of the route was 2∙80 = 160 kilometers.

We calculate the speed:

Decide for yourself:

The car drove at a speed of 50 km/h for the first two hours, at a speed of 100 km/h for the next hour, and at a speed of 75 km/h for two hours. Find the vehicle's IC along the entire route. Give your answer in km/h.

The car drove for the first 120 km at a speed of 60 km/h, for the next 120 km at a speed of 80 km/h, and then for 150 km at a speed of 100 km/h. Find the vehicle's IC along the entire route. Give your answer in km/h.

It is said about three sections of the path. Formula:

The length of the sections is given. Let's determine the time that the car spent on each section: 120/60 hours were spent on the first section, 120/80 hours on the second section, 150/100 hours on the third. We calculate the speed:

Decide for yourself:

The car drove for the first 190 km at a speed of 50 km/h, for the next 180 km at a speed of 90 km/h, and then for 170 km at a speed of 100 km/h. Find the vehicle's IC along the entire route. Give your answer in km/h.

Half the time spent on the road, the car was traveling at a speed of 74 km/h, and the second half of the time at a speed of 66 km/h. Find the vehicle's IC along the entire route. Give your answer in km/h.

*There is a problem about a traveler who crossed the sea. The guys have problems with the solution. If you don't see it, then register on the site! The registration (login) button is located in the MAIN MENU of the site. After registration, log in to the site and refresh this page.

The traveler crossed the sea on a yacht with average speed 17 km/h. He flew back on a sports plane at a speed of 323 km/h. Find the traveler's average speed along the entire journey. Give your answer in km/h.

Sincerely, Alexander.

P.S: I would be grateful if you tell me about the site on social networks.

At school, each of us came across a problem similar to the following. If a car moved part of the way at one speed, and the next part of the road at another, how to find the average speed?

What is this quantity and why is it needed? Let's try to figure this out.

Speed ​​in physics is a quantity that describes the amount of distance traveled per unit of time. That is, when they say that a pedestrian’s speed is 5 km/h, this means that he covers a distance of 5 km in 1 hour.

The formula for finding speed looks like this:
V=S/t, where S is the distance traveled, t is time.

There is no single dimension in this formula, since it describes both extremely slow and very fast processes.

For example, an artificial Earth satellite travels about 8 km in 1 second, and the tectonic plates on which the continents are located, according to scientists’ measurements, diverge by only a few millimeters per year. Therefore, speed dimensions can be different - km/h, m/s, mm/s, etc.

The principle is that the distance is divided by the time required to cover the path. Do not forget about dimensionality if complex calculations are carried out.

In order not to get confused and not make a mistake in the answer, all quantities are given in the same units of measurement. If the length of the path is indicated in kilometers, and some part of it in centimeters, then until we get unity in dimension, we will not know the correct answer.

Constant speed

Description of the formula.

The simplest case in physics is uniform motion. The speed is constant and does not change throughout the entire journey. There are even speed constants tabulated—unchangeable values. For example, sound travels in air at a speed of 340.3 m/s.

And light is the absolute champion in this regard, it has the highest speed in our Universe - 300,000 km/s. These quantities do not change from the starting point of movement to the final point. They depend only on the medium in which they move (air, vacuum, water, etc.).

Uniform movement often occurs to us in everyday life. This is how a conveyor belt works in a plant or factory, a cable car on mountain roads, an elevator (except for very short periods of start and stop).

The graph of such a movement is very simple and represents a straight line. 1 second - 1 m, 2 seconds - 2 m, 100 seconds - 100 m. All points are on the same straight line.

Uneven speed

Unfortunately, it is extremely rare for things to be so ideal both in life and in physics. Many processes occur at an uneven speed, sometimes speeding up, sometimes slowing down.

Let's imagine the movement of a regular intercity bus. At the beginning of the journey, he accelerates, slows down at traffic lights, or even stops altogether. Then it goes faster outside the city, but slower on the ascents, and accelerates again on the descents.

If you depict this process in the form of a graph, you will get a very intricate line. It is possible to determine the speed from the graph only for a specific point, but there is no general principle.

You will need a whole set of formulas, each of which is suitable only for its own section of the drawing. But there's nothing scary. To describe the movement of the bus, an average value is used.

You can find the average speed using the same formula. Indeed, we know the distance between bus stations and travel time has been measured. Divide one by the other and find the required value.

What is it for?

Such calculations are useful to everyone. We plan our day and movements all the time. Having a dacha outside the city, it makes sense to find out the average ground speed when traveling there.

This will make planning your weekend easier. Having learned to find this value, we can be more punctual and stop being late.

Let's return to the example proposed at the very beginning, when a car drove part of the way at one speed, and the other at a different speed. This type of problem is very often used in the school curriculum. Therefore, when your child asks you to help him with a similar issue, it will be easy for you to do it.

By adding up the lengths of the path sections, you get the total distance. By dividing their values ​​by the speeds indicated in the initial data, you can determine the time spent on each of the sections. Adding them up, we get the time spent on the entire journey.

Uneven motion is considered to be movement with varying speed. Speed ​​can vary in direction. We can conclude that any movement NOT along a straight path is uneven. 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 uneven movement, which consists of alternating different types of movements, for example, first a bus accelerates (uniformly accelerated movement), then it moves uniformly for some time, and then stops.

Instantaneous speed

Uneven movement can only be characterized by speed. But the speed always changes! Therefore, we can only talk about speed at a given moment in time. When traveling by car, the speedometer shows you the instantaneous speed of movement every second. But in this case the time must be reduced not to a second, but a much shorter period of time must be considered!

average speed

What is average speed? It is wrong to think that you need to add up all the instantaneous velocities and divide by their number. This is the most common misconception about average speed! Average speed is divide the entire journey by the time taken. And it is not determined in any other way. If you consider the movement of a car, you can estimate its average speeds in the first half of the journey, in the second, and throughout the entire journey. Average speeds may be the same or may be different in these areas.

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

Average moving speed. Average ground speed

If the movement of a body is not rectilinear, then the distance traveled by the body will be greater than its displacement. In this case, the average moving speed differs 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 average and instantaneous speeds;
3) Rule for finding average speed

Often you need to solve a problem where the entire path is divided into equal sections, the average speeds on each section are given, you need to find the average speed along the entire route. 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.

Instantaneous speed can be determined using a motion graph. The instantaneous speed 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 is the tangent of the angle of inclination 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 a 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 of the values ​​of the instantaneous speeds. But the path and time are not given.

What variable speed does the car's speedometer indicate?

Close to instantaneous. Close, since the period of time should be infinitely small, and when taking readings from the speedometer, it is impossible to judge time that way.

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

With uniform movement. Because the speed does not change.

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

Average speed is the speed that is obtained if the entire path is divided by the time it takes the object to cover this path. Average speed formula:

• V av = S/t.

• S = S1 + S2 + S3 = v1*t1 + v2*t2 + v3*t3
• V av = S/t = (v1*t1 + v2*t2 + v3*t3) / (t1 + t2 + t3)

To avoid confusion with hours and minutes, we convert all minutes to hours: 15 minutes. = 0.4 hour, 36 min. = 0.6 hour. Substitute the numerical values ​​into the last formula:

• V av = (20*0.4 + 0.5*6 + 0.6*15) / (0.4 + 0.5 + 0.6) = (8 + 3 + 9) / (0.4 + 0.5 + 0.6) = 20 / 1.5 = 13.3 km/h

Answer: average speed V av = 13.3 km/h.

How to find the average speed of an accelerating motion

If the speed at the beginning of the movement differs from the speed at the end, such movement is called accelerated. Moreover, the body does not always actually move faster and faster. If the movement slows down, they still say that it is moving with acceleration, only the acceleration will be negative.

In other words, if a car, moving away, accelerated to a speed of 10 m/sec in a second, then its acceleration a is equal to 10 m per second per second a = 10 m/sec². If in the next second the car stops, then its acceleration is also equal to 10 m/sec², only with a minus sign: a = -10 m/sec².

The speed of movement with acceleration at the end of the time interval is calculated by the formula:

• V = V0 ± at,

where V0 is the initial speed of movement, a is acceleration, t is the time during which this acceleration was observed. A plus or minus is placed in the formula depending on whether the speed increased or decreased.

The average speed over a period of time t is calculated as the arithmetic mean of the initial and final speeds:

• V av = (V0 + V) / 2.

Finding the average speed: problem

The ball was pushed along a flat plane with an initial speed V0 = 5 m/sec. After 5 sec. the ball stopped. What are the acceleration and average speed?

The final speed of the ball is V = 0 m/sec. The acceleration from the first formula is equal to

• a = (V - V0)/ t = (0 - 5)/ 5 = - 1 m/sec².

Average speed V av = (V0 + V) / 2 = 5 /2 = 2.5 m/sec.