1. Uniform movement is rare. Generally, mechanical motion is motion with varying speed. A movement in which the speed of a body changes over time is called uneven.

For example, traffic is moving unevenly. The bus, starting to move, increases its speed; when braking, its speed decreases. Bodies falling on the Earth's surface also move unevenly: their speed increases with time.

With uneven motion, the coordinate of the body can no longer be determined by the formula x = x 0 + v x t because the speed is not constant. The question arises, what value characterizes the speed of change in the position of the body over time with uneven movement? This value is average speed.

medium speed vWeduneven movement is called a physical quantity equal to the ratio of displacement sbody to time t for which it was made:

v cf = .

The average speed is vector quantity. To determine the modulus of average velocity for practical purposes, this formula can be used only when the body moves along a straight line in one direction. In all other cases, this formula is unsuitable.

Consider an example. It is necessary to calculate the arrival time of the train at each station along the route. However, the movement is not linear. If we calculate the module of the average speed in the section between two stations, using the above formula, then the resulting value will differ from the value of the average speed with which the train moved, since the module of the displacement vector is less than the distance traveled by the train. And the average speed of this train from the starting point to the final point and back in accordance with the above formula is completely equal to zero.

In practice, when determining the average speed, a value equal to path relation l at the time t, for which this path was passed:

v Wed = .

She is often called average ground speed.

2. Knowing the average speed of a body in any part of the trajectory, it is impossible to determine its position at any time. Assume that the car traveled a distance of 300 km in 6 hours. The average speed of the car is 50 km/h. However, at the same time, he could stand for some time, move for some time at a speed of 70 km/h, for some time at a speed of 20 km/h, etc.

Obviously, knowing the average speed of the car for 6 hours, we cannot determine its position after 1 hour, after 2 hours, after 3 hours, etc.

3. When moving, the body passes successively all points of the trajectory. At each point it is at certain points in time and has some speed.

Instantaneous speed is the speed of a body at a given time or at a given point in the trajectory.

Let us assume that the body performs non-uniform rectilinear motion. Let us determine the speed of movement of this body at the point O its trajectory (Fig. 21). Let's select a section on the trajectory AB, inside which there is a point O. moving s 1 in this area the body has committed in time t one . The average speed for this section is v cf 1 = .

Reduce the movement of the body. Let it be equal s 2 , and the time of movement - t 2. Then the average speed of the body during this time: v cf 2 = .Let's further reduce the movement, the average speed in this section: v cf 3 = .

We will continue to reduce the time of motion of the body and, accordingly, its displacement. In the end, the displacement and time will become so small that an instrument, such as a speedometer in a car, will no longer register a change in speed, and the movement over this small period of time can be considered uniform. The average speed in this section is the instantaneous speed of the body at the point O.

In this way,

instantaneous speed - a vector physical quantity equal to the ratio of small displacement D sto a small time interval D t, for which this movement is made:

v = .

Questions for self-examination

1. What movement is called uneven?

2. What is called average speed?

3. What is average ground speed?

4. Is it possible, knowing the trajectory of the body and its average speed for a certain period of time, to determine the position of the body at any time?

5. What is called instantaneous speed?

6. How do you understand the expressions "small displacement" and "small period of time"?

Task 4

1. The car drove through Moscow streets 20 km in 0.5 hours, when leaving Moscow it stood for 15 minutes, and in the next 1 hour 15 minutes it drove 100 km in the Moscow region. What was the average speed of the car in each segment and for the entire journey?

2. What is the average speed of a train on a run between two stations if it traveled the first half of the distance between stations at an average speed of 50 km/h, and the second half at an average speed of 70 km/h?

3. What is the average speed of the train on the run between two stations if it traveled half the time at an average speed of 50 km/h, and the remaining time - at an average speed of 70 km/h?

Uniformly accelerated curvilinear motion

Curvilinear movements - movements, the trajectories of which are not straight, but curved lines. Planets and river waters move along curvilinear trajectories.

Curvilinear motion is always motion with acceleration, even if the absolute value of the speed is constant. Curvilinear motion with constant acceleration always occurs in the plane in which the acceleration vectors and the initial velocities of the point are located. In the case of a curvilinear motion with constant acceleration in the xOy plane, the projections vx and vy of its velocity on the axes Ox and Oy and the coordinates x and y of the point at any time t are determined by the formulas

Uneven movement. Speed ​​with uneven movement

No body moves at a constant speed all the time. Starting the movement, the car moves faster and faster. For a while it can move evenly, but then it slows down and stops. In this case, the car covers different distances in the same time.

A movement in which a body travels unequal segments of the path in equal intervals of time is called uneven. With such a movement, the magnitude of the speed does not remain unchanged. In this case, we can only talk about the average speed.

The average speed shows what is the displacement that the body passes per unit of time. It is equal to the ratio of the movement of the body to the time of movement. The average speed, like the speed of a body in uniform motion, is measured in meters divided by a second. In order to characterize motion more precisely, in physics instantaneous velocity is used.

The speed of a body at a given point in time or at a given point in the trajectory is called instantaneous speed. Instantaneous velocity is a vector quantity and is directed in the same way as the displacement vector. You can measure your instantaneous speed with a speedometer. In the System Internationale, instantaneous speed is measured in meters divided by a second.

point movement speed uneven

The movement of the body in a circle

In nature and technology, curvilinear motion is very common. It is more complicated than a rectilinear one, since there are many curvilinear trajectories; this movement is always accelerated, even when the modulus of speed does not change.

But movement along any curvilinear trajectory can be roughly represented as movement along the arcs of a circle.

When a body moves in a circle, the direction of the velocity vector changes from point to point. Therefore, when they talk about the speed of such a movement, they mean instantaneous speed. The velocity vector is directed along the tangent to the circle, and the displacement vector - along the chords.

Uniform movement in a circle is a movement during which the modulus of the speed of movement does not change, only its direction changes. The acceleration of such a movement is always directed towards the center of the circle and is called centripetal. In order to find the acceleration of a body that moves in a circle, it is necessary to divide the square of the speed by the radius of the circle.

In addition to acceleration, the motion of a body in a circle is characterized by the following quantities:

The rotation period of a body is the time it takes the body to make one complete rotation. The rotation period is denoted by the letter T and is measured in seconds.

Body rotation frequency is the number of revolutions per unit time. The rotational speed is indicated by a letter? and is measured in hertz. In order to find the frequency, it is necessary to divide the unit by the period.

Linear speed - the ratio of the movement of the body to time. In order to find the linear velocity of a body along a circle, it is necessary to divide the circumference by the period (the circumference is 2? times the radius).

Angular velocity is a physical quantity equal to the ratio of the angle of rotation of the radius of the circle along which the body moves to the time of movement. Angular speed is denoted by a letter? and is measured in radians divided by a second. You can find the angular velocity by dividing 2? for a period of. Angular speed and linear speed. In order to find the linear velocity, it is necessary to multiply the angular velocity by the radius of the circle.

Figure 6. Movement in a circle, formulas.

Outline of the lesson on the topic “Uneven movement. Instant Speed"

the date :

Topic: « »

Goals:

educational : Provide and form a conscious assimilation of knowledge about uneven movement and instantaneous speed;

Educational : Continue to develop skills of independent activity, skills of working in groups.

Educational : To form a cognitive interest in new knowledge; cultivate discipline.

Lesson type: a lesson in learning new knowledge

Equipment and sources of information:

Isachenkova, L. A. Physics: textbook. for 9 cells. institutions of general avg. education with Russian lang. education / L. A. Isachenkova, G. V. Palchik, A. A. Sokolsky; ed. A. A. Sokolsky. Minsk: Narodnaya Aveta, 2015

Lesson structure:

Organizational moment (5 min)

Updating of basic knowledge (5min)

Learning new material (14 min)

Physical education (3 min)

Consolidation of knowledge (13min)

Lesson summary (5 min)

Organizing time

Hello, have a seat! (Checking those present).Today in the lesson we have to deal with the concepts of uneven motion and instantaneous speed. And this means thatLesson topic : Uneven movement. Instant Speed

Updating of basic knowledge

We have studied uniform rectilinear motion. However, real bodies - automobiles, ships, aircraft, parts of mechanisms, etc. most often move neither in a straight line nor evenly. What are the laws of such movements?

Learning new material

Consider an example. The car moves along the section of the road shown in Figure 68. On the rise, the movement of the car slows down, when descending it accelerates. car movementand not rectilinear, and not uniform. How to describe such a movement?

First of all, for this it is necessary to clarify the conceptspeed .

From the 7th grade, you know what the average speed is. It is defined as the ratio of the path to the time interval for which this path was traveled:

(1 )

Let's call heraverage travel speed. She shows whatpath on average, the body passed per unit of time.

In addition to the average speed of the path, it is necessary to enter andaverage travel speed:

(2 )

What is the meaning of average travel speed? She shows whatmoving on average performed by the body per unit of time.

Comparing formula (2) with formula (1 ) from § 7, we can conclude:average speed< > is equal to the speed of such a uniform rectilinear motion, at which for a period of time Δ tthe body would move Δ r.

Average travel speed and average travel speed are important characteristics of any movement. The first of them is a scalar quantity, the second is a vector one. Because Δ r < s , then the modulus of the average travel speed is not greater than the average speed of the path |<>| < <>.

The average speed characterizes the movement for the entire period of time as a whole. It does not provide information about the speed of movement at each point of the trajectory (at each moment of time). For this purpose, it introducesinstantaneous speed - the speed of movement at a given time (or at a given point).

How to determine the instantaneous speed?

Consider an example. Let the ball roll down the inclined chute from a point (Fig. 69). The figure shows the position of the ball at different points in time.

We are interested in the instantaneous speed of the ball at the pointO. Dividing the movement of the ball Δr 1 for the corresponding time interval Δ averagetravel speed<>= on site Speed<>can be much different from the instantaneous speed at the pointO. Consider a smaller displacement Δ =AT 2 . It take place in a shorter period of time Δ. average speed<>= although not equal to the speed at the pointO, but closer to her than<>. With a further decrease in displacements (Δ,Δ , ...) and time intervals (Δ, Δ, ...) we will get average speeds that are less and less different from each otherandfrom the instantaneous speed of the ball at the pointO.

This means that a sufficiently accurate value of the instantaneous speed can be found by the formula, provided that the time interval Δt very small:

(3)

Designation ∆ t-» 0 recalls that the speed determined by the formula (3), the closer to the instantaneous speed, the lessΔt .

The instantaneous velocity of the curvilinear motion of the body is found similarly (Fig. 70).

What is the direction of the instantaneous speed? It is clear that in the first example the direction of the instantaneous velocity coincides with the direction of motion of the ball (see Fig. 69). And from the construction in Figure 70 it can be seen that with curvilinear motioninstantaneous speed is directed tangentially to the trajectory at the point where the moving body is at that moment.

Watch the incandescent particles coming off the grindstone (Fig. 71,a). The instantaneous velocity of these particles at the moment of separation is directed tangentially to the circle along which they moved before separation. Similarly, a sports hammer (Fig. 71, b) begins its flight tangentially to the trajectory along which it moved when unwinding by the thrower.

The instantaneous speed is constant only with uniform rectilinear motion. When moving along a curved path, its direction changes (explain why). With uneven movement, its module changes.

If the modulus of instantaneous velocity increases, then the motion of the body is called accelerated , if it decreases - slow.

Give yourself examples of accelerated and slow motions of bodies.

In the general case, when a body moves, both the module of the instantaneous velocity and its direction can change (as in the example with the car at the beginning of the paragraph) (see Fig. 68).

In what follows, we will simply refer to instantaneous speed as speed.

Consolidation of knowledge

The speed of uneven movement on a section of the trajectory is characterized by an average speed, and at a given point of the trajectory - by instantaneous speed.

The instantaneous speed is approximately equal to the average speed determined over a short period of time. The shorter this period of time, the smaller the difference between the average speed and the instantaneous one.

The instantaneous velocity is directed tangentially to the motion trajectory.

If the modulus of instantaneous velocity increases, then the movement of the body is called accelerated, if it decreases, it is called slow.

With uniform rectilinear motion, the instantaneous speed is the same at any point of the trajectory.

Lesson summary

So, let's sum up. What did you learn in class today?

Homework organization

§ 9, ex. 5 #1,2

Reflection.

Continue the phrases:

Today in class I learned...

It was interesting…

The knowledge that I received in the lesson will come in handy

Rolling the body down an inclined plane (Fig. 2);

Rice. 2. Rolling the body down an inclined plane ()

Free fall (Fig. 3).

All these three types of movement are not uniform, that is, the speed changes in them. In this lesson, we will look at non-uniform motion.

Uniform movement - mechanical movement in which the body travels the same distance in any equal time intervals (Fig. 4).

Rice. 4. Uniform movement

Movement is called uneven., at which the body covers unequal distances in equal intervals of time.

Rice. 5. Uneven movement

The main task of mechanics is to determine the position of the body at any time. With uneven movement, the speed of the body changes, therefore, it is necessary to learn how to describe the change in the speed of the body. For this, two concepts are introduced: average speed and instantaneous speed.

It is not always necessary to take into account the fact of a change in the speed of a body during uneven movement; when considering the movement of a body over a large section of the path as a whole (we do not care about the speed at each moment of time), it is convenient to introduce the concept of average speed.

For example, a delegation of schoolchildren travels from Novosibirsk to Sochi by train. The distance between these cities by rail is approximately 3300 km. The speed of the train when it just left Novosibirsk was , does this mean that in the middle of the way the speed was the same, but at the entrance to Sochi [M1]? Is it possible, having only these data, to assert that the time of movement will be (Fig. 6). Of course not, since the residents of Novosibirsk know that it takes about 84 hours to drive to Sochi.

Rice. 6. Illustration for example

When considering the motion of a body over a long section of the path as a whole, it is more convenient to introduce the concept of average velocity.

medium speed called the ratio of the total movement that the body made to the time for which this movement was made (Fig. 7).

Rice. 7. Average speed

This definition is not always convenient. For example, an athlete runs 400 m - exactly one lap. The athlete's displacement is 0 (Fig. 8), but we understand that his average speed cannot be equal to zero.

Rice. 8. Displacement is 0

In practice, the concept of average ground speed is most often used.

Average ground speed- this is the ratio of the full path traveled by the body to the time for which the path has been traveled (Fig. 9).

Rice. 9. Average ground speed

There is another definition of average speed.

average speed- this is the speed with which a body must move uniformly in order to cover a given distance in the same time for which it covered it, moving unevenly.

From the course of mathematics, we know what the arithmetic mean is. For numbers 10 and 36 it will be equal to:

In order to find out the possibility of using this formula to find the average speed, we will solve the following problem.

A task

A cyclist climbs a slope at a speed of 10 km/h in 0.5 hours. Further, at a speed of 36 km / h, it descends in 10 minutes. Find the average speed of the cyclist (Fig. 10).

Rice. 10. Illustration for the problem

Given:; ; ;

Find:

Solution:

Since the unit of measurement for these speeds is km/h, we will find the average speed in km/h. Therefore, these problems will not be translated into SI. Let's convert to hours.

The average speed is:

The full path () consists of the path up the slope () and down the slope () :

The way up the slope is:

The downhill path is:

The time taken to complete the path is:

Answer:.

Based on the answer to the problem, we see that it is impossible to use the arithmetic mean formula to calculate the average speed.

The concept of average speed is not always useful for solving the main problem of mechanics. Returning to the problem about the train, it cannot be argued that if the average speed over the entire journey of the train is , then after 5 hours it will be at a distance from Novosibirsk.

The average speed measured over an infinitesimal period of time is called instantaneous body speed(for example: the speedometer of a car (Fig. 11) shows the instantaneous speed).

Rice. 11. Car speedometer shows instantaneous speed

There is another definition of instantaneous speed.

Instant Speed- the speed of the body at a given moment of time, the speed of the body at a given point of the trajectory (Fig. 12).

Rice. 12. Instant speed

To better understand this definition, consider an example.

Let the car move in a straight line on a section of the highway. We have a graph of the dependence of the displacement projection on time for a given movement (Fig. 13), let's analyze this graph.

Rice. 13. Graph of displacement projection versus time

The graph shows that the speed of the car is not constant. Suppose you need to find the instantaneous speed of the car 30 seconds after the start of observation (at the point A). Using the definition of instantaneous speed, we find the modulus of the average speed over the time interval from to . To do this, consider a fragment of this graph (Fig. 14).

Rice. 14. Graph of displacement projection versus time

In order to check the correctness of finding the instantaneous speed, we find the module of the average speed for the time interval from to , for this we consider a fragment of the graph (Fig. 15).

Rice. 15. Graph of displacement projection versus time

Calculate the average speed for a given period of time:

We received two values ​​of the instantaneous speed of the car 30 seconds after the start of the observation. More precisely, it will be the value where the time interval is less, that is, . If we decrease the considered time interval more strongly, then the instantaneous speed of the car at the point A will be determined more precisely.

Instantaneous speed is a vector quantity. Therefore, in addition to finding it (finding its module), it is necessary to know how it is directed.

(at ) – instantaneous speed

The direction of instantaneous velocity coincides with the direction of movement of the body.

If the body moves curvilinearly, then the instantaneous velocity is directed tangentially to the trajectory at a given point (Fig. 16).

Exercise 1

Can the instantaneous speed () change only in direction without changing in absolute value?

Solution

For a solution, consider the following example. The body moves along a curved path (Fig. 17). Mark a point on the trajectory A and point B. Note the direction of the instantaneous velocity at these points (the instantaneous velocity is directed tangentially to the point of the trajectory). Let the velocities and be identical in absolute value and equal to 5 m/s.

Answer: maybe.

Task 2

Can the instantaneous speed change only in absolute value, without changing in direction?

Solution

Rice. 18. Illustration for the problem

Figure 10 shows that at the point A and at the point B instantaneous speed is directed in the same direction. If the body is moving with uniform acceleration, then .

Answer: maybe.

In this lesson, we began to study uneven movement, that is, movement with a changing speed. Characteristics of non-uniform motion are average and instantaneous speeds. The concept of average speed is based on the mental replacement of uneven motion with uniform motion. Sometimes the concept of average speed (as we have seen) is very convenient, but it is not suitable for solving the main problem of mechanics. Therefore, the concept of instantaneous velocity is introduced.

Bibliography

1. G.Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky. Physics 10. - M .: Education, 2008.
2. A.P. Rymkevich. Physics. Problem book 10-11. - M.: Bustard, 2006.
3. O.Ya. Savchenko. Problems in physics. - M.: Nauka, 1988.
4. A.V. Peryshkin, V.V. Krauklis. Physics course. T. 1. - M .: State. uch.-ped. ed. min. education of the RSFSR, 1957.

1. Internet portal "School-collection.edu.ru" ().
2. Internet portal "Virtulab.net" ().

Homework

1. Questions (1-3, 5) at the end of paragraph 9 (p. 24); G.Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky. Physics 10 (see list of recommended reading)
2. Is it possible, knowing the average speed for a certain period of time, to find the movement made by the body for any part of this interval?
3. What is the difference between instantaneous speed in uniform rectilinear motion and instantaneous speed in non-uniform motion?
4. While driving a car, speedometer readings were taken every minute. Is it possible to determine the average speed of the car from these data?
5. The cyclist rode the first third of the route at a speed of 12 km per hour, the second third at a speed of 16 km per hour, and the last third at a speed of 24 km per hour. Find the average speed of the bike for the entire journey. Give your answer in km/h

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?