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

Rice. 2. Rolling the body along 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 uneven motion.

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

Rice. 4. Uniform movement

Movement is called uneven., in which the body travels unequal paths for equal periods of time.

Rice. 5. Uneven movement

The main task of mechanics is to determine the position of the body at any given time. With an 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 with 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 on the way to Sochi [M1]? Is it possible, with 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 get to Sochi.

Rice. 6. Illustration for example

When considering the movement of a body over a large section of the path as a whole, it is more convenient to introduce the concept of average speed.

Average speed is called the ratio of the total movement that the body has made to the time during which this movement is completed (Fig. 7).

Rice. 7. Average speed

This definition is not always convenient. For example, an athlete runs 400 meters - exactly one lap. The athlete's movement is equal to 0 (Fig. 8), however, 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 total path traversed by the body to the time during which the path traversed (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 travel a given distance in the same time it took it, moving unevenly.

We know from the course of mathematics what the arithmetic mean is. For numbers 10 and 36, it will be:

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

Task

The cyclist climbs the slope at a speed of 10 km / h, spending 0.5 hours on it. Then it descends at a speed of 36 km / h 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 also find the average speed in km / h. Therefore, these problems will not be translated into SI. Let's translate into hours.

Average speed is:

The full path () consists of the uphill path () and downhill path ():

The path of ascent to the slope is:

The descent path from the slope is:

The time taken to complete the full path is equal to:

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 along the entire path of the train is equal, then in 5 hours it will be at a distance from Novosibirsk.

The average speed measured over an infinitely small period of time is called instantaneous body speed(for example: car speedometer (fig. 11) shows instantaneous speed).

Rice. 11. Car speedometer shows instantaneous speed

There is another definition of instantaneous speed.

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

Rice. 12. Instantaneous speed

To better understand this definition, consider an example.

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

Rice. 13. Graph of the dependence of the projection of displacement on time

The graph shows that the vehicle speed is not constant. Suppose it is necessary to find the instantaneous vehicle speed 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 for the time interval from to. To do this, consider a fragment of this graph (Fig. 14).

Rice. 14. Graph of the dependence of the projection of displacement on time

In order to check the correctness of finding the instantaneous velocity, let us find the modulus of the average velocity for the time interval from to, for this we will consider a fragment of the graph (Fig. 15).

Rice. 15. Graph of the dependence of the projection of displacement on time

We calculate the average speed for a given time interval:

Received two values ​​of the instantaneous vehicle speed 30 seconds after the start of the observation. More precise 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 accurately.

Instantaneous velocity 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 velocity () change only in direction, without changing in absolute value?

Solution

For a solution, consider the following example. The body moves along a curved trajectory (Fig. 17). Let's mark the point on the trajectory A and point B... Let us mark 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 the same in absolute value and equal to 5 m / s.

Answer: maybe.

Assignment 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 the instantaneous speed is directed in the same way. If the body moves uniformly accelerated, then.

Answer: maybe.

In this lesson, we began to study uneven movement, that is, movement with a varying speed. The characteristics of uneven movement are average and instantaneous speeds. The concept of average speed is based on the mental replacement of uneven movement with uniform movement. 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 speed 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. Physics tasks. - 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 the 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 with uniform rectilinear motion and instantaneous speed with uneven motion?
4. While driving the car, the speedometer readings were taken every minute. Is it possible to determine the average speed of the vehicle from this 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 along the way. Give your answer in km / hour

Equally accelerated curvilinear motion

Curvilinear movements are movements whose trajectories are not straight lines, but curved lines. Planets and river waters move along curvilinear trajectories.

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

Irregular movement. Irregular motion speed

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

The movement, in which the body passes unequal segments of the path at 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 average speed.

The average speed shows what the displacement that the body passes per unit of time is equal to. It is equal to the ratio of the movement of the body to the time of movement. 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 accurately, instantaneous speed is used in physics.

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

point movement speed uneven

Body movement in a circle

Curvilinear movement is very common in nature and technology. It is more difficult than rectilinear, since there are many curvilinear trajectories; this movement is always accelerated, even when the speed module does not change.

But movement along any curved path can be roughly represented as movement along the arcs of a circle.

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

Uniform movement along a circle is a movement during which the modulus of the movement speed 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 movement of a body in a circle is characterized by the following quantities:

The period of rotation of the body is the time during which the body makes one complete revolution. The period of rotation is indicated by the letter T and is measured in seconds.

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

Linear velocity is the ratio of body movement to time. In order to find the linear velocity of a body in a circle, it is necessary to divide the circumference by the period (the circumference is equal to 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. The angular velocity is indicated by the letter? and is measured in radians divided by a second. You can find the angular velocity by dividing 2? for a period of. Angular velocity and linear velocity among themselves. In order to find the linear velocity, the angular velocity must be multiplied by the radius of the circle.

Figure 6. Circular motion, formulas.

Movement with varying speed is considered to be uneven. The speed can vary in direction. It can be concluded 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 be changed numerically. This movement will also be uneven. A special case of such a movement is uniformly accelerated movement.

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

Instant speed

Uneven movement can only be characterized by speed. But the speed is always changing! 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 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 instantaneous speeds and divide by their number. This is the most common misconception about average speed! Average speed is divide the entire path by the elapsed time... And it is not determined in any other way. If we consider the movement of the car, we can estimate its average speeds in the first half of the journey, in the second, along the entire journey. Average speeds can be the same, or they can be different in these areas.

A horizontal line is drawn on top of the averages.

Average travel speed. Average ground speed

If the movement of the body is not rectilinear, then the path traversed by the body will be greater than its movement. In this case, the average travel 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 the average and instantaneous speeds;
3) The rule of finding the average speed of movement

It is often required to solve a problem where the entire path is divided into equal sections, the average speeds are given for each section, it is required to find the average speed of movement along 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 similar problems.

Instantaneous speed can be determined using the driving 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 slope of the tangent to the graph of the function.

Exercises

While driving the car, the speedometer readings were taken every minute. Is it possible to determine the average speed of the vehicle from this 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 instantaneous speeds. And the way and time are not given.

What is the speed of the variable movement shown by the car speedometer?

Close to instant. Close, since the time interval should be infinitely small, and when taking readings from the speedometer, you cannot judge the time like that.

When are instantaneous and average speeds equal? Why?

With uniform movement. Because the speed doesn't change.

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

Uniform movement- this is movement with constant speed, that is, when the speed does not change (v = const) and acceleration or deceleration does not occur (a = 0).

Straight motion is movement in a straight line, that is, the trajectory of a rectilinear movement is a straight line.

This is a movement in which the body makes the same movements for any equal intervals of time. For example, if we divide some time interval into segments of one second, then with uniform motion the body will move the same distance for each of these segments of time.

The speed of uniform rectilinear movement does not depend on time and at each point of the trajectory is directed in the same way as the movement of the body. That is, the displacement vector coincides in direction with the velocity vector. In this case, the average speed for any period of time is equal to the instantaneous speed:

vcp = v

Uniform straight motion speed is a physical vector quantity equal to the ratio of the body's displacement over any time interval to the value of this interval t:

= / t

Thus, the speed of uniform rectilinear motion shows how much a material point moves per unit of time.

Moving with uniform rectilinear motion is determined by the formula:

Distance traveled in rectilinear motion it is equal to the displacement modulus. If the positive direction of the OX axis coincides with the direction of motion, then the projection of the velocity onto the OX axis is equal to the magnitude of the velocity and is positive:

vx = v, that is, v> 0

The projection of displacement on the OX axis is equal to:

s = vt = x - x0

where x 0 is the initial coordinate of the body, x is the final coordinate of the body (or the coordinate of the body at any time)

Equation of motion, that is, the dependence of the coordinates of the body on time x = x (t) takes the form:

x = x0 + vt

If the positive direction of the OX axis is opposite to the direction of motion of the body, then the projection of the body's velocity onto the OX axis is negative, the velocity is less than zero (v< 0), и тогда уравнение движения принимает вид:

x = x0 - vt

Uniform rectilinear movement is a special case of uneven movement.

Uneven movement- this is a movement in which a body (material point) makes unequal displacements for equal periods of time. For example, a city bus moves unevenly, as its movement consists mainly of acceleration and deceleration.

Equivalent motion- this is a movement in which the speed of a body (material point) for any equal time intervals changes in the same way.

Acceleration of a body with equal motion remains constant in absolute value and in direction (a = const).

Equally variable motion can be uniformly accelerated or equally slowed down.

Equally accelerated movement- this is the movement of a body (material point) with a positive acceleration, that is, with such a movement, the body accelerates with constant acceleration. In the case of uniformly accelerated motion, the body's velocity module increases with time, the direction of acceleration coincides with the direction of the motion velocity.

Equal slow motion- this is the movement of a body (material point) with negative acceleration, that is, with such a movement, the body evenly slows down. With equally slow motion, the vectors of speed and acceleration are opposite, and the modulus of speed decreases with time.

In mechanics, any rectilinear motion is accelerated, therefore decelerated motion differs from accelerated only by the sign of the projection of the acceleration vector onto the selected axis of the coordinate system.

Average speed of variable movement is determined by dividing the movement of the body by the time during which this movement was made. The unit of measurement for the average speed is m / s.

vcp = s / t

This is the speed of a body (material point) at a given moment in time or at a given point of the trajectory, that is, the limit to which the average speed tends with an infinite decrease in the time interval Δt:

Instantaneous velocity vector equidistant motion can be found as the first derivative of the time displacement vector:

= "

Velocity vector projection on the OX axis:

vx = x ’

it is a derivative of the coordinate with respect to time (similarly, the projections of the velocity vector onto other coordinate axes are obtained).

This is the value that determines the rate of change in the speed of the body, that is, the limit to which the change in speed tends with an infinite decrease in the time interval Δt:

Acceleration vector of equal motion can be found as the first derivative of the velocity vector with respect to time or as the second derivative of the displacement vector with respect to time:

= "=" Considering that 0 is the speed of the body at the initial moment of time (initial speed), is the speed of the body at a given moment of time (final speed), t is the time interval during which the change in speed has occurred, will be as follows:

From here formula for the speed of uniform motion at any given time:

0 + T

vx = v0x ± axt

The “-” (minus) sign in front of the projection of the acceleration vector refers to equal deceleration motion. Equations of the projections of the velocity vector on other coordinate axes are written in a similar way.

Since the acceleration is constant with equal motion (a = const), the acceleration graph is a straight line parallel to the 0t axis (time axis, Fig. 1.15).

Rice. 1.15. Time dependence of the acceleration of the body.

Speed ​​versus time is a linear function whose graph is a straight line (Fig. 1.16).

Rice. 1.16. Time dependence of body speed.

Speed ​​versus time graph(fig. 1.16) shows that

In this case, the displacement is numerically equal to the area of ​​the figure 0abc (Fig. 1.16).

The area of ​​the trapezoid is equal to the product of the half-sum of the lengths of its bases by the height. The bases of the trapezoid 0abc are numerically equal:

0a = v0 bc = v

The height of the trapezoid is t. Thus, the area of ​​the trapezoid, and hence the projection of the displacement on the OX axis, is equal to:

In the case of equally slow motion, the projection of the acceleration is negative, and in the formula for the projection of the displacement, a “-” (minus) sign is put before the acceleration.

The graph of the body's velocity versus time at various accelerations is shown in Fig. 1.17. The graph of the dependence of displacement on time at v0 = 0 is shown in Fig. 1.18.

Rice. 1.17. Time dependence of body speed for different values ​​of acceleration.

Rice. 1.18. Time dependence of body movement.

The speed of the body at a given time t 1 is equal to the tangent of the angle of inclination between the tangent to the graph and the time axis v = tg α, and the displacement is determined by the formula:

If the time of movement of the body is unknown, you can use a different formula for the movement, solving a system of two equations:

It will help us derive a formula for the projection of displacement:

Since the coordinate of the body at any time is determined by the sum of the initial coordinate and the projection of the displacement, it will look like this:

The plot of the x (t) coordinate is also a parabola (like the displacement plot), but the vertex of the parabola generally does not coincide with the origin. For a x< 0 и х 0 = 0 ветви параболы направлены вниз (рис. 1.18).

Lesson outline on the topic “Uneven movement. Instant speed "

date :

Theme: « »

Goals:

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

Developing : Continue the development of skills for independent work, skills of working in groups.

Educational : Form a cognitive interest in new knowledge; foster discipline of behavior.

Lesson type: a lesson in the assimilation of new knowledge

Equipment and sources of information:

Isachenkova, L.A. Physics: textbook. for 9 cl. institutions total. wednesday education with rus. lang. training / L. A. Isachenkova, G. V. Palchik, A. A. Sokolsky; ed. A. A. Sokolsky. Minsk: Narodnaya asveta, 2015

Lesson structure:

Organizational moment (5 min)

Basic knowledge update (5min)

Learning new material (14 min)

Physical education (3 minutes)

Consolidation of knowledge (13min)

Lesson summary (5 min)

Organizing time

Hello, sit down! (Checking those present).Today in the lesson we must deal with the concepts of uneven motion and instantaneous speed. This means thatLesson topic : Irregular movement. Instant speed

Updating basic knowledge

We studied uniform rectilinear motion. However, real bodies - cars, ships, airplanes, parts of mechanisms, etc. most often move both not in a straight line, and not evenly. What are the patterns of such movements?

Learning new material

Let's look at an example. The car moves along the section of the road, shown in Figure 68. On the rise, the car's movement slows down, while on the descent, it accelerates. Car movementand not straight, 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 average speed is. It is defined as the ratio of the path to the time interval for which this path is traversed:

(1 )

We will call itaverage travel speed. She shows whatway on average, a 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, the body performed 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 over a period of time Δ tthe body would move Δ r.

Average track 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 module of the average speed of movement is not more 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,instant speed - the speed of movement at a given moment in time (or at a given point).

How to determine instantaneous speed?

Let's look at an example. Let the ball roll down an inclined chute from a point (fig. 69). The figure shows the position of the ball at different times.

We are interested in the instantaneous speed of the ball at a pointO. Dividing the movement of the ball Δr 1 for the corresponding time interval Δ the averagetravel speed<>= on site Speed<>can be much different from the instantaneous speed at the pointO. Consider a smaller displacement Δ =V 2 . It occurs in a shorter time interval 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 otherandon the instantaneous velocity of the ball at the pointO.

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

(3)

Δ designation t- »0 reminds that the speed determined by the formula (3), the closer to the instantaneous speed, the lessΔt .

The instantaneous velocity of the curvilinear movement of the body is found in the same way (Fig. 70).

How is the instantaneous velocity directed? 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 a curvilinear movementinstantaneous speed is directed tangentially to the trajectory at the point where the moving body is at this moment.

Observe the glowing 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, the sports hammer (Fig. 71, b) begins its flight tangentially to the trajectory along which it moved when untwisted by the thrower.

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

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

Give yourself examples of accelerated and decelerated body movements.

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

In what follows, the instantaneous velocity will be referred to simply as velocity.

Consolidation of knowledge

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

The instantaneous speed is directed tangentially to the trajectory of motion.

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

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

Lesson summary

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

Organization of homework

§ 9, exercise. 5 No. 1.2

Reflection.

Continue phrases:

Today in the lesson I learned ...

It was interesting…

The knowledge I learned in the lesson will come in handy