Uneven movement instantaneous speed plan abstract. Lesson summary: Solving problems "Average speed with uneven movement"

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 changing 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 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.

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:; ; ;

To find:

Decision:

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?

Decision

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?

Decision

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

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

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

Homework

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)
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?
What is the difference between instantaneous speed in uniform rectilinear motion and instantaneous speed in non-uniform motion?
While driving a car, speedometer readings were taken every minute. Is it possible to determine the average speed of the car from these data?
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

Sections: Physics

Class: 7

Lesson type: learning new material.

Goals and objectives of the lesson:

Educational:
- introduce the basic concepts of mechanical motion: relativity of motion, trajectory, distance traveled, uniform and non-uniform motion;
- introduce the concept of speed as a physical quantity, the formula and units of its measurement.
Educational:
- develop cognitive interests, intellectual and creative abilities, interest in the study of physics;
Educational:
- develop the skills of self-acquisition of knowledge, organization of educational activities, setting goals, planning;
- to form the ability to systematize, classify and generalize the acquired knowledge;
- develop students' communication skills.

DURING THE CLASSES

I. Organizational moment

II. Homework:§§13-14, ex. 3 (oral).

III. Explanation of new material

1. We begin the lesson by declaring a new lesson topic and try to answer the question: “What allows us to judge whether the body is moving or at rest?”. After the students' answers, we quote an excerpt from A.S. Pushkin's poem "Movement" (see Fig. 1).
In the passage, a very important point was made, necessary for reasoning about whether the body is moving or at rest. Namely, relative to which bodies the movement occurs or does not occur. How can you tell if a body is moving or at rest?

Rice. one ( Presentation, slide 2)

2. Relativity of motion.

In order to isolate such a characteristic sign of mechanical motion as relativity, let us consider and analyze a simple experiment with a cart moving on a table. Let us consider in relation to which topics it moves, and in relation to which it rests (see Fig. 2, 3).

Rice. 2 (Slides 4-10).

Rice. 3 (Slide 11).

IV. In order to consolidate the material, we solve the following tasks:

Task 1. Indicate in relation to which bodies the following bodies are at rest and in relation to which - in motion: a passenger in a moving truck; a car following a truck at the same distance, a load in a car trailer.

Task 2. Relative to what bodies is the person standing on the pavement at rest and relative to what bodies does he move?

Rice. 4 (Slide 12).

Task 3. List the bodies in relation to which the driver of a moving tram is at rest.

Pupils usually answer that a person is at rest relative to the sidewalk, tree, traffic light, house and moving relative to a car driving along the road. In this situation, students should pay attention to the fact that a person, like the Earth, moves at a speed of 30 km / s relative to the Sun.

3. Trajectory of movement.

Next, we introduce the concept of a trajectory and, depending on its shape, distinguish two types of motion: rectilinear and curvilinear. First of all, we draw the attention of students to the movement of such bodies, the trajectories of which are clearly visible (see Fig. 5). Here we introduce the concept of the distance traveled as a physical quantity measured by the length of the trajectory along which the body moves for a certain period of time. In this regard, we repeat the basic units of length measurement known from the course of mathematics.

Rice. 5 (Slide 15).

Task 4. Match the example of mechanical movement with the type of toolpath.

EXAMPLE TYPE OF TRAJECTORY

A) meteor fall 1) circle
B) movement of the stopwatch hand 2) curve
C) a raindrop falling into a calm 3) a straight line
weather.

Task 5. Express the distance traveled in meters:

65 km
0.54 km
4 km 300 m
2300 cm
4 m 10 cm

(Slide 16).

4. Rectilinear uniform motion

Consider further what types of movement exist? Let's define what kind of motion is called uniform. A movement in which a body travels equal distances in equal intervals of time. Consider an example of rectilinear uniform motion (see Fig. 6).

Lesson

Topic: Rectilinear uniformly accelerated motion. Speed in uneven motion.

Lesson Objectives:

Educational:

1. to form the concept of rectilinear uniformly accelerated motion, instantaneous speed, acceleration;

2. build an acceleration graph;

3. develop skills in solving graphic and calculation problems

Developing:

1. develop the practical skills of students: the ability to analyze, generalize, highlight the main idea from the teacher’s story and draw conclusions;

2. develop the ability to apply acquired knowledge in new conditions.

Educators:

1. to expand the horizons of students about the types of mechanical movement (in particular, about rectilinear uniformly variable (uniformly accelerated) motion);

2. develop curiosity, interest in studying physics and, attentiveness, discipline

Lesson type: Combined lesson.

During the classes.

1) Organizing time

Establish class readiness for the lesson.

2) Motivation

Movement is life. Each body moves differently: with its own purpose, trajectory, speed. your movements - development, which is impossible without obtaining new knowledge. So today, we will discover a new characteristic of movement, which is an integral part of our life.

3) Knowledge update

Independent work (20 min)

4) Learning new material

We have studied the uniform motion of a body when its speed remains unchanged and at any moment of time and at any distance can be found as the ratio of the distance traveled to the time.

Please give examples of uniform motion.

(students give examples).

How often can we observe such a movement?

(general opinion of students: rarely, almost always, the speed of the body changes for some reason)

Indeed, such a movement is actually very rare and, as a rule, in mechanisms. But in the world around us, another movement is widespread.

fast motion is a fairly common type of movement. An example of such a movement is the movement of a load thrown from a certain height, the movement of a braking bus or a starting elevator.

In order to somehow characterize the accelerated movement, a quantity is introduced, which is calledacceleration body.

Acceleration is a physical quantity equal to the ratio of the change in speed to the time span for which it happened.

In addition, you can use the everyday definition: acceleration is the rate of change of speed.

Often, we consider acceleration in projection onto some axis (for example, onto the axis ), while the acceleration projection will take the form:

Note that the acceleration in all cases isvector magnitude, that is, it has not only magnitude, but also direction. Acceleration in the SI system is measured in meters divided by a second squared.

One meter per second squared is the acceleration at which for every second the speed of the body changes by one meter per second.

We have figured out how to determine the acceleration module, now we will figure out how to determine the direction of acceleration. To do this, we depict the change in speed in vector form (Fig. 1).

Rice. 1. Change in body speed during accelerated movement

Accordingly, the acceleration of the body will be directed in the same direction as the vector .

One of the simplest types of non-uniform motion is uniformly accelerated motion.

Uniformly accelerated is a movement in which for any equal intervals of time the speed of the body increases by the same amount.In uniformly accelerated motion, the acceleration of the body is constant.

In addition, sometimes allocate the so-called equally slow motion. Uniformly decelerated motion is a movement in which the speed of the body is opposite to its acceleration.

Let's draw graphs of the dependence of the acceleration of the body on time for uniformly accelerated motion. Since the acceleration is constant during uniformly accelerated motion (Fig. 2):

Rice. 2. Acceleration of the body during uniformly accelerated motion

The red graph corresponds to the case when the acceleration projection is positive. The green graph corresponds to the case when the acceleration projection is zero. Blue - negative projection of acceleration.

In order to solve the main problem of kinematics, that is, to find the position of the body at any time, you must first find the speed of the body at any time. For this, we should write down the law of change of instantaneous speed from time for uniformly accelerated motion. This can be done by simply expressing the speed from the acceleration formula.

where is the initial speed of the body, - acceleration. The law of speed change, written in vector form, is the most general, but it is rather inconvenient to use it to determine the speed at any point in time. Therefore, let us consider the law of change of instantaneous speed from time in the projection on the axis chosen along the direction of motion.

Consider four possible cases (Fig. 3):

Rice. 3. Four possible cases of directivity of the initial velocity and acceleration

in case a)the speed of the body and its acceleration are directed along the positive direction of the coordinate axis, and the law of change in speed will take the form:

in case in) the speed of the body is directed along the positive direction of the coordinate axis, and the acceleration is directed along the negative direction of the coordinate axis, we previously called such a movement uniformly slowed down, and its law of change in speed:

It can be seen from the form of the laws of change in speed over time that the projection of speed depends linearly on time, and accordingly, the graph of the dependence of the projection of speed on time will be a straight line (Fig. 4).

Rice. 4. Graphs of the dependence of the speed of the body on time for uniformly accelerated motion

The graph (Fig. 4a) shows the dependence of the velocity projection on time. The green straight line corresponds to the case, the body was at rest, and at the initial moment of time it began to move in the positive direction of the coordinate axis with increasing speed. The red straight line corresponds to the case when at the initial moment of time the body had some speed directed in the positive direction of the coordinate axis, and increases with time.

Figure 4b shows the relationship between the slope of the graph of the dependence of the speed of the body on time and the acceleration of the body during uniformly accelerated motion.

Finally, let's consider one singular point on the graph of the dependence of the projection of the body's velocity on time. Figure 5 shows the point at which the speed of the body changes its direction to the opposite. Such a point is calledturning point (Fig. 5).

Rice. 5. Turning point

So, in this lesson we learned about the concept of body acceleration. In addition, we considered the laws of change in the speed of the body from time to time. Next, we learned how to build graphs of body speed versus time, and finally introduced the concept of a turning point.

Homework