The topic of the lesson is “Uniform and uneven movement. Speed"
Lesson Objectives:
introduce the concepts of uniform and uneven
movement;introduce the concept of speed as a physical
quantities, formula and units of its measurement.develop cognitive interests,
intellectual and creative abilities,
interest in studying physics;develop independent skills
acquisition of knowledge, organization of training
activities, goal setting, planning;develop the ability to organize
classify and generalize the acquired knowledge;develop communication skills
students
Educational:
Educational:
Educational:
During the classes:
1. Repetition
What is mechanical movement? Give examples
What is a trajectory? What are they?
What is a path? How is it denoted, in what units is it measured?
Translate:
in m 80cm, 5cm, 2km, 3dm, 12dm, 1350cm, 25000mm, 67km
in cm 2 dm, 5 km, 30 mm
2. Assimilation of new knowledge
Uniform movement A movement in which a body travels equal distances in equal intervals of time.
Uneven movement A movement in which a body travels unequal distances in equal intervals of time.
Examples of uniform and non-uniform motion
Speed of rectilinear uniform motion- physical quantity, equal to the ratio the path to the time it took to complete the path.
Let's check if our knowledge is enough to solve the following problem. Two cars started moving simultaneously from the village with the same speed of 60 km/h. Is it possible to say that in an hour they will be in the same place?
Conclusion: speed should be characterized not only by number, but also by direction. Such quantities, which, in addition to a numerical value, also have a direction, are called vector quantities.
Speed is a vector physical quantity.
Scalar quantities are quantities that are characterized only by a numerical value (for example, path, time, length, etc.)
To characterize non-uniform motion, the concept of average speed is introduced.
To determine the average speed of the body during uneven movement, it is necessary to divide the entire distance traveled by the entire time of movement:
Working with the textbook table p.37
3. Checking the assimilation of new knowledge
Problem solving
1. Convert speed units to basic SI units:
36 km/h = _________________________________________________________________
120 m/min = ________________________________________________________________
18 km/h = _________________________________________________________________
90 m/min = _______________________________________________________________
2. The balloon is moving east at a speed of 30 km/h. Plot the velocity vector using the scale: 1 cm=10 km/h
Algorithm for solving problems in physics:
1. Carefully read the condition of the problem and understand the main question; present the processes and phenomena described in the condition of the problem.
2. Re-read the content of the problem in order to clearly present the main question of the problem, the purpose of its solution, the known values, based on which you can search for a solution.
3. Make a brief note of the problem condition using generally accepted letter designations.
4. Make a drawing or drawing for the task.
5. Determine which method will solve the problem; make a plan to solve it.
6. Write down the basic equations describing the processes proposed by the task system.
7. Record the solution in general view, expressing the required quantities in terms of the given ones.
8. Check the correctness of the solution of the problem in general terms by performing actions with the names of the quantities.
9. Perform calculations with the specified accuracy.
10. Make an assessment of the reality of the resulting solution.
11. Write down the answer in the required form
3. Find the speed of the French athlete Roman Zaballo, who in 1981 ran the distance between the French cities of Florence and Montpellier (510 km) in 60 hours.
4. Find the speed of a cheetah (the fastest mammal) if it runs 210 meters in 7 seconds.
5. V.I.Lukashik Problems No. 117,118,119
6. Homework: §14,15, exercise 4(4)
To develop the mental abilities of students, the ability to analyze, highlight common and distinctive properties; to develop the ability to apply theoretical knowledge in practice when solving problems of finding the average speed of uneven movement.
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Lesson in grade 9 on the topic: "Average and instantaneous speeds of uneven movement"
Teacher - Malyshev M.E.
Date -17.10.2013
Lesson Objectives:
Educational Purpose:
- Repeat the concept - average and instantaneous speeds,
- learn to find the average speed under various conditions, using tasks from the GIA materials and USE past years.
Development goal:
- develop the mental abilities of students, the ability to analyze, highlight common and distinctive properties; develop the ability to apply theoretical knowledge in practice; develop memory, attention, observation.
educational goal:
- to bring up a steady interest in the study of mathematics and physics through the implementation of interdisciplinary connections;
Lesson type:
- a lesson on generalization and systematization of knowledge and skills on a given topic.
Equipment:
- computer, multimedia projector;
- notebooks;
- set of equipment L-micro in the section "Mechanics"
During the classes
1. Organizational moment
Mutual greeting; checking the readiness of students for the lesson, organizing attention.
2. Communication of the topic and objectives of the lesson
Screen slide: “ Practice is born only from a close connection of physics and mathematics” Bacon F.
The topic and objectives of the lesson are reported.
3. Entrance control (repetition of theoretical material)(10 min)
Organization of oral front work with class on repetition.
Physics teacher:
1. What is the simplest type of movement you know? (uniform movement)
2. How to find the speed with uniform motion? (displacement divided by time v= s / t )? 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.
3. How to find the speed with uneven movement? What is it called? (Average speed, v cp = s / t)
In practice, when determining the average speed, a value equal tothe ratio of the path s to the time t during which this path was traveled: v cf = s/t . She is often calledaverage ground speed.
4. What are the features of the average speed? (Average speed is a vector quantity. To determine the modulus of average speed in 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).
5. What is instantaneous speed? What is the direction of the instantaneous velocity vector? (Instantaneous speed is the speed of the body at a given point in time or at a given point in the trajectory. The vector of instantaneous speed at each point coincides with the direction of motion at a given point.)
6. What is the difference between instantaneous speed with uniform rectilinear motion and instantaneous speed with uneven motion? (In the case of uniform rectilinear motion, the instantaneous speed at any point and at any time is the same; in the case of uneven rectilinear motion, the instantaneous speed is different).
7. Is it possible to determine the position of the body at any moment of time knowing the average speed of its movement in any part of the trajectory? (it is impossible to determine its position at any point in time).
Suppose that the car traveled a distance of 300 km in 6 hours. What is equal to average speed movement? The average speed of the car is 50 km/h. However, at the same time, he could stand for some time, for some time move 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. of time.
1. Verbally find the speed of the car if it traveled 180 km in 3 hours.
2. A car traveled for 1 hour at a speed of 80 km/h and 1 hour at a speed of 60 km/h. Find your average speed. Indeed, the average speed is (80+60)/2=70 km/h. In this case, the average speed is equal to the arithmetic mean of speeds.
3. Let's change the condition. The car traveled 2 hours at a speed of 60 km/h and 3 hours at a speed of 80 km/h. What is the average speed for the whole journey?
(60 2+80 3)/5=72 km/h. Tell me, is the average speed equal to the arithmetic mean of the speeds now? No.
The most important thing to remember when finding average speed is that it is an average, not an arithmetic average. Of course, when you hear the problem, you immediately want to add the speeds and divide by 2. This is the most common mistake.
The average speed is equal to the arithmetic mean of the velocities of the body during movement only if the body with these velocities travels all the way in the same time intervals.
4. Problem solving (15 min)
Task number 1. The speed of the boat with the current is 24 km per hour, against the current 16 km per hour. Find the average speed.(Checking the assignments at the blackboard.)
Solution. Let S be the path from the starting point to the final point, then the time taken to travel downstream is S/24, and upstream is S/16, the total travel time is 5S/48. Since the entire journey, round trip, is 2S, therefore, the average speed is 2S/(5S/48)=19.2 km per hour.
Pilot study“Uniformly accelerated motion, initial velocity is zero”(Experiment conducted by students)
Before starting to execute practical work remember the rules of TB:
- Before starting work: carefully study the content and procedure for conducting a laboratory workshop, prepare workplace and remove foreign objects, place devices and equipment in such a way as to prevent their falling and overturning, check the serviceability of equipment and devices.
- During work : accurately follow all the instructions of the teacher, without his permission, do not do any work on your own, monitor the serviceability of all fasteners in devices and fixtures.
- Upon completion of work: tidy up the workplace, hand over the instruments and equipment to the teacher.
Investigation of the dependence of speed on time with uniformly accelerated motion (the initial speed is zero).
Target: study of uniformly accelerated motion, plotting v=at dependence on the basis of experimental data.
From the definition of acceleration it follows that the speed of the body v, moving in a straight line with constant acceleration, after some time tafter the start of movement can be determined from the equation: v\u003d v 0 +at. If the body began to move without an initial velocity, that is, at v0 = 0, this equation becomes simpler: v= a t. (1)
The speed at a given point of the trajectory can be determined by knowing the movement of the body from rest to this point and the time of movement. Indeed, when moving from a state of rest ( v0 = 0 ) with constant acceleration, the displacement is determined by the formula S= at 2 /2, whence, a=2S/ t 2 (2). After substituting formula (2) into (1): v=2 S/t (3)
To perform work, the rail guide is set with a tripod in an inclined position.
Its upper edge should be at a height of 18-20 cm from the table surface. A plastic mat is placed under the bottom edge. The carriage is installed on the guide in the uppermost position, and its protrusion with the magnet should be facing the sensors. The first sensor is placed near the carriage magnet so that it starts the stopwatch as soon as the carriage starts to move. The second sensor is installed at a distance of 20-25 cm from the first one. Further work is performed in this order:
- They measure the movement that the carriage will make when moving between the sensors - S 1
- They start the carriage and measure the time of its movement between the sensors t 1
- According to formula (3), the speed with which the carriage moved at the end of the first section v 1 \u003d 2S 1 / t 1
- Increase the distance between the sensors by 5 cm and repeat a series of experiments to measure the speed of the body at the end of the second section: v 2 \u003d 2 S 2 /t 2 The carriage in this series of experiments, as in the first, is allowed from its uppermost position.
- Two more series of experiments are carried out, increasing the distance between the sensors by 5 cm in each series. This is how the values \u200b\u200bof the speed v h and v 4
- Based on the data obtained, a graph of the dependence of speed on the time of movement is built.
- Summing up the lesson
Homework with comments:Choose any three tasks:
1. A cyclist, having traveled 4 km at a speed of 12 km/h, stopped and rested for 40 minutes. He traveled the remaining 8 km at a speed of 8 km/h. Find the average speed (in km/h) of the cyclist for the whole journey?
2. The cyclist traveled 35 m in the first 5 s, 100 m in the next 10 s, and 25 m in the last 5 s. Find the average speed for the entire journey.
3. For the first 3/4 of the time of its movement, the train traveled at a speed of 80 km / h, the rest of the time - at a speed of 40 km / h. What is the average speed (in km/h) of the train for the entire journey?
4. The car traveled the first half of the way at a speed of 40 km/h, the second - at a speed of 60 km/h. Find the average speed (in km/h) of the car for the whole journey?
5. The car drove the first half of the way at a speed of 60 km/h. He drove the rest of the way at a speed of 35 km/h, and the last section at a speed of 45 km/h. Find the average speed (in km/h) of the car for the entire journey.
“Practice is born only from the close connection of physics and mathematics” Bacon F.
a) “Acceleration” (initial speed is less than final) b) “Deceleration” (final speed is less than initial)
Orally 1. Find the speed of the car if it traveled 180 km in 3 hours. 2. The car drove 1 hour at a speed of 80 km/h and 1 hour at a speed of 60 km/h. Find your average speed. Indeed, the average speed is (80+60)/2=70 km/h. In this case, the average speed is equal to the arithmetic mean of speeds. 3. Let's change the condition. The car traveled 2 hours at a speed of 60 km/h and 3 hours at a speed of 80 km/h. What is the average speed for the whole journey?
(60*2+80*3)/5=72 km/h. Tell me, is the average speed equal to the arithmetic mean of the speeds now?
Task The speed of the boat with the current is 24 km per hour, against the current 16 km per hour. Find the average speed of the boat.
Solution. Let S be the path from the starting point to the final point, then the time spent on the path along the stream is S / 24, and against the current - S / 16, the total travel time is 5S / 48. Since the entire journey, round trip, is 2S, therefore, the average speed is 2S/(5S/48)=19.2 km per hour.
Solution. Vav = 2s / t 1 + t 2 t 1 = s / V 1 and t 2 = s / V 2 Vav = 2s / V 1 + s / V 2 = 2 V 1 V 2 / V 1 + V 2 V av = 19.2 km/h
To the house: The cyclist rode the first third of the track 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 kilometers per hour.
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 during 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:; ; ;
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
- 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
Preparation for ZNO. Physics.
Synopsis 2. Uneven movement.
5. Univariate (uniformly accelerated) movement
Uneven movement– movement with variable speed.
Definition. Instant Speed- the speed of the body at a given point of the trajectory, at a given time. It is found by the ratio of the movement of the body to the time interval ∆t, during which this movement was made, if the time interval tends to zero.
Definition. Acceleration 
- a value showing how much the speed changes over a time interval ∆t.
Where is the final, and is the initial speed for the considered time interval.
Definition. Equally variable rectilinear motion (uniformly accelerated)- this is a movement in which for any equal time intervals the speed of the body changes by an equal value, i.e. is a movement with constant acceleration.
Comment. Saying that the motion is uniformly accelerated, we consider that the speed increases, i.e. acceleration projection when moving along the reference direction (velocity and acceleration coincide in direction), and speaking - uniformly slowed down, we consider that the speed decreases, i.e. (velocity and acceleration are directed towards each other). In school physics, both of these movements are usually called uniformly accelerated.
Displacement equations, m:
Graphs of uniformly variable (uniformly accelerated) rectilinear motion:
The graph is a straight line parallel to the time axis.
The graph is a straight line that is built “by points”.
Comment. The speed graph always starts from the initial speed.
SPEED IN IRREGULAR MOVEMENT
Unevenis called a movement in which the speed of the body changes with time.
The average speed of uneven movement is equal to the ratio of the displacement vector to the travel time
Then the displacement with uneven motion
instantaneous speed called the speed of the body at a given time or at a given point in the trajectory.
Speed- This quantitative characteristic body movements.
average speed is a physical quantity equal to the ratio of the point displacement vector to the time interval Δt during which this displacement occurred. The direction of the average velocity vector coincides with the direction of the displacement vector . The average speed is determined by the formula:
Instant Speed , that is, the speed at a given moment of time is a physical quantity equal to the limit to which the average speed tends with an infinite decrease in the time interval Δt:
In other words, the instantaneous speed at a given moment of time is the ratio of a very small movement to a very small period of time during which this movement occurred.
The instantaneous velocity vector is directed tangentially to the trajectory of the body (Fig. 1.6).
Rice. 1.6. Instantaneous velocity vector.
In the SI system, speed is measured in meters per second, that is, the unit of speed is considered to be the speed of such uniform rectilinear motion, in which in one second the body travels a distance of one meter. The unit of speed is denoted m/s. Often speed is measured in other units. For example, when measuring the speed of a car, train, etc. The commonly used unit of measure is kilometers per hour:
1 km/h = 1000 m / 3600 s = 1 m / 3.6 s
or
1 m/s = 3600 km / 1000 h = 3.6 km/h
Addition of speeds
The velocities of the body in different reference systems are connected by the classical law of addition of speeds.
body speed relative to fixed frame of reference is equal to the sum of the velocities of the body in moving frame of reference and the most mobile frame of reference relative to the fixed one.
For example, a passenger train is moving along a railroad at a speed of 60 km/h. A person is walking along the carriage of this train at a speed of 5 km/h. If we consider the railway to be motionless and take it as a frame of reference, then the speed of a person relative to the frame of reference (that is, relative to railway), will be equal to the addition of the speeds of the train and the person, that is, 60 + 5 = 65, if the person goes in the same direction as the train; and 60 - 5 = 55 if the person and the train are moving in different directions. However, this is only true if the person and the train are moving along the same line. If a person moves at an angle, then this angle will have to be taken into account, remembering that speed is vector quantity.
Now let's look at the example described above in more detail - with details and pictures.
So, in our case, the railway is fixed frame of reference. The train that is moving along this road is moving frame of reference. The car on which the person is walking is part of the train.
The speed of a person relative to the car (relative to the moving frame of reference) is 5 km/h. Let's call it C.
The speed of the train (and hence the wagon) relative to a fixed frame of reference (that is, relative to the railway) is 60 km/h. Let's denote it with the letter B. In other words, the speed of the train is the speed of the moving reference frame relative to the fixed frame of reference.
The speed of a person relative to the railway (relative to a fixed frame of reference) is still unknown to us. Let's denote it with a letter.
Let's associate the XOY coordinate system with the fixed reference system (Fig. 1.7), and the X P O P Y P coordinate system with the moving reference system (see also the Reference System section). And now let's try to find the speed of a person relative to a fixed frame of reference, that is, relative to the railway.
For a short period of time Δt, the following events occur:
Then for this period of time the movement of a person relative to the railway:
H+B
This displacement addition law. In our example, the movement of a person relative to the railway is equal to the sum of the movements of a person relative to the wagon and the wagon relative to the railway.
The law of addition of displacements can be written as follows:
= ∆ H ∆t + ∆ B ∆t
