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 railway 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 stationary and take it as a reference frame, then the speed of a person relative to the reference system (that is, relative to the 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

Lesson #3

Topic. Uniform rectilinear motion. Speed. The law of addition of speeds. Movement charts.

Target: the formation of knowledge about rectilinear motion, speed as a physical quantity, the classical law of adding speeds, the solution of the main problem of mechanics for rectilinear uniform motion; consideration of graphs of the dependence of speed, coordinates of rectilinear uniform motion on time.

Lesson type: combined lesson.

1. 
   Organizational stage
2. 
   ^ Examination homework.

The teacher selectively checks the written homework of three or four students or involves students with a high level of preparation in such a check.

front poll.

• 
  What is a reference system?
• 
  What is a trajectory? What types of fissile motion depending on the trajectory?
• 
  What is called a path? moving?
• 
  What is the difference between path and movement?
• 
  What is the essence of the concept of relativity of motion?

1. 
   Reporting the topic, purpose and tasks of the lesson

Topic study plan

1. 
   Uniform rectilinear motion.
2. 
   The speed of uniform rectilinear motion as a physical quantity.
3. 
   The law of adding speeds.
4. 
   Moving rectilinear uniform motion. Solution of the main problem of mechanics for rectilinear uniform motion.
5. 
   Movement charts.

1. 
   Learning new material

1. Uniform rectilinear motion

The simplest type of motion is uniform rectilinear motion.

Uniform rectilinear movement called such a movement of the body, in which the body for any equal time intervals performs the same movement and the trajectory of its movement is a straight line.

Question for students:

1. 
   Give examples of uniform rectilinear motion.
2. 
   What do you think, do we often encounter cases of rectilinear uniform motion?
3. 
   Why study this type of movement, be able to describe its patterns?

^ 2. The speed of uniform rectilinear motion as a physical quantity

One of the characteristics of uniform rectilinear motion is its speed. The teacher invites students to characterize the speed as a physical quantity according to the generalized plan of the characteristic physical quantity.

Generalized plan for the characteristics of a physical quantity:

1. 
   A phenomenon that characterizes a value.
2. 
   Definition, designation.
3. 
   Formulas that relate a given quantity to other quantities.
4. 
   Units.
5. 
   Measurement methods.

The speed of uniform rectilinear motion as a physical quantity

1. 
   direct measurements (using a speedometer, radar);
2. 
   indirect measurements (by formula)

We designate:

- velocity vector;

υ x , υ y - projections of the velocity vector on the coordinate axes Ox, Oy;

υ - speed modulus.

Question:

Can the velocity projection be negative? (The velocity projection can be either positive or negative depending on how the body is moving (Figure 1).)

1. 
   ^ The law of addition of speeds

As we already know, speed is a relative value and depends on the chosen frame of reference.

If moving the same material point consider with respect to two reference systems associated with a fixed body and a moving one (for example, a person who is standing on the bank of the river along which this boat is sailing, and a person who is at the same time on the boat is watching the movement of a person on the deck of a boat), then we can formulate the classical law of adding velocities.

The law of adding speeds: the speed of the body relative to the fixed frame of reference is equal to the vector sum of the speed of the body relative to the moving frame and the actual speed of the moving frame relative to the fixed one:

where and are the velocities of the body relative to the fixed and moving frames of reference, respectively, and is the speed of the moving frame of reference relative to the fixed one (Fig. 2).

1. 
   ^ Moving rectilinear uniform motion. Solution of the main problem of mechanics for rectilinear uniform motion

From the formula
you can determine the displacement modulus for rectilinear uniform motion:
.

If a material point, moving along the OX axis, has moved from a point with coordinate x 0 to a point with coordinate X , then for the time t she moved:
(Fig. 3).

Since the main task of mechanics is to determine the position of a body at a given moment of time according to known initial conditions, the equation
and is a solution to the main problem of mechanics.

This equation is also called the basic law of uniform rectilinear motion.

1. 
   Movement charts

1. 
   Velocity vs. Time Plot

Function Graph
is a straight line parallel to the time axis t (Fig. 4, a).

If > 0, then this line passes above the time axis t , and if t.

The area of ​​the figure bounded by the graph and the axis t , is numerically equal to the displacement modulus (Fig. 4, b).

1. 
   Graph of displacement projection versus time

schedule
is a straight line passing through the origin. If > 0, then s x increases with time, if s x decreases with time (Fig. 5, a). The slope of the graph is greater, the greater the velocity modulus (Fig. 5, b).

If in question about the path graph, it should be remembered that the path is the length of the trajectory, therefore it cannot decrease, but can only grow with time, therefore, this graph cannot approach the time axis (Fig. 5, c).

1. 
   ^ Plot of coordinate versus time

Schedule
different from chart
only by shifting x 0 along the coordinate axis.

The point of intersection of graphs 1 and 2 corresponds to the moment when the coordinates of the bodies are equal, that is, this point determines the moment in time and the coordinate of the meeting of two bodies (Fig. 6).

1. 
   Application of acquired knowledge

Problem solving (oral)

1. 
   Moving objects are given in random order: pedestrian; sound waves in the air; oxygen molecule at 0 °C; weak wind; electromagnetic waves in a vacuum; storm wind.

Try to arrange objects in descending order according to speeds (velocities of objects are not given, students use pre-acquired knowledge, intuition).

Answer:

1. 
   electromagnetic waves in vacuum (300,000 km/s);
2. 
   oxygen molecule at 0 °C (425 m/s);
3. 
   sound waves in the air (330 m/s);
4. 
   storm wind (21 m/s);
5. 
   light wind (4 m/s);
6. 
   pedestrian (1.3 m/s).

1. 
   Summing up the lesson and reporting homework

The teacher sums up the lesson, evaluates the activities of students.

Homework

1. 
   Learn the theoretical material from the textbook.
2. 
   Solve problems.

Test

Find the correct answer.

1. 
   Which of the following examples of motion can be considered uniform?

1. 
   Car is braking
2. 
   The passenger goes down the subway escalator
3. 
   Airplane takes off

1. 
   Rectilinear uniform movement is called, in which:

1. 
   the modulus of the body's velocity remains unchanged
2. 
   the speed of the body changes by the same value in any equal intervals of time
3. 
   the body performs the same movements for any time intervals

1. 
   A passenger train, moving uniformly, covered a distance of 30 km in 20 minutes. Find the speed of the train.

BUT 10 m/s B 15 m/s IN 25 m/s

1. 
   A motorcycle is moving at a speed of 36 km/h. How far will it travel in 20 seconds?

BUT 200 m B 720 km IN 180 m

1. 
   On fig. Figure 7 shows a graph of the path of uniform motion versus time. What is the speed of the body?

BUT 5 m/s B 10 m/s IN 20 m/s

1. 
   On fig. Figure 8 shows a graph of the speed of uniform motion versus time. What is the distance traveled by the body in 3 s?

BUT 4 m B 18 m IN 36 m

acceleration is called a vector physical quantity equal to the ratio of a very small change in the velocity vector to a small period of time during which this change occurred, i.e. is a measure of the rate of change of speed:

;
.

A meter per second per second is such an acceleration at which the speed of a body moving in a straight line and uniformly accelerated changes by 1 m / s in a time of 1 s.

The direction of the acceleration vector coincides with the direction of the velocity change vector (
) at very small values ​​of the time interval during which the velocity changes.

If the body moves in a straight line and its speed increases, then the direction of the acceleration vector coincides with the direction of the velocity vector; when the speed decreases, it is opposite to the direction of the speed vector.

When moving along a curvilinear trajectory, the direction of the velocity vector changes in the process of movement, the acceleration vector can be directed at any angle to the velocity vector.

Uniform, uniformly accelerated rectilinear motion

Moving at a constant speed is called uniform rectilinear motion. In uniform rectilinear motion, the body moves in a straight line and for any equal intervals of time covers the same path.

A movement in which a body makes unequal movements in equal intervals of time is called uneven movement. With such a movement, the speed of the body changes with time.

equivariable is called such a movement in which the speed of the body for any equal time intervals changes by the same amount, i.e. movement with constant acceleration.

uniformly accelerated called uniformly variable motion, in which the magnitude of the speed increases. equally slow- uniformly variable motion, in which the magnitude of the speed decreases.

Addition of speeds

Consider the movement of a body in a moving coordinate system. Let be – movement of the body in a moving coordinate system, - movement of the moving coordinate system relative to the fixed one, then – the movement of the body in a fixed coordinate system is equal to:

.

If displacement And happen at the same time, then:

.

In this way

.

We have found that the speed of a body relative to a fixed frame of reference is equal to the sum of the speed of a body in a moving frame of reference and the speed of a moving frame of reference relative to a fixed one. This statement is called the classical law of addition of velocities.

Graphs of the dependence of kinematic quantities on time in uniform and uniformly accelerated motion

With uniform motion:

Velocity graph - straight line y=b;

Acceleration graph - straight line y= 0;

The displacement graph is a straight line y=kx+b.

With uniformly accelerated motion:

Velocity graph - straight line y=kx+b;

Acceleration graph - straight line y=b;

Movement graph - parabola:

if a>0, branches up;

the greater the acceleration, the narrower the branches;

the vertex coincides in time with the moment when the speed of the body is zero;

usually passes through the origin.

Free fall of bodies. Acceleration of gravity

Free fall is the movement of a body when only the force of gravity acts on it.

In free fall, the acceleration of the body is directed vertically downward and is approximately equal to 9.8 m/s 2 . This acceleration is called free fall acceleration and the same for all bodies.

Uniform movement around the circumference

With uniform motion in a circle, the value of the speed is constant, and its direction changes in the process of motion. The instantaneous velocity of a body is always directed tangentially to the trajectory of motion.

Because If the direction of the velocity is constantly changing during uniform motion in a circle, then this motion is always uniformly accelerated.

The time interval for which the body makes a complete revolution when moving in a circle is called the period:

.

Because the circumference s is equal to 2R, the period of revolution for a body moving uniformly at a speed v along a circle with radius R is equal to:

.

The reciprocal of the period of revolution is called the frequency of revolution and shows how many revolutions the body makes in a circle per unit time:

.

The angular velocity is the ratio of the angle through which the body has turned to the time of rotation:

.

Angular velocity is numerically equal to the number of revolutions in 2 seconds.

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'll look at uneven movement.

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.

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:; ; ;

To 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