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 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 traveled 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 average speed 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 instantaneous speed.
Instant Speed is the speed of the body in this moment 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 started to study uniform motion, i.e. moving at a variable 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
Section 1 MECHANICS
Chapter 1: Fundamentals of kinematics
mechanical movement. Trajectory. Path and movement. Addition of speeds
mechanical movement of the body called the change in its position in space relative to other bodies over time.
The mechanical movement of bodies studies Mechanics. A section of mechanics that describes the geometric properties of motion without taking into account the masses of bodies and active forces, is called kinematics .
Mechanical movement is relative. To determine the position of a body in space, you need to know its coordinates. To determine the coordinates material point it is necessary, first of all, to choose a reference body and associate a coordinate system with it.
Reference bodya body is called, relative to which the position of other bodies is determined. The reference body is chosen arbitrarily. It can be anything: land, building, car, ship, etc.
The coordinate system, the body of reference with which it is associated, and the indication of the time reference form reference system , relative to which the motion of the body is considered (Fig. 1.1).
A body whose dimensions, shape and structure can be neglected when studying a given mechanical movement, is called material point . A material point can be considered a body whose dimensions are much smaller than the distances characteristic of the motion considered in the problem.
Trajectoryis the line along which the body moves.
Depending on the type of trajectory of movement, they are divided into rectilinear and curvilinear.
Pathis the length of the trajectory ℓ(m) ( fig.1.2)
The vector drawn from the initial position of the particle to its final position is called moving this particle for a given time.
Unlike the path, the displacement is not a scalar, but a vector quantity, since it shows not only how far, but also in what direction the body has moved in a given time.
Displacement vector modulus(that is, the length of the segment that connects the start and end points of the movement) can be equal to the distance traveled or less than the distance traveled. But the displacement module can never be greater than the distance traveled. For example, if a car moves from point A to point B along a curved path, then the absolute value of the displacement vector is less than the distance traveled ℓ. The path and the displacement module are equal only in one single case, when the body moves in a straight line.
Speedis a vector quantitative characteristic of the movement of the body
average speedis a physical quantity equal to the ratio of the point displacement vector to the time interval
The direction of the average velocity vector coincides with the direction of the displacement vector.
instant speed, that is, the speed at a given time is a vector physical quantity, equal to the limit, to which the average speed tends with an infinite decrease in the time interval Δt.
The instantaneous velocity vector is directed tangentially to the motion trajectory (Fig. 1.3).
In the SI system, speed is measured in meters per second (m / s), that is, the unit of speed is considered to be the speed of such a uniform rectilinear motion, at which in one second the body travels a distance of one meter. Speed is often measured in kilometers per hour.
or 1 
Addition of speeds
Any mechanical phenomena are considered in some frame of reference: movement makes sense only relative to other bodies. When analyzing the motion of the same body in different systems reference, all kinematic characteristics of motion (path, trajectory, movement, speed, acceleration) are different.
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 stationary and take it as a frame of reference, then the speed of a person is relatively railway, will be equal to the addition of the speeds of the train and the person, that is
60km/h + 5km/h = 65km/h if the person is walking in the same direction as the train and
60km/h - 5km/h = 55km/h if the person is walking against the direction of the train.
However, this is only true in this case, if the person and the train are moving along the same line. If a person moves at an angle, then this angle must be taken into account, and the fact that speed is a vector quantity.
Let's consider the example described above in more detail - with details and pictures.
So, in our case, the railway is a fixed frame of reference. The train that moves along this road is a 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 denote it with a letter. 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 a letter. In other words, the speed of the train is the speed of the moving frame relative to the fixed frame.
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 with the fixed frame of reference (Fig. 1.4) the XOY coordinate system, and with the moving frame of reference - X p O p Y p. Let us now determine the speed of a person relative to the fixed frame of reference, that is, relative to the railway.
For a short period of time Δt, the following events occur:
The person moves relative to the car at a distance
The wagon moves relative to the railway for a distance
Then for this period of time the movement of a person relative to the railway:
it 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.
Dividing both parts of the equality by a small period of time Dt, during which the movement occurred:
We get:
|
Reference system.
reference system- this is a set of a reference body, an associated coordinate system and a time reference system, in relation to which the movement (or equilibrium) of any material points or bodies is considered
Trajectory, path and displacement.
Displacement vector- vector starting point which coincides with the initial position of the moving point and the end of the vector with its final position.
Trajectory of movement of a material point- a line described by this point in space (rectilinear or curvilinear).
way point is the sum of the lengths of all parts of the trajectory passed by the point during the considered time interval.
Material point.
Material point- a body that has mass and speed, but whose dimensions and shapes are not significant under the conditions of this problem.
Average speed.
The average speed of a moving point over a period of time t- a vector quantity equal to the ratio of the displacement vector to the time interval during which this displacement occurred.
Average (ground) speed
Average moving speed (vector mean)
Relativity of motion.
Relativity of mechanical motion- this is the dependence of the trajectory of the body, the distance traveled, displacement and speed on the choice of the reference system.
The law of addition of velocities in classical mechanics.
Vabs = Vrel + Vtrans
The absolute velocity of a material point is equal to the vector sum of the translational and relative velocity.
Rectilinear uniform motion.
Rectilinear uniform motion- movement with a constant modulus and direction speed.
Equations of motion and graphs x(t), vx(t), s(t) for uniform rectilinear motion.
equation of uniform rectilinear motion of a material point:
(17)
Or
Formulas for uniform rectilinear motion
| = const |  = const |
|
 |  |
|
| S \u003d v (t - t 0) | ||
Graphs of speed, projections of speed, path and coordinates versus time for uniform rectilinear motion
Velocity graph v = v(t)
Graph of the speed of uniform motion is a straight line parallel to the x-axis (t-axis).
On schedule v = v(t) you can find the distance traveled for the time interval t: it is numerically equal to the area of \u200b\u200bthe OABS figure (rectangle):
q(area of rectangle OABC) = OA OC v 1 t 1 S
Path chart S = S(t)
The uniform motion path graph is a straight line that forms an angle with the time axis.
On this chart, but v~tg(the speed of uniform motion is proportional to the tangent of the angle that the path graph makes with the time axis).
Graph of point coordinates versus time: x = x(t)
Equation x \u003d x 0 + v x (t - t 0) - linear function, so the graph x = x(t) is a straight line that forms an angle with the time axis.