Irregular movement. average speed

There are very few examples of uniform motion in nature. The Earth moves almost evenly around the Sun, raindrops are dripping, bubbles in soda pop up, the clock hand is moving.

There are many examples of uneven movement Flying a ball while playing football, moving a cat while hunting a bird, moving a car

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:

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:

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

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:

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:

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 modulus of the body's velocity increases with time, the direction of acceleration coincides with the direction of the speed of motion.

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.

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:

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

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 in case of uniform 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:

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

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 track the speed was the same, and 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 of these speeds is km / h, we will also find the average speed in km / h. Therefore, we will not translate these problems 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 drive in a straight line along a stretch of 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 precisely.

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 velocity 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

1. Uniform movement is rare. Typically, mechanical movement is movement with varying speed. The movement in which the speed of the body changes over time is called uneven.

For example, transport is moving unevenly. The bus, starting movement, increases its speed; when braking, its speed decreases. Bodies falling to the surface of the Earth also move unevenly: their speed increases over time.

With uneven movement, the coordinate of the body can no longer be determined by the formula x = x 0 + v x t, since the speed of movement is not constant. The question arises, what value characterizes the rate of change in body position over time with uneven movement? This value is average speed.

Average speed vWeduneven movement is called a physical quantity equal to the ratio of displacement sbodies by time t, for which it was committed:

Average speed is vector quantity... To determine the modulus of the average velocity for practical purposes, this formula can be used only in the case when the body moves along a straight line in one direction. In all other cases, this formula is unusable.

Let's look at an example. It is necessary to calculate the time of arrival of the train at each station along the route. Moreover, its movement is not rectilinear. If we calculate the modulus of the average speed in the section between two stations using the above formula, then the obtained value will differ from the value of the average speed with which the train was moving, since the modulus of the displacement vector is less than the distance traveled by the train. And the average speed of movement of this train from the starting point to the final point and back, in accordance with the above formula, is completely zero.

In practice, when determining the average speed, a value equal to relation of the way l In time t, for which this path was passed:

v Wed = .

She is often called average ground speed.

2. Knowing the average speed of a body on any part of the trajectory, it is impossible to determine its position at any time. Suppose that the car traveled 300 km in 6 hours. 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 a car for 6 hours, we cannot determine its position after 1 hour, after 2 hours, after 3 hours, etc.

3. When moving, the body passes sequentially all points of the trajectory. At each point, it is at certain points in time and has some kind of speed.

Instantaneous speed is the speed of a body at a given moment in time or at a given point on the trajectory.

Suppose that the body makes an uneven rectilinear motion. Let us determine the speed of movement of this body at the point O its trajectory (fig. 21). Select a section on the trajectory AB inside which the point is O... Moving s 1 in this area, the body has completed in time t 1 . Average speed of movement in this section - v Wed 1 =.

Let's reduce the movement of the body. Let it be equal s 2, and the time of movement is t 2. Then the average speed of the body during this time: v cf 2 =. Let's decrease the displacement again, the average speed in this section: v Wed 3 =.

We will continue to reduce the time of body movement and, accordingly, its movement. Eventually, the movement and time will become so small that a device, for example, a speedometer in a car, will no longer register the change in speed and the movement in this short period of time can be considered uniform. The average speed in this area is the instantaneous speed of the body at the point O.

Thus,

instantaneous velocity is a vector physical quantity equal to the ratio of small displacement D sto a small time interval D t, for which this movement was made:

Self-test questions

1. What movement is called uneven?

2. What is called average speed?

3. What does the average ground speed show?

4. Is it possible, knowing the trajectory of the body and its average speed for a certain period of time, to determine the position of the body at any time?

5. What is called instantaneous speed?

6. How do you understand the expressions "small displacement" and "small time interval"?

Assignment 4

1. The car drove through Moscow streets 20 km in 0.5 hour, when leaving Moscow it stood for 15 minutes, and in the next 1 hour and 15 minutes traveled 100 km in the Moscow region. What was the average speed of the vehicle on each section and all the way?

2. What is the average speed of a train on the stretch between two stations, if it traveled the first half of the distance between stations at an average speed of 50 km / h, and the second - at an average speed of 70 km / h?

3. What is the average speed of a train on the stretch between two stations if it traveled half the time at an average speed of 50 km / h, and the remaining time at an average speed of 70 km / h?