Instantaneous movement speed. Instantaneous speed: concept, calculation formula, recommendations for finding

As we have already noted, uniform motion is the simplest model mechanical movement. If such a model is not applicable, then more complex ones should be used. To construct them, we need to introduce and consider the concept of speed in the case of non-uniform motion.
Let the material point move in such a way that its law of motion has the form of a smooth curve DIA(Fig. 40).

Rice. 40
For the time interval from t o before t1 point coordinate changed from x o before x 1. If we calculate the speed according to the previous rule
v cp \u003d Δx / Δt \u003d (x 1 - x o) / (t 1 - t o). (1)
and write the equation of the law of motion as for uniform motion
x \u003d x o + v cp (t - t o), (2)
then this function will coincide with the real law of motion only at the extreme points of the interval, where the straight line AB(which is described by equation (2)) intersects with the curve DIA. If we want to calculate by formula (2) the coordinate of a point at an intermediate point in time, then we get the value X //, which may differ markedly from the true value X /.
Thus, the speed (it is called the average speed), calculated by formula (1), in this case characterizes the speed of moving the point over the entire interval on average, but it does not allow calculating the coordinates of the point at an arbitrary point in time.
The average speed is a physical quantity equal to the ratio of the change in the coordinate of a point to the time interval during which this change occurred.
The geometric meaning of the average speed is the slope coefficient of the secant AB graphics of the law of motion.
For a more detailed, more accurate description of the movement, two values ​​of the average speed can be set:
a) during the time period t o before t /
v cp1 = (x / − x o)/(t / − t o);
b) in the time interval from t / before t1
v cp2 = (x 1 - x /)/(t 1 - t /).
If a law of motion is constructed from these two average velocities, then it will be represented by a broken line DIA, which more accurately describes the actual movement of the point. And if such accuracy does not suit us, then it is necessary to split the time intervals further - into four, eight, etc. parts. In this case, it is necessary to set four, eight, etc. values ​​of average speeds, respectively. Agree, such a description becomes cumbersome and inconvenient. A way out of this situation has long been found - it lies in the fact that you need to consider the speed as a function of time.
Let's see how it will change average speed with a decrease in the time interval for which we calculate this speed. We will calculate the average speed for the time interval from t o before t1, successively approximating the value to t o. In this case, the family of secants A o A 1, A o A 1 /, A o A 1 //(Fig. 41)

rice. 41
will tend to some limit position of the line A o B, which is tangent to the graph of the law of motion.
Let us give another example of the law of motion to show that the instantaneous speed can be either greater or less than the average speed (Fig. 42 with the same notation as in Fig. 41).

rice. 42
The procedure for refining the description of motion can also be shown algebraically by successively calculating the ratios
v cp = (x 1 - x o)/(t 1 - t o), v cp / = (x 1 / - x o)/(t 1 / - t o), v cp // = (x 1 // - x o) /(t 1 // − t o).
In this case, it turns out that these quantities approach some well-defined value. This limit is called the instantaneous speed.
Instantaneous speed is the ratio of the change in the coordinate of a point to the time interval during which this change occurred, with a time interval tending to zero 1:
v = ∆x/∆t at ∆t → 0. (3)
The geometric meaning of the instantaneous speed is the slope coefficient of the tangent to the graph of the law of motion.
Thus, we “tied” the value of the instantaneous speed to a specific moment in time - we set the value of the speed in this moment time at a given point in space. Thus, we have the opportunity to consider the speed of the body as a function of time, or a function of coordinates.
From a mathematical point of view, this is much more convenient than setting the values ​​of average speeds over many small time intervals. Let's think about it: does the speed have a physical meaning at a given moment of time? Speed ​​is a characteristic of movement, in this case, the movement of a body in space. In order to fix the movement, it is necessary to observe the movement for a certain period of time. To measure the speed, a period of time is also needed. Even the most advanced speed meters - radar installations - measure the speed of moving vehicles, albeit for a small (on the order of one millionth of a second) interval, but not at some point in time. Therefore, the expression "velocity at a given time" from the point of view of physics is incorrect. Nevertheless, in mechanics they constantly use the concept of instantaneous speed, which is very convenient in mathematical calculations. Mathematically, logically, we can consider the passage to the limit ∆t → 0, and physically there is the minimum possible value of the interval Δt, for which you can measure the speed.
However, if we study the movement of a car for several hours, then a period of time of one second can be considered infinitely small.
Thus, the concept of instantaneous velocity is a reasonable compromise between the simplicity of the mathematical description and the strict physical meaning. Such "compromises" we will meet in the course of studying physics constantly.
In the future, speaking of speed, we will have in mind exactly the instantaneous speed. Note that with uniform motion, the instantaneous speed is equal to the previously determined speed because, with uniform motion, the ratio ∆x/∆t does not depend on the value of the time interval, therefore it remains unchanged even for an arbitrarily small Δt.
Since speed can depend on time, it should be considered as a function of time and depicted as a graph.
With uniform motion at a constant speed y, the graph of the dependence of speed on time is a straight line parallel to the time axis (in Fig. 43 - a straight line AB).
Consider the time interval from t o before t1. The product of the value of this interval ( t 1 − t o) for speed v o equals, on the one hand, a change in the coordinate Δx, and on the other hand, the area of ​​the rectangle under the graph of velocity versus time.

rice. 43
The area under the graph should be understood, again, in physical sense, as a product of physical quantities having different dimensions, and not in a purely geometric sense− as a product of the lengths of the segments.
Let us show that the area under the graph of the dependence of speed on time is equal to the change in coordinate for any dependence of speed on time v(t). Let's divide the travel time from t o before t into small intervals of Δt; on each interval we determine the average speed v1. Then the area of ​​the rectangle with base Δt and height v1(in Fig. 44 it is marked with denser shading) will be equal to the change in the coordinate over this short period of time. The sum of the areas of all such rectangles (shaded in Fig. 44)

rice. 44
will be equal to the change in the coordinate of the point for the considered period of time of movement from t o before t1. If now all time intervals Δt decrease (respectively increasing their number), then the sums of the areas of the rectangles will tend to the area curvilinear trapezoid under the graph of the function v(t).
Let us supplement our definition of the area under the curve with one more convention: we will assume that if the curve lies t under the time axis (that is, the speed is negative), then we will consider the corresponding area as negative (Fig. 45).

rice. 45

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 is railway 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:; ; ;

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 main task 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

In order to better understand this definition Let's look at an example.

Let the car move in a straight line on a section of the highway. We have a plot of displacement projection versus time for this movement(Fig. 13), let's analyze this chart.

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.

On the this lesson we began to study non-uniform motion, that is, motion with varying 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

Its coordinates change. Coordinates can change quickly or slowly. Physical quantity, which characterizes the rate of change of coordinates, is called speed.

Example

The average speed is a vector quantity numerically equal to the displacement per unit time and co-directional with the displacement vector: $\left\langle v\right\rangle =\frac(\triangle r)(\triangle t)$ ; $\left\langle v\right\rangle \uparrow \uparrow \triangle r$

Figure 1. Average speed is co-directed to movement

The modulus of the average speed along the path is: $\left\langle v\right\rangle =\frac(S)(\triangle t)$

Instantaneous speed gives accurate information about the movement at a certain point in time. The expression "velocity of a body at a given moment of time" from the point of view of physics is not correct. However, the concept of instantaneous speed is very convenient in mathematical calculations, and it is constantly used.

Instantaneous velocity (or simply velocity) is the limit to which the average velocity $\left\langle v\right\rangle $ tends as the time interval $\triangle t$ tends to zero:

$v=(\mathop(lim)_(\triangle t) \frac(\triangle r)(\triangle t)\ )=\frac(dr)(dt)=\dot(r)$ (1)

The vector $v$ is directed tangentially to the curvilinear trajectory, since the infinitesimal (elementary) displacement dr coincides with the infinitesimal element of the trajectory ds.

Figure 2. Instantaneous velocity vector $v$

AT Cartesian coordinates equation (1) is equivalent to three equations

$\left\( \begin(array)(c) v_x=\frac(dx)(dt)=\dot(x) \\ v_y=\frac(dy)(dt)=\dot(y) \\ v_z =\frac(dz)(dt)=\dot(z) \end(array) \right.$ (2)

The modulus of the vector $v$ in this case is equal to:

$v=\left|v\right|=\sqrt(v^2_x+v^2_y+v^2_z)=\sqrt(x^2+y^2+z^2)$ (3)

The transition from Cartesian rectangular coordinates to curvilinear ones is carried out according to the rules of differentiation complex functions. Let the radius vector r be a function of curvilinear coordinates: $r=r\left(q_1,q_2,q_3\right)\ $. Then the speed $v=\frac(dr)(dt)=\sum^3_(i=1)(\frac(\partial r)(\partial q_i)\frac(\partial q_i)(\partial t))= \sum^3_(i=1)(\frac(\partial r)(\partial q_i))\dot(q_i)$

Figure 3. Displacement and instantaneous velocity in curvilinear coordinate systems

In spherical coordinates, setting $q_1=r;\ \ q_2=\varphi ;\ \ q_3=\theta $, we obtain a representation of $v$ in the following form:

$v=v_re_r+v_(\varphi )e_(\varphi )+v_(\theta )e_(\theta )$ where $v_r=\dot(r);\ \ v_(\varphi )=r\dot( \varphi )sin\theta ;;\ \ v_(\theta )=r\dot(\theta )\ ;;$ \[\dot(r)=\frac(dr)(dt);;\ \ \dot( \varphi )=\frac(d\varphi )(dt);;\ \ \dot(\theta )=\frac(d\theta )(dt); v=r\sqrt(1+(\varphi )^2sin^2\theta +(\theta )^2)\]

Instantaneous speed is the value of the derivative of the function of displacement with respect to time in given moment time, and is related to the elementary displacement by the following relationship: $dr=v\left(t\right)dt$

Task 1

The law of motion of a point along a straight line: $x\left(t\right)=0.15t^2-2t+8$. Find the instantaneous speed of the point 10 seconds after the start of movement.

The instantaneous velocity of a point is the first derivative of the radius vector with respect to time. Therefore, for the instantaneous speed, we can write:

Answer: 10 seconds after the start of movement, the instantaneous speed of the point is 1 m/s.

Task 2

Traffic material point given by the equation~ $x=4t-0.05t^2$. Determine the time $t_(stop)$ at which the point will stop and the average ground speed $\left\langle v\right\rangle $.

Let's find the instantaneous velocity equation: $v\left(t\right)=\dot(x)\left(t\right)=4-0.1t$

Answer: The point will stop 40 seconds after the start of movement. The average speed of its movement is 0.1 m/s.

More on the topic § 1.7. AVERAGE SPEED WITH NON-UNIFORM RECTIOLINEAR MOVEMENT. INSTANT SPEED:

1. 3.2.1 Average flame propagation speed in the main combustion phase.
2. 3.2.2 Average flame propagation speed in the second phase of combustion.
3. 3.2.3 Average flame propagation speed in the third phase of combustion
4. 4.2.3 Semi-empirical dependence of the average speed of flame propagation in the second phase of combustion
5. 4.2.2 Semi-empirical formula for the average flame propagation velocity in the main combustion phase
6. Theorem 27. The third rule. If two bodies are equal in mass, but B is moving a little faster than A, then not only will A be reflected in the opposite direction, but B will also transfer half of its excess speed to A, and both will continue to move at the same speed in the same direction.

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).

Let's assume that the car traveled a distance of 300 km in 6 hours. What is the average speed of 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, starting speed zero"(Experiment conducted by students)

Before starting to execute practical work remember the rules of TB:

1. 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.
2. 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.
3. 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: the study uniformly accelerated motion, plotting the dependence v=at 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. (one)

Speed ​​in given point trajectories 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:

1. They measure the movement that the carriage will make when moving between the sensors - S 1
2. They start the carriage and measure the time of its movement between the sensors t 1
3. According to the formula (3) determine the speed with which the carriage moved at the end of the first section v 1 \u003d 2S 1 / t 1
4. 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.
5. 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
6. Based on the data obtained, a graph of the dependence of speed on the time of movement is built.
7. 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.