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, their speed changes. In this lesson we will look at uneven motion.

Uniform movement - mechanical movement in which a body travels the same distance in any equal periods of time (Fig. 4).

Rice. 4. Uniform movement

Movement is called uneven, in which the body travels unequal paths in equal periods of time.

Rice. 5. Uneven movement

The main task of mechanics is to determine the position of the body at any moment in time. When the body moves unevenly, the speed of the body changes, therefore, it is necessary to learn to describe the change in the speed of the body. To do this, two concepts are introduced: average speed and instantaneous speed.

The fact of a change in the speed of a body during uneven movement does not always need to be taken into account; when considering the movement of a body over a large section of the path as a whole (the speed at each moment of time is not important to us), 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 3,300 km. The speed of the train when it just left Novosibirsk was , does this mean that in the middle of the journey the speed was like this same, but at the entrance to Sochi [M1]? Is it possible, having only these data, to say that the travel time will be (Fig. 6). Of course not, since residents of Novosibirsk know that it takes approximately 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.

Medium speed they call the ratio of the total movement that the body has made to the time during which this movement has been 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 zero.

Rice. 8. Displacement is 0

In practice, the concept of average ground speed is most often used.

Average ground speed is the ratio of the total path traveled by the body to the time during which the path was 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 in which it passed it, moving unevenly.

From the mathematics course 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, let's solve the following problem.

Task

A cyclist climbs a slope at a speed of 10 km/h, spending 0.5 hours. Then it goes down 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 for these speeds is km/h, we will find the average speed in km/h. Therefore, we will not convert these problems 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 path to climb the slope is:

The path of descent from the slope is:

The time it takes to travel the full 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 said that if the average speed along the entire journey of the train is equal to , 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 speed of the body(for example: a car’s speedometer (Fig. 11) shows instantaneous speed).

Rice. 11. Car speedometer shows instantaneous speed

There is another definition of instantaneous speed.

Instantaneous speed– speed of body movement in at the 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, let's look at an example.

Let the car move straight along a section of highway. We have a graph of the projection of displacement versus 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. Let's say you need to find the instantaneous speed of a car 30 seconds after the start of observation (at the point A). Using the definition of instantaneous speed, we find the magnitude 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, let’s find the average speed module 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

We calculate the average speed over a given period of time:

We obtained two values of the instantaneous speed of the car 30 seconds after the start of observation. More accurate will be the value where the time interval is smaller, that is. If we decrease the time interval under consideration more strongly, then the instantaneous speed of the car at the point A will be determined more accurately.

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 a body moves curvilinearly, then the instantaneous speed is directed tangentially to the trajectory at a given point (Fig. 16).

Task 1

Can instantaneous speed () change only in direction, without changing in magnitude?

Solution

To solve this, consider the following example. The body moves along a curved path (Fig. 17). Let's mark a point on the trajectory of movement A and period B. Let us note the direction of the instantaneous velocity at these points (the instantaneous velocity is directed tangentially to the trajectory point). Let the velocities and be equal in magnitude and equal to 5 m/s.

Answer: Maybe.

Task 2

Can instantaneous speed change only in magnitude, 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 in the same direction. If a body moves uniformly accelerated, then .

Answer: Maybe.

In this lesson, we began to study uneven movement, that is, movement with varying speed. The characteristics of uneven 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 speed is introduced.

References

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. Physics problems. - M.: Nauka, 1988.
A.V. Peryshkin, V.V. Krauklis. Physics course. T. 1. - M.: State. teacher 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 (page 24); G.Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky. Physics 10 (see list of recommended readings)
Is it possible, knowing the average speed over a certain period of time, to find the displacement made by a body during any part of this interval?
What is the difference between instantaneous speed during uniform linear motion and instantaneous speed during uneven motion?
While driving a car, speedometer readings were taken every minute. Is it possible to determine the average speed of a 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 over the entire journey. Give your answer in km/hour

Instant Speed:

In the world around us, uniform motion is rare. Usually the speed of a body changes over time. This movement is called uneven. To characterize uneven movement, a physical quantity is called equal to the ratio of the movement of the body to the time during which this movement occurred, and is called high-speed movement.

On the graph, the slope of a straight line connecting two points is represented by a ratio and shows how quickly the position of the body changes over time.

If the movement of a body is not rectilinear, then the distance traveled by the body will be greater than its displacement. Therefore, to calculate the average speed, find the ratio of the distance traveled by the body to the time:

In this case, the average speed is called travel. Unlike travel speed, ground speed is a scalar. For example, the average speed (movement) of a car returning to the starting point is zero. But at the same time, its average ground speed is different from zero.

Knowing the average speed of a body on any part of the path, it is impossible to determine its position at any time. When moving, the body passes sequentially all points of the trajectory. At each point it is at certain times and has a certain speed. The speed of a body at a given moment or at a given point on the trajectory is called instantaneous speed.

Instantaneous speed can be thought of as the average speed over a short period of time. Instantaneous speed is equal to the ratio of a small movement on a section of a trajectory to a small period of time during which this movement was completed.

Instantaneous speed can also be determined using a motion graph. The instantaneous speed of a body at any point on the graph is determined by the slope of the tangent to the curve at the corresponding point. To determine the instantaneous speed at a certain point, you need to take any two points on a straight line that is tangent to the motion graph and calculate the average speed for the selected segment. The instantaneous speed of the body at a given point will be numerically equal to the tangent of the angle of inclination of the tangent to the graph of the function.

The tangent of the tangent angle is numerically equal to the instantaneous speed at this point

With uniform motion, the displacement module is numerically equal to the area under the velocity graph. With uneven movement, this equality is also satisfied. You can consider the movement of a body at separate time intervals. If you choose less and less, then the speed at each interval will change less and less. Then, for each period of time, the area under the graph is equal to the product of the height (speed) and the base (time period), that is, the area is equal to the displacement of the body over this period of time. And the area under the entire graph is equal to the sum of the areas for each time period. Thus, the displacement during uneven motion is numerically equal to the area under the velocity graph.

Often the average speed is found from a graph of the speed modulus versus time. The area under the velocity graph determines the distance traveled by the body. Therefore, in accordance with the determination of the average speed according to the graph, it is possible to select a constant speed value that will allow you to travel the same distance and in the same time as when moving at a variable speed.

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, their speed changes. In this lesson we will look at uneven motion.

Uniform movement - mechanical movement in which a body travels the same distance in any equal periods of time (Fig. 4).

Rice. 4. Uniform movement

Movement is called uneven, in which the body travels unequal paths in equal periods of time.

Rice. 5. Uneven movement

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.

Medium speed they call the ratio of the total movement that the body has made to the time during which this movement has been 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 zero.

Rice. 8. Displacement is 0

In practice, the concept of average ground speed is most often used.

Average ground speed is the ratio of the total path traveled by the body to the time during which the path was 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 in which it passed it, moving unevenly.

From the mathematics course 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, let's solve the following problem.

Task

A cyclist climbs a slope at a speed of 10 km/h, spending 0.5 hours. Then it goes down 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 for these speeds is km/h, we will find the average speed in km/h. Therefore, we will not convert these problems 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 path to climb the slope is:

The path of descent from the slope is:

The time it takes to travel the full 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 said that if the average speed along the entire journey of the train is equal to , 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 speed of the body(for example: a car’s speedometer (Fig. 11) shows instantaneous speed).

Rice. 11. Car speedometer shows instantaneous speed

There is another definition of instantaneous speed.

Instantaneous 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. Instant speed

To better understand this definition, let's look at an example.

Let the car move straight along a section of highway. We have a graph of the projection of displacement versus time for a given movement (Fig. 13), let’s analyze this graph.

Rice. 13. Graph of displacement projection versus time

Rice. 14. Graph of displacement projection versus time

In order to check the correctness of finding the instantaneous speed, let’s find the average speed module 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

We calculate the average speed over a given period of time:

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 a body moves curvilinearly, then the instantaneous speed is directed tangentially to the trajectory at a given point (Fig. 16).

Task 1

Can instantaneous speed () change only in direction, without changing in magnitude?

Solution

Answer: Maybe.

Task 2

Can instantaneous speed change only in magnitude, 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 in the same direction. If a body moves uniformly accelerated, then .

Answer: Maybe.

References

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. Physics problems. - M.: Nauka, 1988.
A.V. Peryshkin, V.V. Krauklis. Physics course. T. 1. - M.: State. teacher 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 (page 24); G.Ya. Myakishev, B.B. Bukhovtsev, N.N. Sotsky. Physics 10 (see list of recommended readings)
Is it possible, knowing the average speed over a certain period of time, to find the displacement made by a body during any part of this interval?
What is the difference between instantaneous speed during uniform linear motion and instantaneous speed during uneven motion?
While driving a car, speedometer readings were taken every minute. Is it possible to determine the average speed of a 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 over the entire journey. Give your answer in km/hour

Kinematics- part of mechanics in which the movement of a material point is studied without considering the reasons that cause this movement.

Mechanical body movement is called the change in its position in space relative to other bodies over time.

The main task of mechanics- determine the position of the body in space at any time.

A movement in which all points of the body move equally is called forward movement bodies.

A body whose dimensions can be neglected under the conditions of the motion being studied is called material point

Reference body- this is any body conventionally accepted as motionless, relative to which the movement of other bodies is considered.

Watch- an instrument in which periodic motion is used to measure periods of time.

Reference system represents a reference body, an associated coordinate system and a clock.

TRAJECTORY, PATH AND MOVEMENT

Trajectory- a line that a material point describes during its movement.

The path is the length of the trajectory of the body.

By moving the body is a vector connecting the initial position of a body to its final position.

DISPLACEMENT AND SPEED DURING RIGHT LINEAR UNIFORM MOTION

Straight-line movement- a movement whose trajectory is a straight line.

A movement in which a body makes equal movements at any equal intervals of time is called uniform movement.

Speed of uniform rectilinear motion- the ratio of the vector of movement of a body over any period of time to the value of this interval:

Knowing the speed, you can find the displacement over a known period of time using the formula

In rectilinear uniform motion, the velocity and displacement vectors have the same direction.

Projection of movement onto the axis X: s x = x t . Since s x = x - x 0, then the body coordinate x = x 0 + s x. Similarly for the y-axis: y = y 0 + s y.

As a result, we obtain the equations of rectilinear uniform motion of a body in projections on the x and y axes:

RELATIVITY OF MOTION

The position of the body is relative, that is, it is different in different reference systems. Therefore, its motion is also relative.

SPEED WITH UNEVEN MOTION

Uneven is a movement in which the speed of a body changes over time.

The average speed of uneven movement is equal to the ratio of the displacement vector to the travel time

Then the displacement during uneven movement

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

ACCELERATION. UNIFORMLY ACCELERATED MOTION

Uniformly accelerated is a movement in which the speed of a body changes equally over any equal intervals of time.

Acceleration of the body is the ratio of the change in the speed of a body to the time during which this change occurred.

Acceleration characterizes the rate of change in speed.

Acceleration is a vector quantity. It shows how the instantaneous speed of a body changes per unit time.

Knowing the initial speed of the body and its acceleration, from formula (1) you can find the speed at any time:

To do this, the equation must be written in projections onto the selected axis:

V x =V 0x + a x t

Speed graph at uniformly accelerated motion is straight.

DISPLACEMENT AND PATH IN RECTILINEAR UNIFORM ACCELERATED MOTION

Let us assume that the body has moved in time t, moving with acceleration. If the speed changes from to and given that,

Using a speed graph, you can determine the distance traveled by a body in known time path - it is numerically equal to the area of the shaded surface.

FREE FALL OF BODIES

The movement of bodies in airless space under the influence of gravity is called free fall.

Free fall is uniformly accelerated motion. The acceleration of gravity in a given place on the Earth is constant for all bodies and does not depend on the mass of the falling body: g = 9.8 m/s 2 .

To solve various problems from the "Kinematics" section, two equations are needed:

Example: A body, moving uniformly accelerated from a state of rest, covered a distance of 18 m in the fifth second. What is the acceleration and how far did the body travel in 5 s?

In the fifth second, the body has traveled the distance s = s 5 - s 4 and s 5 and s 4 are the distances traveled by the body in 4 and 5 s, respectively.

Answer: a body moving with an acceleration of 4 m/s2 travels 50 m in 5 s.

Problems and tests on the topic "Topic 1. "Mechanics. Fundamentals of kinematics."

Material point (Reference system)
Lessons: 3 Assignments: 9 Tests: 1
Graphs of the dependence of kinematic quantities on time during uniformly accelerated motion - Laws of interaction and motion of bodies: basics of kinematics, grade 9
Lessons: 2 Assignments: 9 Tests: 1

Lessons: 1 Assignments: 9 Tests: 1

To complete assignments on the topic "Mechanics" you need to know Newton's laws, laws of universal gravitation, Hooke's laws, conservation of momentum and energy, as well as the basic formulas of kinematics (equations of coordinates, velocity and displacement).

Strictly follow the order of study theoretical material, proposed in the recommendations for the Physics course.

When completing tasks in the Mechanics course, pay attention to the signs of the projection of vectors in the selected reference system. This is a common mistake that high school students make.

Don’t be lazy to draw diagrams (drawings) of problems - this can make solving the problem much easier for you.

Analyze the conditions of each specific task, compare the answers with the conditions and reality.

Don't invent your own problems using the original data!

Uniformly accelerated curvilinear motion

Curvilinear movements are movements whose trajectories are not straight, but curved lines. Planets and river waters move along curvilinear trajectories.

Curvilinear motion is always motion with acceleration, even if the absolute value of the velocity is constant. Curvilinear motion with constant acceleration always occurs in the plane in which the acceleration vectors and initial velocities of the point are located. In the case of curvilinear motion with constant acceleration in the xOy plane, the projections vx and vy of its velocity on the Ox and Oy axes and the x and y coordinates of the point at any time t are determined by the formulas

Uneven movement. Rough speed

No body moves at a constant speed all the time. When the car starts moving, it moves faster and faster. It can move steadily for a while, but then it slows down and stops. In this case, the car travels different distances in the same time.

Movement in which a body travels unequal lengths of path in equal intervals of time is called uneven. With such movement, the speed does not remain unchanged. In this case, we can only talk about average speed.

Average speed shows how much movement a body goes through per unit time. It is equal to the ratio of the displacement of the body to the time of movement. Average speed, like the speed of a body during uniform motion, is measured in meters divided by a second. In order to characterize motion more accurately, instantaneous speed is used in physics.

The speed of a body at a given moment in time or at a given point in the trajectory is called instantaneous speed. Instantaneous speed is a vector quantity and is directed in the same way as the displacement vector. You can measure instantaneous speed using a speedometer. In the International System, instantaneous speed is measured in meters divided by second.

point movement speed uneven

Movement of a body in a circle

Curvilinear motion is very common in nature and technology. It is more complex than a straight line, since there are many curved trajectories; this movement is always accelerated, even when the velocity module does not change.

But movement along any curved path can be approximately represented as movement along the arcs of a circle.

When a body moves in a circle, the direction of the velocity vector changes from point to point. Therefore, when they talk about the speed of such movement, they mean instantaneous speed. The velocity vector is directed tangentially to the circle, and the displacement vector is directed along the chords.

Uniform circular motion is a motion during which the module of the motion velocity does not change, only its direction changes. The acceleration of such motion is always directed towards the center of the circle and is called centripetal. In order to find the acceleration of a body moving in a circle, it is necessary to divide the square of the speed by the radius of the circle.

In addition to acceleration, the motion of a body in a circle is characterized by the following quantities:

The period of rotation of a body is the time during which the body makes one complete revolution. The rotation period is designated by the letter T and is measured in seconds.

The frequency of rotation of a body is the number of revolutions per unit time. Is the rotation speed indicated by a letter? and is measured in hertz. In order to find the frequency, you need to divide one by the period.

Linear speed is the ratio of the movement of a body to time. In order to find the linear speed of a body in a circle, it is necessary to divide the circumference by the period (the circumference is equal to 2? multiplied by the radius).

Angular velocity is a physical quantity equal to the ratio of the angle of rotation of the radius of the circle along which the body moves to the time of movement. Angular velocity is indicated by a letter? and is measured in radians divided per second. Can you find the angular velocity by dividing 2? for a period. Angular velocity and linear velocity among themselves. In order to find the linear speed, it is necessary to multiply the angular speed by the radius of the circle.

Figure 6. Circular motion, formulas.

Formula for average speed of rectilinear uneven motion. Rectilinear uneven movement

Problems and tests on the topic "Topic 1. "Mechanics. Fundamentals of kinematics."

Uniformly accelerated curvilinear motion

Uneven movement. Rough speed

Movement of a body in a circle