Uneven movement. Instantaneous speed

IN real life It is very difficult to encounter uniform motion, since objects of the material world cannot move with such great accuracy, and even for a long period of time, so usually in practice a more realistic physical concept is used that characterizes the movement of a certain body in space and time.

Note 1

Uneven motion is characterized by the fact that a body can travel the same or different paths in equal periods of time.

To fully understand this type of mechanical movement, an additional concept is introduced average speed.

average speed

Definition 1

The average speed is physical quantity, which is equal to the ratio of the entire path traveled by the body to the total time of movement.

This indicator is considered in a specific area:

$\upsilon = \frac(\Delta S)(\Delta t)$

By this definition, average speed is a scalar quantity, since time and distance are scalar quantities.

The average speed can be determined by the displacement equation:

The average speed in such cases is considered a vector quantity, since it can be determined through the ratio of the vector quantity to the scalar quantity.

The average speed of movement and the average speed of travel characterize the same movement, but they are different quantities.

An error is usually made in the process of calculating average speed. It consists in the fact that the concept of average speed is sometimes replaced by the arithmetic mean speed of the body. This defect is allowed in different areas of body movement.

The average speed of a body cannot be determined through the arithmetic mean. To solve problems, the equation for average speed is used. Using it you can find the average speed of a body in a certain area. To do this, divide the entire path traveled by the body into total time movements.

The unknown quantity $\upsilon$ can be expressed in terms of others. They are designated:

$L_0$ and $\Delta t_0$.

We get a formula according to which the search for an unknown quantity is carried out:

$L_0 = 2 ∙ L$, and $\Delta t_0 = \Delta t_1 + \Delta t_2$.

When solving a long chain of equations, one can arrive at the original version of searching for the average speed of a body in a certain area.

With continuous movement, the speed of the body also continuously changes. Such a movement gives rise to a pattern in which the speed at any subsequent points of the trajectory differs from the speed of the object at the previous point.

Instantaneous speed

Instantaneous speed is the speed in a given period of time at a certain point on the trajectory.

The average speed of a body will differ more from the instantaneous speed in cases where:

• it is greater than the time interval $\Delta t$;
• it is less than a period of time.

Definition 2

Instantaneous speed is a physical quantity that is equal to the ratio of a small movement on a certain section of the trajectory or the path traveled by a body to the short period of time during which this movement was made.

Instantaneous speed becomes a vector quantity when talking about the average speed of movement.

Instantaneous speed becomes a scalar quantity when talking about the average speed of a path.

With uneven motion, a change in the speed of a body occurs over equal periods of time by an equal amount.

Uniform motion of a body occurs at the moment when the speed of an object changes by an equal amount over any equal periods of time.

Types of uneven movement

With uneven movement, the speed of the body constantly changes. There are main types of uneven movement:

• movement in a circle;
• the movement of a body thrown into the distance;
• uniformly accelerated motion;
• uniform slow motion;
• uniform motion
• uneven movement.

The speed can vary by numerical value. Such movement is also considered uneven. No special case uniform motion consider uniformly accelerated motion.

Definition 3

Unequally variable motion is the movement of a body when the speed of the object does not change by a certain amount over any unequal periods of time.

Equally variable motion is characterized by the possibility of increasing or decreasing the speed of a body.

Motion is called uniformly slow when the speed of a body decreases. Uniformly accelerated motion is a motion in which the speed of a body increases.

Acceleration

For uneven motion, one more characteristic has been introduced. This physical quantity is called acceleration.

Acceleration is a vector physical quantity equal to the ratio of the change in the speed of a body to the time when this change occurred.

$a=\frac(\upsilon )(t)$

With uniformly alternating motion, there is no dependence of acceleration on the change in the speed of the body, as well as on the time of change of this speed.

Acceleration indicates the quantitative change in the speed of a body over a certain unit of time.

In order to obtain a unit of acceleration, it is necessary to substitute the units of speed and time into the classical formula for acceleration.

In projection onto coordinate axis 0X equation will take the following form:

$υx = υ0x + ax ∙ \Delta t$.

If you know the acceleration of a body and its initial speed, you can find the speed in advance at any time. this moment time.

A physical quantity that is equal to the ratio of the path traveled by a body in a specific period of time to the duration of such an interval is the average ground speed. Average ground speed is expressed as:

• scalar quantity;
• non-negative value.

The average speed is represented in vector form. It is directed to where the movement of the body is directed over a certain period of time.

The average speed module is equal to the average ground speed in cases where the body has been moving in one direction all this time. The module of the average speed decreases to the average ground speed if, during the process of movement, the body changes the direction of its movement.

Uniform movement- this is movement at a constant speed, that is, when the speed does not change (v = const) and acceleration or deceleration does not occur (a = 0).

Straight-line movement- this is movement in a straight line, that is, the trajectory of rectilinear movement is a straight line.

This is a movement in which a body makes equal movements at any equal intervals of time. For example, if we divide a certain time interval into one-second intervals, then with uniform motion the body will move the same distance for each of these time intervals.

The speed of uniform rectilinear motion 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:

vcp = v

Speed ​​of uniform rectilinear motion is a physical vector quantity equal to the ratio of the movement of a body over any period of time to the value of this interval t:

=/t

Thus, the speed of uniform rectilinear motion shows how much movement a material point makes per unit time.

Moving with uniform straight motion is determined by the formula:

Distance traveled in linear motion is equal to the displacement module. If the positive direction of the OX axis coincides with the direction of movement, then the projection of the velocity onto the OX axis is equal to the magnitude of the velocity and is positive:

vx = v, that is v > 0

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

s = vt = x - x0

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 body coordinates on time x = x(t), takes the form:

x = x0 + vt

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 speed is less than zero (v< 0), и тогда уравнение движения принимает вид:

x = x0 - vt

Uniform linear movement- This is a special case of uneven motion.

Uneven movement- this is a movement in which a body (material point) makes unequal movements over equal periods of time. For example, a city bus moves unevenly, since its movement consists mainly of acceleration and deceleration.

Equally alternating motion is a movement in which the speed of the body ( material point) changes equally over any equal periods of time.

Acceleration of a body during uniform motion remains constant in magnitude and direction (a = const).

Uniform motion can be uniformly accelerated or uniformly decelerated.

Uniformly accelerated motion- this is the movement of a body (material point) with positive acceleration, that is, with such movement the body accelerates with constant acceleration. In the case of uniformly accelerated motion, the modulus of the body’s velocity increases over time, and the direction of acceleration coincides with the direction of the speed of movement.

Equal slow motion- this is the movement of a body (material point) with negative acceleration, that is, with such movement the body uniformly slows down. In uniformly slow motion, the velocity and acceleration vectors are opposite, and the velocity modulus decreases over time.

In mechanics, any rectilinear motion is accelerated, therefore slow motion differs from accelerated motion only in the sign of the projection of the acceleration vector onto the selected axis of the coordinate system.

Average variable speed is determined by dividing the movement of the body by the time during which this movement was made. The unit of average speed is m/s.

vcp = s/t

This is the speed of a body (material point) at a given moment of 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 uniformly alternating motion can be found as the first derivative of the displacement vector with respect to time:

= "

Velocity vector projection on the OX axis:

vx = x’

this is the derivative of the coordinate with respect to time (the projections of the velocity vector onto other coordinate axes are similarly obtained).

This is a quantity that determines the rate of change in the speed of a body, that is, the limit to which the change in speed tends with an infinite decrease in the time interval Δt:

Acceleration vector of uniformly alternating 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 period of time during which the change in speed occurred, will be as follows:

From here uniform speed formula at any time:

0 + t If a body moves rectilinearly along the OX axis of a rectilinear Cartesian coordinate system, coinciding in direction with the body’s trajectory, then the projection of the velocity vector onto this axis is determined by the formula:

vx = v0x ± axt

The “-” (minus) sign in front of the projection of the acceleration vector refers to uniformly slow motion. The equations for projections of the velocity vector onto other coordinate axes are written similarly.

Since in uniform motion the acceleration is constant (a = const), the acceleration graph is a straight line parallel to the 0t axis (time axis, Fig. 1.15).

Rice. 1.15. Dependence of body acceleration on time.

Dependence of speed on time is a linear function, the graph of which is a straight line (Fig. 1.16).

Rice. 1.16. Dependence of body speed on time.

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 ​​a trapezoid is equal to the product of half the sum of the lengths of its bases and its height. The bases of the trapezoid 0abc are numerically equal:

0a = v0 bc = v

The height of the trapezoid is t. Thus, the area of ​​the trapezoid, and therefore the projection of displacement onto the OX axis is equal to:

In the case of uniformly slow motion, the acceleration projection is negative and in the formula for the displacement projection a “-” (minus) sign is placed before the acceleration.

A graph of the velocity of a body versus time at various accelerations is shown in Fig. 1.17. The graph of displacement versus time for v0 = 0 is shown in Fig. 1.18.

Rice. 1.17. Dependence of body speed on time for different acceleration values.

Rice. 1.18. Dependence of body movement on time.

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 another displacement formula by solving a system of two equations:

It will help us derive the formula for displacement projection:

Since the coordinate of the body at any moment in time is determined by the sum of the initial coordinate and the displacement projection, it will look like this:

The graph of the coordinate x(t) is also a parabola (like the graph of displacement), but the vertex of the parabola in the general case does not coincide with the origin. When a x< 0 и х 0 = 0 ветви параболы направлены вниз (рис. 1.18).

Uneven motion is considered to be movement with varying speed. Speed ​​can vary in direction. We can conclude that any movement NOT along a straight path is uneven. For example, the movement of a body in a circle, the movement of a body thrown into the distance, etc.

The speed can vary by numerical value. This movement will also be uneven. A special case of such motion is uniformly accelerated motion.

Sometimes there is uneven movement, which consists of alternating various types movements, for example, first the bus accelerates (uniformly accelerated motion), then moves uniformly for some time, and then stops.

Instantaneous speed

Uneven movement can only be characterized by speed. But the speed always changes! Therefore, we can only talk about speed at a given moment in time. When traveling by car, the speedometer shows you the instantaneous speed of movement every second. But in this case the time must be reduced not to a second, but a much shorter period of time must be considered!

average speed

What is average speed? It is wrong to think that you need to add up all the instantaneous velocities and divide by their number. This is the most common misconception about average speed! Average speed is divide the entire journey by the time taken. And it is not determined in any other way. If you consider the movement of a car, you can estimate its average speeds in the first half of the journey, in the second, and throughout the entire journey. Average speeds may be the same or may be different in these areas.

For average values, a horizontal line is drawn on top.

Average moving speed. Average ground speed

If the movement of a body is not rectilinear, then the distance traveled by the body will be greater than its displacement. In this case, the average moving speed differs from the average ground speed. Ground speed is a scalar.

The main thing to remember

1) Definition and types of uneven movement;
2) The difference between average and instantaneous speeds;
3) Rule for finding average speed

Often you need to solve a problem where the entire path is divided into equal sections, the average speeds on each section are given, you need to find the average speed along the entire route. The wrong decision will be if you add up the average speeds and divide by their number. Below is a formula that can be used to solve such problems.

Instantaneous speed can 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. Instantaneous speed is the tangent of the angle of inclination of the tangent to the graph of the function.

Exercises

While driving a car, speedometer readings were taken every minute. Is it possible to determine the average speed of a car from these data?

It is impossible, since in the general case the value of the average speed is not equal to the arithmetic mean of the values ​​of the instantaneous speeds. But the path and time are not given.

What variable speed does the car's speedometer indicate?

Close to instantaneous. Close, since the period of time should be infinitely small, and when taking readings from the speedometer, it is impossible to judge time that way.

In what case are the instantaneous and average speeds equal? Why?

With uniform movement. Because the speed does not change.

The speed of movement of the hammer upon impact is 8 m/s. What speed is it: average or instantaneous?

Mechanical motion is the change in the position of a body in space over time relative to other bodies.

Based on the definition, the fact of motion of a body can be established by comparing its positions at successive moments of time with the position of another body, which is called the body of reference.

Thus, watching a ball on a football field, we can say that it changes its position relative to the goal or relative to the foot of a football player. A ball that rolls on the floor changes its position relative to the floor. A residential building is at rest relative to the Earth, but changes its position relative to the Sun.

Mechanical motion path

Trajectory- This is the line along which the body moves. For example, the trace of an airplane in the sky and the trace of a tear on the cheek are all trajectories of body movement. Movement trajectories can be straight, curved or broken. But the length of the trajectory, or the sum of the lengths, is the path traveled by the body.

The path is designated by the letter S. And is measured in meters, centimeters and kilometers.

There are other units of measurement of length.

Types of mechanical movement: uniform and uneven movement

Uniform movement- mechanical movement in which a body travels the same distance at any equal intervals of time

Uneven movement- mechanical movement in which a body travels a different distance in any equal intervals of time

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

There are many examples of uneven motion: the flight of a ball during a game of football, the movement of a cat while hunting for a bird, the movement of a car

Lesson plan on the topic “Uneven movement. Instant Speed"

date :

Subject: « »

Goals:

Educational : Provide and form a conscious assimilation of knowledge about uneven movement and instantaneous speed;

Developmental : Continue developing independent activity skills and group work skills.

Educational : To form cognitive interest in new knowledge; develop behavioral discipline.

Lesson type: lesson in learning new knowledge

Equipment and sources of information:

Isachenkova, L. A. Physics: textbook. for 9th grade. public institutions avg. education with Russian language training / L. A. Isachenkova, G. V. Palchik, A. A. Sokolsky; edited by A. A. Sokolsky. Minsk: People's Asveta, 2015

Lesson structure:

Organizational moment (5 min)

Updating basic knowledge (5 min)

Learning new material (14 min)

Physical education minute (3 min)

Consolidation of knowledge (13min)

Lesson summary (5 min)

Organizing time

Hello, sit down! (Checking those present).Today in the lesson we must understand the concepts of uneven motion and instantaneous speed. And this means thatLesson topic : Uneven movement. Instantaneous speed

Updating of reference knowledge

We studied uniform linear motion. However, real bodies - cars, ships, airplanes, machine parts, etc. most often move neither rectilinearly nor uniformly. What are the patterns of such movements?

Learning new material

Let's look at an example. A car is moving along the section of road shown in Figure 68. On an ascent, the car’s movement slows down, and on a descent it accelerates. Car movementneither straight nor uniform. How to describe such a movement?

First of all, for this it is necessary to clarify the conceptspeed .

From 7th grade you know what average speed is. It is defined as the ratio of the path to the period of time during which this path is traveled:

(1 )

Let's call heraverage travel speed. She shows whatpath on average the body passed per unit of time.

In addition to the average travel speed, you must also enteraverage moving speed:

(2 )

What is the meaning of average moving speed? She shows whatmoving on average performed by the body per unit of time.

Comparing formula (2) with formula (1 ) from § 7, we can conclude:average speed< > equal to the speed of such uniform rectilinear motion, at which in a period of time Δ tthe body would move Δ r.

The average speed of the path and the average speed of movement are important characteristics of any movement. The first of them is a scalar quantity, the second is a vector quantity. Because Δ r < s , then the module of the average speed of movement is not greater than the average speed of the path |<>| < <>.

Average speed characterizes movement over the entire period of time as a whole. It does not provide information about the speed of movement at each point of the trajectory (at each moment in time). For this purpose, it is introducedinstantaneous speed - speed of movement at a given time (or at a given point).

How to determine instantaneous speed?

Let's look at an example. Let the ball roll down an inclined chute from a point (Fig. 69). The figure shows the positions of the ball at different times.

We are interested in the instantaneous speed of the ball at the pointABOUT. Dividing the movement of the ball Δr 1 for the corresponding period of time Δ averagetravel speed<>= on the section Speed<>can be much different from the instantaneous speed at a pointABOUT. Consider a smaller displacement Δ =IN 2 . It will occur in a shorter period of time Δ. average speed<>= although not equal to the speed at the pointABOUT, but already closer to her than<>. With a further decrease in displacement (Δ,Δ , ...) and time intervals (Δ, Δ, ...) we will obtain average speeds that differ less and less from each otherAndfrom the instantaneous speed of the ball at a pointABOUT.

This means that a fairly accurate value of the instantaneous speed can be found using the formula, provided that the time interval Δt very small:

(3)

Designation Δ t-» 0 reminds that the speed determined by formula (3), the closer to the instantaneous speed, the smallerΔt .

The instantaneous speed of curvilinear motion of a body is found in a similar way (Fig. 70).

What is the direction of the instantaneous speed? It is clear that in the first example the direction of the instantaneous velocity coincides with the direction of motion of the ball (see Fig. 69). And from the construction in Figure 70 it is clear that with curvilinear movementinstantaneous speed is directed tangentially to the trajectory at the point where the moving body is located at that moment.

Observe the hot particles coming off the grindstone (Fig. 71,A). The instantaneous speed of these particles at the moment of separation is directed tangentially to the circle along which they moved before separation. Similarly, the sports hammer (Fig. 71, b) begins its flight tangentially to the trajectory along which it moved when untwisted by the thrower.

Instantaneous speed is constant only with uniform linear motion. When moving along a curved path, its direction changes (explain why). With uneven movement, its module changes.

If the module of instantaneous speed increases, then the motion of the body is called accelerated , if it decreases - slow

Give yourself examples of accelerated and decelerated movements of bodies.

In the general case, when a body moves, both the magnitude of the instantaneous velocity and its direction can change (as in the example with a car at the beginning of the paragraph) (see Fig. 68).

In what follows we will simply call instantaneous speed speed.

Consolidation of knowledge

The speed of uneven movement on a section of a trajectory is characterized by average speed, and at a given point of the trajectory by instantaneous speed.

Instantaneous speed is approximately equal to the average speed determined over a short period of time. The shorter this period of time, the smaller the difference between the average speed and the instantaneous speed.

Instantaneous speed is directed tangentially to the trajectory of motion.

If the module of instantaneous speed increases, then the movement of the body is called accelerated, if it decreases, it is called slow.

With uniform rectilinear motion, the instantaneous speed is the same at any point of the trajectory.

Lesson summary

So, let's summarize. What did you learn in class today?

Organization homework

§ 9, ex. 5 No. 1,2

Reflection.

Continue the phrases:

Today in class I learned...

It was interesting…

The knowledge I gained in the lesson will be useful