EntranceUneven straight motion. Types of movements Average value of the velocity module in uneven movement
Uniform linear movement- This is a special case of uneven movement.
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. When uniformly accelerated motion the modulus of the body's velocity increases over time, 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. Unit average speed– m/s.
V cp = s / t is the speed of the body (material point) in this moment 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:
V x = x’ is the derivative of the coordinate with respect to time (the projections of the velocity vector onto other coordinate axes are similarly obtained).
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 period 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 trajectory of the body, then the projection of the velocity vector onto this axis is determined by the formula: v x = v 0x ± a x t The “-” (minus) sign before 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 = v 0 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
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. Instantaneous speed 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?
1. 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 moves 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 over time.
With uneven movement, the coordinate of the body can no longer be determined using the formula x = x 0 + v x t, since the speed of movement is not constant. The question arises: what value characterizes the speed of change in body position over time with uneven movement? This quantity is average speed.
Medium speed vWeduneven movement is called physical quantity, equal to the displacement ratio sbodies by time t for which it was committed:
|
v cf = . |
Average speed is vector quantity. To determine the average velocity module 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 unsuitable.
Let's look at an example. It is necessary to calculate the time of arrival of the train at each station along the route. However, the movement is not linear. If you calculate the module of the average speed in the section between two stations using the above formula, the resulting value will differ from the value of the average speed at which the train was moving, since the module 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 path relation l In time t, during which this path was passed:
v Wed = .
She is often called average ground speed.
2. Knowing the average speed of a body at any part of the trajectory, it is impossible to determine its position at any time. Let's assume 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, move for some time 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 in 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 times and has some speed.
Instantaneous speed is the speed of a body at a given moment in time or at a given point in the trajectory.
Let us assume that the body makes uneven linear motion. Let us determine the speed of movement of this body at the point O its trajectory (Fig. 21). Let us select a section on the trajectory AB, inside which there is a point O. Moving s 1 in this area the body has completed in time t 1 . The average speed in this section is v avg 1 = .
Let's reduce body movement. Let it be equal s 2, and the movement time is t 2. Then the average speed of the body during this time: v avg 2 = .Let us further reduce the movement, the average speed in this section is: v cf 3 = .
We will continue to reduce the time of movement of the body and, accordingly, its displacement. Eventually, the movement and time will become so small that a device, such as a speedometer in a car, will no longer record the change in speed and the movement over 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 speed is a vector physical quantity equal to the ratio of small displacement D sto a short period of time D t, during which this movement was completed:
|
v = . |
Self-test questions
1. What kind of movement is called uneven?
2. What is average speed?
3. What does average ground speed indicate?
4. Is it possible, knowing the trajectory of a body and its average speed over a certain period of time, to determine the position of the body at any moment in time?
5. What is instantaneous speed?
6. How do you understand the expressions “small movement” and “short period of time”?
Task 4
1. The car drove along Moscow streets 20 km in 0.5 hours, when leaving Moscow it stood for 15 minutes, and in the next 1 hour 15 minutes it drove 100 km around the Moscow region. At what average speed did the car move in each section and along the entire route?
2. What is the average speed of a train on a 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 half at an average speed of 70 km/h?
3. What is the average speed of a train on a 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?
With uneven motion, a body can travel both equal and different paths in equal periods of time.
To describe uneven motion, the concept is introduced average speed.
Average speed, by this definition, is a scalar quantity because the path and time are scalar quantities.
However, the average speed can also be determined through displacement according to the equation
The average speed of a path and the average speed of movement are two different quantities that can characterize the same movement.
When calculating average speed, a mistake is often made in that the concept of average speed is replaced by the concept of the arithmetic mean of the speed of the body in different areas of movement. To show the illegality of such a substitution, consider the problem and analyze its solution.
From point A train leaves for point B. For half the entire journey the train moves at a speed of 30 km/h, and for the second half of the journey at a speed of 50 km/h.
What is the average speed of the train on section AB?
|
|
|
The movement of the train on section AC and section CB is uniform. Looking at the text of the problem, you often immediately want to give the answer: υ av = 40 km/h.
Yes, because it seems to us that the formula used to calculate the arithmetic average is quite suitable for calculating the average speed.
Let's see: is it possible to use this formula and calculate the average speed by finding the half-sum of the given speeds.
To do this, let's consider a slightly different situation.
Let's say we're right and the average speed is really 40 km/h.
Then let's solve another problem.
|
|
|
As you can see, the problem texts are very similar, there is only a “very small” difference.
If in the first case we are talking about half the journey, then in the second case we are talking about half the time.
Obviously, point C in the second case is somewhat closer to point A than in the first case, and it is probably impossible to expect the same answers in the first and second problems.
If, when solving the second problem, we also give the answer that the average speed is equal to half the sum of the speeds in the first and second sections, we cannot be sure that we solved the problem correctly. What should I do?
The way out of the situation is as follows: the fact is that average speed is not determined through the arithmetic mean. There is a defining equation for average speed, according to which, to find the average speed in a certain area, the entire path traveled by the body must be divided by the entire time of movement:
We need to start solving the problem with the formula that determines the average speed, even if it seems to us that in some case we can use a simpler formula.
We will move from the question to known quantities.
We express the unknown quantity υ avg through other quantities – L 0 and Δ t 0 .
It turns out that both of these quantities are unknown, so we must express them in terms of other quantities. For example, in the first case: L 0 = 2 ∙ L, and Δ t 0 = Δ t 1 + Δ t 2.
Let us substitute these values, respectively, into the numerator and denominator of the original equation.
In the second case we do exactly the same. We don't know the whole path and all the time. We express them: and
It is obvious that the travel time on section AB in the second case and the travel time on section AB in the first case are different.
In the first case, since we do not know the times and we will try to express these quantities: and in the second case we express and:
We substitute the expressed quantities into the original equations.
Thus, in the first problem we have:
After transformation we get: 
In the second case we get 
and after the transformation:
The answers, as predicted, are different, but in the second case we found that the average speed is indeed equal to half the sum of the speeds.
The question may arise: why can’t we immediately use this equation and give such an answer?
The point is that, having written down that the average speed in section AB in the second case is equal to half the sum of the speeds in the first and second sections, we would imagine not a solution to a problem, but a ready-made answer. The solution, as you can see, is quite long, and it begins with the defining equation. The fact that in this case we received the equation that we wanted to use initially is pure coincidence.
With uneven movement, the speed of a body can continuously change. With such movement, the speed at any subsequent point of the trajectory will differ from the speed at the previous point.
The speed of a body at a given moment of time and at a given point of the trajectory is called instantaneous speed.
The longer the time period Δt, the more the average speed differs from the instantaneous one. And, conversely, the shorter the time period, the less the average speed differs from the instantaneous speed of interest to us.
Let us define the instantaneous speed as the limit to which the average speed tends over an infinitesimal period of time:
If we are talking about the average speed of movement, then the instantaneous speed is a vector quantity:
If we are talking about the average speed of a path, then the instantaneous speed is a scalar quantity:
There are often cases when, during uneven motion, the speed of a body changes over equal periods of time by the same amount.
With uniform motion, the speed of a body can either decrease or increase.
If the speed of a body increases, then the movement is called uniformly accelerated, and if it decreases, it is called uniformly slow.
A characteristic of uniformly alternating motion is a physical quantity called acceleration.
Knowing the acceleration of the body and its initial speed, you can find the speed at any predetermined moment in time: 
In projection onto coordinate axis 0X the equation will take the form: υ x = υ 0 x + a x ∙ Δ t.
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 motion, the additional concept of average speed is introduced.
average speed
Definition 1
Average speed is a physical quantity that 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. Uniformly accelerated motion is considered a special case of uneven 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 the 0X coordinate axis, the 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 at any given moment in advance.
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.