In this lesson, the topic of which is: “The equation of motion with constant acceleration. Progressive movement”, we will remember what movement is, how it happens. We also recall what acceleration is, consider the equation of motion with constant acceleration and how to use it to determine the coordinates of a moving body. Let's consider an example of a problem for fixing the material.
The main task of kinematics is to determine the position of the body at any time. The body can rest, then its position will not change (see Fig. 1).
Rice. 1. Body at rest
A body can move in a straight line at a constant speed. Then its displacement will change uniformly, that is, equally in equal time intervals (see Fig. 2).
Rice. 2. Movement of the body when moving at a constant speed
Movement, speed multiplied by time, we have been able to do this for a long time. The body can move with constant acceleration, consider such a case (see Fig. 3).
Rice. 3. Body motion with constant acceleration
Acceleration
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Acceleration is the change in speed per unit of time(see fig. 4) :
Rice. 4. Acceleration Speed is a vector quantity, therefore, the change in speed, i.e., the difference between the vectors of the final and initial speed, is a vector. Acceleration is also a vector directed in the same direction as the velocity difference vector (see Fig. 5).
We are considering a rectilinear motion, so we can choose a coordinate axis along the straight line along which the motion occurs, and consider the projections of the velocity and acceleration vectors on this axis:
|
Then its speed uniformly changes: (if its initial speed was equal to zero). How to find the move now? Multiplying speed by time is impossible: the speed was constantly changing; which one to take? How to determine where the body will be at any time during such a movement - today we will solve this problem.
Let's immediately define the model: we are considering a rectilinear translational motion of the body. In this case, we can apply the material point model. The acceleration is directed along the same straight line along which the material point moves (see Fig. 6).
translational movement
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Translational motion is such a motion in which all points of the body move in the same way: at the same speed, making the same movement (see Fig. 7).
Rice. 7. Forward movement How else can it be? Wave your hand and follow: it is clear that the palm and shoulder moved differently. Look at the Ferris wheel: points near the axis hardly move, and the booths move at a different speed and along different trajectories (see Fig. 8).
Rice. 8. Movement of selected points on the Ferris wheel Look at a moving car: if you do not take into account the rotation of the wheels and the movement of parts of the motor, all points of the car move in the same way, we consider the movement of the car to be translational (see Fig. 9).
Rice. 9. Vehicle movement Then it makes no sense to describe the movement of each point, you can describe the movement of one. The car is considered a material point. Please note that during translational movement, the line connecting any two points of the body during movement remains parallel to itself (see Fig. 10).
Rice. 10. The position of the line connecting two points |
The car drove straight for an hour. At the beginning of the hour, his speed was 10 km/h, and at the end - 100 km/h (see Fig. 11).
Rice. 11. Drawing for the problem
The speed changed uniformly. How many kilometers has the car traveled?
Let's analyze the condition of the problem.
The speed of the car changed uniformly, that is, its acceleration was constant throughout the journey. Acceleration is by definition equal to:
The car was driving in a straight line, so we can consider its movement in the projection on one coordinate axis:
Let's find a move.
Increasing Speed Example
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Nuts are placed on the table, one nut per minute. It is clear: how many minutes pass, so many nuts will be on the table. Now let's imagine that the speed of putting nuts increases evenly from zero: no nuts are put in the first minute, one nut is put in the second, then two, three, and so on. How many nuts will be on the table after some time? It is clear that it is less than if the maximum speed was always maintained. Moreover, it is clearly seen that it is less than 2 times (see Fig. 12).
Rice. 12. The number of nuts at different laying speeds It is the same with uniformly accelerated motion: let's say that at first the speed was equal to zero, at the end it became equal (see Fig. 13).
Rice. 13. Speed change If the body were constantly moving at such a speed, its displacement would be equal, but since the speed increased uniformly, it would be 2 times less. |
We are able to find the displacement with UNIFORM motion: . How to get around this problem? If the speed does not change much, then the movement can be approximately considered uniform. The change in speed will be small over a short period of time (see Fig. 14).
Rice. 14. Speed change
Therefore, we divide the travel time T into N small segments of duration (see Fig. 15).
Rice. 15. Splitting a segment of time
Let's calculate the displacement at each time interval. The speed increases at each interval by:
On each segment, we will consider the movement to be uniform and the speed approximately equal to the initial speed on the given time interval. Let's see if our approximation does not lead to an error if we assume that the motion is uniform over a small interval. The maximum error will be:
and the total error for the entire journey -> . For large N, we assume that the error is close to zero. We will see this on the graph (see Fig. 16): there will be an error on each interval, but the total error for a sufficiently large number of intervals will be negligible.
Rice. 16. Error on intervals
So, each next speed value is one and the same value greater than the previous one. We know from algebra that this is an arithmetic progression with a progression difference:
The path on the sections (with uniform rectilinear motion (see Fig. 17) is equal to:
Rice. 17. Consideration of areas of body movement
On the second section:
On the nth section, the path is equal to:
Arithmetic progression
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Arithmetic progression such a numerical sequence is called in which each subsequent number differs from the previous one by the same amount. An arithmetic progression is given by two parameters: the initial term of the progression and the difference of the progression. Then the sequence is written like this: The sum of the first terms of an arithmetic progression is calculated by the formula:
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Let's sum up all the paths. This will be the sum of the first N members of the arithmetic progression:
Since we have divided the movement into many intervals, we can assume that , then:
We had a lot of formulas, and in order not to get confused, we did not write x indices each time, but considered everything in projection onto the coordinate axis.
So, we have obtained the main formula of uniformly accelerated motion: displacement with uniformly accelerated motion in time T, which we, along with the definition of acceleration (change in speed per unit time), will use to solve problems:
We were working on a car problem. Substitute the numbers into the solution and get the answer: the car drove 55.4 km.
Mathematical part of the problem solution
We have dealt with movement. And how to determine the coordinate of the body at any time?
By definition, the movement of a body in time is a vector whose beginning is at the starting point of the movement, and whose end is at the end point where the body will be in time. We need to find the coordinate of the body, so we write an expression for the projection of the displacement onto the coordinate axis (see Fig. 18):
Rice. 18. Movement projection
Let's express the coordinate:
That is, the coordinate of the body at the moment of time is equal to the initial coordinate plus the projection of the movement that the body made during the time . We have already found the projection of displacement during uniformly accelerated motion, it remains to substitute and write down:
This is the equation of motion with constant acceleration. It allows you to find out the coordinate of a moving material point at any time. It is clear that we choose the moment of time within the interval when the model works: the acceleration is constant, the movement is rectilinear.
Why the equation of motion cannot be used to find a path
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In what cases can we consider modulo movement to be equal to the path? When a body moves along a straight line and does not change direction. For example, with uniform rectilinear motion, we do not always clearly stipulate whether we find the path or the movement, they still coincide. With uniformly accelerated motion, the speed changes. If the speed and acceleration are directed in opposite directions (see Fig. 19), then the speed modulus decreases, and at some point it will become zero and the speed will change direction, that is, the body will start moving in the opposite direction.
Rice. 19. Velocity modulus decreases And then, if in this moment time the body is at a distance of 3 m from the beginning of the observation, then its displacement is 3 m, but if the body first passed 5 m, then turned around and passed another 2 m, then the path will be 7 m. And how to find it if you don’t know these numbers? You just need to find the moment when the speed is zero, that is, when the body turns around, and find the path to and from this point (see Fig. 20).
Rice. 20. The moment when the speed is 0 |
Bibliography
- Sokolovich Yu.A., Bogdanova GS Physics: A Handbook with Examples of Problem Solving. - 2nd edition redistribution. - X .: Vesta: Publishing house "Ranok", 2005. - 464 p.
- Landsberg G.S. Elementary textbook of physics; v.1. Mechanics. Heat. Molecular physics - M .: Publishing house "Nauka", 1985.
- Internet portal "kaf-fiz-1586.narod.ru" ()
- Internet portal "Study - Easy" ()
- Internet portal "Knowledge Hypermarket" ()
Homework
- What is an arithmetic progression?
- What kind of movement is progressive?
- What is a vector quantity?
- Write down the formula for acceleration in terms of change in speed.
- What is the equation of motion with constant acceleration?
- The acceleration vector is directed towards the movement of the body. How will the body change its speed?
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ABC of physics
Almighty friction
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steam engines
"This mighty giant was three meters tall: the giant easily pulled a van with five passengers. The Steam Man had a chimney pipe on his head, from which thick black smoke poured ... everything, even the face, was made of iron, and all this constantly gnashed and rumbled ... "Who is this about? Who are these praises for? ......... read
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Thales of Miletus endowed him with a soul, Plato compared him with a poet, Orpheus found him like a bridegroom ... In the Renaissance, a magnet was considered a reflection of the sky and attributed to it the ability to bend space. The Japanese believed that a magnet is a force that will help turn fortune towards you ......... read
On the other side of the mirror
Do you know how many interesting discoveries the "mirror" can give? The image of your face in the mirror has the right and left halves swapped. But faces are rarely completely symmetrical, so others see you completely different. Have you thought about it? ......... read
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"The realization that the miraculous was near us comes too late." - A. Blok.
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“I want to create miracles!” he said and asked himself: “But tell me, have you done anything at all?” Leonardo da Vinci wrote his treatises in cryptography using an ordinary mirror, so his encrypted manuscripts could only be read for the first time three centuries later.........
Motion with constant acceleration is a motion in which the acceleration vector remains constant both in magnitude and in direction. An example of this type of movement is the movement of a point in the field of gravity (both vertically and at an angle to the horizon).
Using the definition of acceleration, we obtain the following relation
After integration, we have the equality 
.
Given that the instantaneous velocity vector is 
, we will have the following expression
Integration of the last expression gives the following relation
. From where we get the equation of motion of a point with constant acceleration
.
Examples of vector equations of motion of a material point
Uniform rectilinear motion ( 
):
. (1.7)
Movement with constant acceleration ( 
):
. (1.8)
The dependence of speed on time when a point moves with constant acceleration has the form:
.
(1.9)
Questions for self-control.
Formulate the definition of mechanical motion.
Define a material point.
How is the position of a material point in space determined in the vector way of describing motion?
What is the essence of the vector method for describing mechanical motion? What characteristics are used to describe this movement?
Give definitions of vectors of average and instantaneous speed. How is the direction of these vectors determined?
Define the mean and instantaneous acceleration vectors.
Which of the relations is the equation of motion of a point with constant acceleration? What relationship determines the dependence of the velocity vector on time?
§1.2. Coordinate way of describing motion
In the coordinate method, a coordinate system (for example, Cartesian) is chosen to describe the movement. The reference point is rigidly fixed with the selected body ( reference body). Let be 
unit vectors directed to the positive sides of the axes OX, OY and OZ, respectively. The position of the point is given by the coordinates 
.
The instantaneous velocity vector is defined as follows:
where 
projections of the velocity vector on the coordinate axes, and 
derivatives of coordinates with respect to time.
The length of the velocity vector is related to its projections by the relation:
. (1.11)
For the instantaneous acceleration vector, the relation is true:
where 
projections of the acceleration vector on the coordinate axes, and 
time derivatives of velocity vector projections.
The length of the instantaneous acceleration vector is found by the formula:
. (1.13)
Examples of equations of point motion in a Cartesian coordinate system
. (1.14)
Motion equations: 
. (1.15)
Dependences of the projections of the velocity vector on the coordinate axes on time:
(1.16)
Questions for self-control.
What is the essence of the coordinate method of describing motion?
What ratio determines the instantaneous velocity vector? What formula is used to calculate the magnitude of the velocity vector?
What ratio determines the instantaneous acceleration vector? What formula is used to calculate the magnitude of the instantaneous acceleration vector?
What relations are called the equations of uniform motion of a point?
What relationships are called equations of motion with constant acceleration? What formulas are used to calculate the projections of the instantaneous velocity of a point on the coordinate axes?
Kinematics is the study of classical mechanical motion in physics. Unlike dynamics, science studies why bodies move. She answers the question of how they do it. In this article, we will consider what acceleration and movement with constant acceleration are.
The concept of acceleration
When a body moves in space, in some time it overcomes a certain path, which is the length of the trajectory. To calculate this path, use the concepts of speed and acceleration.
Speed as a physical quantity characterizes the speed of change in time of the distance traveled. The speed is directed tangentially to the trajectory in the direction of body movement.
Acceleration is a slightly more complex quantity. In short, it describes the change in speed at a given point in time. The math looks like this:
To understand this formula more clearly, let's give a simple example: suppose that in 1 second of motion the body's speed increased by 1 m/s. These figures, substituted into the expression above, lead to the result: the acceleration of the body during this second was equal to 1 m/s 2 .
The direction of acceleration is completely independent of the direction of velocity. Its vector coincides with the vector of the resultant force that causes this acceleration.
An important point in the above definition of acceleration should be noted. This value characterizes not only the change in speed modulo, but also in direction. The latter fact should be taken into account in the case of curvilinear motion. Further in the article only rectilinear motion will be considered.
Speed when moving with constant acceleration
Acceleration is constant if it retains its modulus and direction during motion. Such a motion is called uniformly accelerated or uniformly slowed down - it all depends on whether the acceleration leads to an increase in speed or to its decrease.
In the case of a body moving with constant acceleration, the speed can be determined by one of the following formulas:
The first two equations characterize uniformly accelerated movement. The difference between them is that the second expression is applicable for the case of a non-zero initial velocity.
The third equation is an expression for the speed at uniformly slow motion with constant acceleration. The acceleration is directed against the speed.
The graphs of all three functions v(t) are straight lines. In the first two cases, the straight lines have a positive slope relative to the x-axis, in the third case this slope is negative.
Distance formulas
For a path in the case of movement with a constant acceleration (acceleration a = const), it is not difficult to obtain formulas if you calculate the integral of the speed over time. Having done this mathematical operation for the above three equations, we get the following expressions for the path L:
L \u003d v 0 * t + a * t 2 / 2;
L \u003d v 0 * t - a * t 2 / 2.
The graphs of all three path-time functions are parabolas. In the first two cases, the right branch of the parabola increases, and for the third function it gradually reaches a certain constant, which corresponds to the distance traveled until the body stops completely.
The solution of the problem
Moving at a speed of 30 km / h, the car began to accelerate. In 30 seconds he walked a distance of 600 meters. What was the acceleration of the car?
First of all, let's convert the initial speed from km/h to m/s:
v 0 \u003d 30 km / h \u003d 30000/3600 \u003d 8.333 m / s.
Now we write the equation of motion:
L \u003d v 0 *t + a*t 2 /2.
From this equality, we express the acceleration, we get:
a = 2*(L - v 0 *t)/t 2 .
All physical quantities in this equation are known from the conditions of the problem. We substitute them into the formula and get the answer: a ≈ 0.78 m / s 2. Thus, moving with a constant acceleration, the car increased its speed by 0.78 m/s every second.
We also calculate (for interest) what speed he acquired after 30 seconds of accelerated movement, we get:
v \u003d v 0 + a * t \u003d 8.333 + 0.78 * 30 \u003d 31.733 m / s.
The resulting speed is 114.2 km/h.
Lesson Objectives:
Educational:
Developing:
Vos nutritious
Lesson type : Combined lesson.
View document content
Lesson topic: “Acceleration. Rectilinear motion with constant acceleration.
Prepared by - physics teacher MBOU "Secondary School No. 4" Pogrebnyak Marina Nikolaevna
Class -11
Lesson 5/4 Lesson topic: “Acceleration. Rectilinear motion with constant acceleration».
Lesson Objectives:
Educational: To acquaint students with the characteristic features of rectilinear uniformly accelerated motion. Give the concept of acceleration as the main physical quantity characterizing non-uniform motion. Enter the formula for determining the instantaneous speed of a body at any time, calculate the instantaneous speed of a body at any time,
to improve the ability of students to solve problems in analytical and graphical ways.
Developing: development of theoretical, creative thinking among schoolchildren, the formation of operational thinking aimed at choosing optimal solutions
Vosnutritious : to cultivate a conscious attitude to learning and interest in the study of physics.
Lesson type : Combined lesson.
Demos:
1. Uniformly accelerated motion of a ball on an inclined plane.
2. Multimedia application "Fundamentals of kinematics": fragment "Uniformly accelerated motion".
Working process.
1. Organizational moment.
2. Knowledge check: Independent work ("Movement." "Graphs of rectilinear uniform motion") - 12 min.
3. Learning new material.
Plan for presenting new material:
1. Instantaneous speed.
2. Acceleration.
3. Speed in rectilinear uniformly accelerated motion.
1. Instantaneous speed. If the speed of the body changes with time, to describe the movement, you need to know what the speed of the body is at a given time (or at a given point in the trajectory). This speed is called instantaneous speed.
You can also say that the instantaneous speed is the average speed over a very small interval of time. When driving at a variable speed, the average speed measured over different time intervals will be different.
However, if smaller and smaller time intervals are taken when measuring the average speed, the value of the average speed will tend to some specific value. This is the instantaneous speed at a given time. In the future, speaking of the speed of a body, we will mean its instantaneous speed.
2. Acceleration. With uneven movement, the instantaneous speed of the body is a variable; it is different in modulus and (or) in direction at different moments of time and at different points of the trajectory. All car and motorcycle speedometers show us only the instantaneous speed module.
If the instantaneous speed of non-uniform movement changes unequally over the same time intervals, then it is very difficult to calculate it.
Such complex uneven movements are not studied at school. Therefore, we will consider only the simplest non-uniform motion - uniformly accelerated rectilinear motion.
Rectilinear motion, in which the instantaneous speed changes in the same way for any equal time intervals, is called uniformly accelerated rectilinear motion.
If the speed of a body changes as it moves, the question arises: what is the “rate of change of speed”? This quantity, called acceleration, plays the most important role in all mechanics: we will soon see that the acceleration of a body is determined by the forces acting on this body.
Acceleration is the ratio of a change in the speed of a body to the time interval during which this change occurred.
Unit of acceleration in SI: m/s 2 .
If a body moves in one direction with an acceleration of 1 m/s 2, its speed changes every second by 1 m/s.
The term "acceleration" is used in physics when it comes to any change in speed, including when the modulus of speed decreases or when the modulus of speed remains unchanged and the speed changes only in direction.
3. Speed in rectilinear uniformly accelerated motion.
It follows from the definition of acceleration that v = v 0 + at.
If we direct the x-axis along the straight line along which the body moves, then in projections onto the x-axis we get v x \u003d v 0 x + a x t.
Thus, in a rectilinear uniformly accelerated motion, the velocity projection linearly depends on time. This means that the graph of v x (t) is a straight line segment.
Movement formula:
Accelerating car speed chart:
Decelerating car speed chart
4. Consolidation of new material.
What is the instantaneous velocity of a stone thrown vertically upward at the top of the trajectory?
What speed - average or instantaneous - are we talking about in the following cases:
a) the train traveled between stations at a speed of 70 km/h;
b) the speed of the hammer on impact is 5 m/s;
c) the speedometer on the electric locomotive shows 60 km/h;
d) a bullet flies out of a rifle at a speed of 600 m/s.
TASKS SOLVED IN THE LESSON
The OX axis is directed along the trajectory of the rectilinear motion of the body. What can you say about the movement, in which: a) v x 0, and x 0; b) v x 0, a x v x x 0;
d) v x x v x x = 0?
1. The hockey player lightly hit the puck with a stick, giving it a speed of 2 m / s. What will be the speed of the puck 4 s after the impact if, as a result of friction against the ice, it moves with an acceleration of 0.25 m / s 2?
2. The train, 10 seconds after the start of movement, acquires a speed of 0.6 m/s. How long will it take for the speed of the train to reach 3 m/s?
5.HOMEWORK: §5,6, ex. 5 No. 2, ex. 6 #2.
