Determining the speed according to the traffic schedule. Graphical representation of uniform rectilinear motion - document

This video lesson is devoted to the topic “Speed ​​of rectilinear uniformly accelerated motion. Speed ​​Graph. During the lesson, students will need to remember such a physical quantity as acceleration. Then they will learn how to determine the speeds of a uniformly accelerated rectilinear motion. After the teacher will tell you how to build a speed graph correctly.

Let's remember what acceleration is.

Definition

Acceleration is a physical quantity that characterizes the change in speed over a certain period of time:

That is, acceleration is a quantity that is determined by the change in speed over the time during which this change occurred.

Once again about what uniformly accelerated motion is

Let's consider the problem.

The car increases its speed by . Is the car moving with uniform acceleration?

At first glance, it seems so, because for equal periods of time, the speed increases by equal amounts. Let's take a closer look at the movement for 1 s. It is possible that the car moved uniformly for the first 0.5 s and increased its speed by 0.5 s in the second. There could be another situation: the car accelerated to the first yes, and the remaining ones moved evenly. Such a movement will not be uniformly accelerated.

By analogy with uniform motion, we introduce the correct formulation of uniformly accelerated motion.

uniformly accelerated called such a movement in which the body for ANY equal intervals of time changes its speed by the same amount.

Often called uniformly accelerated is such a movement in which the body moves with constant acceleration. The simplest example of uniformly accelerated motion is the free fall of a body (the body falls under the influence of gravity).

Using the equation that determines the acceleration, it is convenient to write a formula for calculating the instantaneous speed of any interval and for any moment of time:

The velocity equation in projections is:

This equation makes it possible to determine the speed at any moment of the movement of the body. When working with the law of change of speed from time, it is necessary to take into account the direction of speed in relation to the selected CO.

On the question of the direction of velocity and acceleration

In uniform motion, the direction of velocity and displacement always coincide. In the case of uniformly accelerated motion, the direction of velocity does not always coincide with the direction of acceleration, and the direction of acceleration does not always indicate the direction of motion of the body.

Let's consider the most typical examples of the direction of velocity and acceleration.

1. Velocity and acceleration are directed in the same direction along one straight line (Fig. 1).

Rice. 1. Velocity and acceleration are directed in the same direction along one straight line

In this case, the body accelerates. Examples of such movement can be free fall, the start of the movement and acceleration of the bus, the launch and acceleration of the rocket.

2. Speed ​​and acceleration are directed in different directions along one straight line (Fig. 2).

Rice. 2. Speed ​​and acceleration are directed in different directions along the same straight line

Such a movement is sometimes called uniformly slow. In this case, the body is said to be slowing down. Eventually it will either stop or start moving in the opposite direction. An example of such a movement is a stone thrown vertically upwards.

3. Velocity and acceleration are mutually perpendicular (Fig. 3).

Rice. 3. Velocity and acceleration are mutually perpendicular

Examples of such motion are the motion of the Earth around the Sun and the motion of the Moon around the Earth. In this case, the trajectory of motion will be a circle.

Thus, the direction of acceleration does not always coincide with the direction of velocity, but always coincides with the direction of change of velocity.

Speed ​​Graph(projection of speed) is the law of change of speed (projection of speed) from time for uniformly accelerated rectilinear motion, presented graphically.

Rice. 4. Graphs of the dependence of the projection of speed on time for uniformly accelerated rectilinear motion

Let's analyze different charts.

First. Velocity projection equation: . As the time increases, the speed also increases. Please note that on a graph where one of the axes is time and the other is speed, there will be a straight line. This line starts from the point , which characterizes the initial speed.

The second is the dependence with a negative value of the acceleration projection, when the movement is slow, that is, the modulo speed first decreases. In this case, the equation looks like this:

The graph starts at the point and continues until the point , the intersection of the time axis. At this point, the speed of the body becomes zero. This means that the body has stopped.

If you look closely at the velocity equation, you will remember that there was a similar function in mathematics:

Where and are some constants, for example:

Rice. 5. Function Graph

This is the equation of a straight line, which is confirmed by the graphs we have examined.

To finally understand the speed graph, let's consider special cases. In the first graph, the dependence of speed on time is due to the fact that the initial speed, , is equal to zero, the acceleration projection is greater than zero.

Write this equation. And the type of chart itself is quite simple (chart 1).

Rice. 6. Various cases of uniformly accelerated motion

Two more cases uniformly accelerated motion are shown in the following two graphs. The second case is a situation when at first the body moved with a negative projection of acceleration, and then began to accelerate in the positive direction of the axis.

The third case is the situation where the acceleration projection is less than zero and the body is continuously moving in the direction opposite to the positive axis direction. At the same time, the modulus of speed is constantly increasing, the body is accelerating.

Graph of acceleration versus time

Uniformly accelerated motion is a motion in which the acceleration of the body does not change.

Let's look at the charts:

Rice. 7. Graph of dependence of projections of acceleration on time

If any dependence is constant, then on the graph it is depicted as a straight line parallel to the x-axis. Lines I and II - direct movements for two different bodies. Note that line I lies above the abscissa line (positive acceleration projection), and line II lies below (negative acceleration projection). If the motion were uniform, then the acceleration projection would coincide with the abscissa axis.

Consider Fig. 8. The area of ​​\u200b\u200bthe figure bounded by the axes, the graph and the perpendicular to the x-axis is:

The product of acceleration and time is the change in speed over a given time.

Rice. 8. Speed ​​change

The area of ​​the figure bounded by the axes, dependence and perpendicular to the abscissa axis is numerically equal to the change in the speed of the body.

We used the word "number" because the units for area and change in speed are not the same.

In this lesson, we got acquainted with the equation of speed and learned how to graphically represent this equation.

Bibliography

1. Kikoin I.K., Kikoin A.K. Physics: A textbook for the 9th grade of high school. - M.: "Enlightenment".
2. Peryshkin A.V., Gutnik E.M., Physics. Grade 9: textbook for general education. institutions / A.V. Peryshkin, E.M. Gutnik. - 14th ed., stereotype. - M.: Bustard, 2009. - 300 p.
3. Sokolovich Yu.A., Bogdanova G.S. Physics: Handbook with examples of problem solving. - 2nd edition redistribution. - X .: Vesta: Publishing house "Ranok", 2005. - 464 p.

1. Internet portal "class-fizika.narod.ru" ()
2. Internet portal "youtube.com" ()
3. Internet portal "fizmat.by" ()
4. Internet portal "sverh-zadacha.ucoz.ru" ()

Homework

1. What is uniformly accelerated motion?

2. Describe the movement of the body and determine the distance traveled by the body according to the graph for 2 s from the beginning of the movement:

3. Which of the graphs shows the dependence of the projection of the body's velocity on time during uniformly accelerated motion at ?

Questions.

1. Write down the formula by which you can calculate the projection of the instantaneous velocity vector of rectilinear uniformly accelerated motion, if you know: a) the projection of the initial velocity vector and the projection of the acceleration vector; b) the projection of the acceleration vector, given that the initial velocity is zero.

2. What is the graph of the projection of the velocity vector of uniformly accelerated motion at an initial speed: a) equal to zero; b) not equal to zero?

3. How are the movements, the graphs of which are presented in Figures 11 and 12, similar and different from each other?

In both cases, the movement occurs with acceleration, but in the first case, the acceleration is positive, and in the second, it is negative.

Exercises.

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 skier moves down the mountain from rest with an acceleration equal to 0.2 m/s 2 . After what time interval will its speed increase to 2 m/s?

3. In the same coordinate axes, plot the projections of the velocity vector (on the X axis, co-directed with the initial velocity vector) for rectilinear uniformly accelerated motion for the cases: a) v ox \u003d 1m / s, a x \u003d 0.5 m / s 2 ; b) v ox \u003d 1m / s, a x \u003d 1 m / s 2; c) v ox \u003d 2 m / s, a x \u003d 1 m / s 2.
The scale is the same in all cases: 1cm - 1m/s; 1cm - 1s.

4. In the same coordinate axes, build graphs of the projection of the velocity vector (on the X axis, co-directed with the initial velocity vector) for rectilinear uniformly accelerated motion for the cases: a) v ox = 4.5 m/s, a x = -1.5 m / s 2; b) v ox \u003d 3 m / s, a x \u003d -1 m / s 2
Choose your own scale.

5. Figure 13 shows the graphs of the module of the velocity vector versus time for rectilinear motion of two bodies. What is the modulus of acceleration of body I? body II?

3.1. Uniform movement in a straight line.

3.1.1. Uniform movement in a straight line- movement in a straight line with a constant modulus and direction of acceleration:

3.1.2. Acceleration()- a physical vector quantity showing how much the speed will change in 1 s.

In vector form:

where is the initial speed of the body, is the speed of the body at the moment of time t.

In the projection on the axis Ox:

where is the projection of the initial speed on the axis Ox, - projection of the body velocity on the axis Ox at the time t.

The signs of the projections depend on the direction of the vectors and the axis Ox.

3.1.3. Graph of projection of acceleration versus time.

With uniformly variable motion, acceleration is constant, therefore it will be straight lines parallel to the time axis (see Fig.):

3.1.4. Speed ​​in uniform motion.

In vector form:

In the projection on the axis Ox:

For uniformly accelerated motion:

For slow motion:

3.1.5. Velocity projection plot versus time.

The graph of the projection of speed against time is a straight line.

Direction of movement: if the graph (or part of it) is above the time axis, then the body moves in the positive direction of the axis Ox.

Acceleration value: the greater the tangent of the angle of inclination (the steeper it goes up or down), the greater the acceleration module; where is the change in speed over time

Intersection with the time axis: if the graph crosses the time axis, then the body slowed down before the intersection point (equally slow movement), and after the intersection point it began to accelerate in the opposite direction (equally accelerated movement).

3.1.6. The geometric meaning of the area under the graph in the axes

Area under the graph when on the axis Oy speed is delayed, and on the axis Ox Time is the path traveled by the body.

On fig. 3.5 the case of uniformly accelerated motion is drawn. The path in this case will be equal to the area of ​​the trapezoid: (3.9)

3.1.7. Formulas for calculating the path

Uniformly accelerated motion	Uniformly slow motion
(3.10)	(3.12)
(3.11)	(3.13)
(3.14)

All formulas presented in the table work only while maintaining the direction of movement, that is, until the intersection of the straight line with the time axis on the graph of the dependence of the projection of speed on time.

If the intersection has occurred, then the movement is easier to break into two stages:

before crossing (braking):

After crossing (acceleration, movement in the opposite direction)

In the formulas above - the time from the beginning of the movement to the intersection with the time axis (time to stop), - the path that the body has traveled from the beginning of the movement to the intersection with the time axis, - the time elapsed from the moment of crossing the time axis to the present moment t, - the path that the body has traveled in the opposite direction during the time elapsed from the moment of crossing the time axis to the present moment t, - the module of the displacement vector for the entire time of movement, L- the path traveled by the body during the entire movement.

3.1.8. Move in -th second.

In time, the body will travel the path:

In time, the body will travel the path:

Then, in the i-th interval, the body will cover the path:

The interval can be any length of time. Most often with

Then in 1 second the body travels the path:

For 2nd second:

For the 3rd second:

If we look carefully, we will see that, etc.

Thus, we arrive at the formula:

In words: the paths covered by the body in successive periods of time correlate with each other as a series of odd numbers, and this does not depend on the acceleration with which the body moves. We emphasize that this relation is valid for

3.1.9. Body coordinate equation for uniformly variable motion

Coordinate equation

The signs of the projections of the initial velocity and acceleration depend on the relative position of the corresponding vectors and the axis Ox.

To solve problems, it is necessary to add to the equation the equation for changing the velocity projection on the axis:

3.2. Graphs of kinematic quantities for rectilinear motion

3.3. Free fall body

Free fall means the following physical model:

1) The fall occurs under the influence of gravity:

2) There is no air resistance (in tasks it is sometimes written “neglect air resistance”);

3) All bodies, regardless of mass, fall with the same acceleration (sometimes they add - “regardless of the shape of the body”, but we consider the movement of only a material point, so the shape of the body is no longer taken into account);

4) The acceleration of free fall is directed strictly downward and is equal on the surface of the Earth (in problems we often take it for convenience of calculations);

3.3.1. Equations of motion in the projection onto the axis Oy

Unlike movement along a horizontal straight line, when far from all tasks change the direction of movement, in free fall it is best to immediately use the equations written in projections onto the axis Oy.

Body coordinate equation:

Velocity projection equation:

As a rule, in problems it is convenient to choose the axis Oy in the following way:

Axis Oy directed vertically upwards;

The origin of coordinates coincides with the level of the Earth or the lowest point of the trajectory.

With this choice, the equations and are rewritten in the following form:

3.4. Movement in a plane Oxy.

We have considered the motion of a body with acceleration along a straight line. However, the uniform movement is not limited to this. For example, a body thrown at an angle to the horizon. In such tasks, it is necessary to take into account the movement along two axes at once:

Or in vector form:

And changing the projection of speed on both axes:

3.5. Application of the concept of derivative and integral

We will not give here a detailed definition of the derivative and integral. To solve problems, we need only a small set of formulas.

Derivative:

where A, B and that is the constants.

Integral:

Now let's see how the concept of derivative and integral is applicable to physical quantities. In mathematics, the derivative is denoted by """, in physics, the time derivative is denoted by "∙" over a function.

Speed:

that is, the speed is a derivative of the radius vector.

For velocity projection:

Acceleration:

that is, acceleration is a derivative of speed.

For acceleration projection:

Thus, if the law of motion is known, then we can easily find both the speed and acceleration of the body.

We now use the concept of an integral.

Speed:

that is, the speed can be found as the time integral of the acceleration.

Radius vector:

that is, the radius vector can be found by taking the integral of the velocity function.

Thus, if the function is known, then we can easily find both the speed and the law of motion of the body.

The constants in the formulas are determined from the initial conditions - the value and at the moment of time

3.6. Velocity Triangle and Displacement Triangle

3.6.1. speed triangle

In vector form, at constant acceleration, the law of velocity change has the form (3.5):

This formula means that the vector is equal to the vector sum of vectors and the vector sum can always be depicted in the figure (see figure).

In each task, depending on the conditions, the velocity triangle will have its own form. Such a representation makes it possible to use geometric considerations in solving, which often simplifies the solution of the problem.

3.6.2. Movement Triangle

In vector form, the law of motion at constant acceleration has the form:

When solving the problem, you can choose the reference system in the most convenient way, therefore, without losing generality, we can choose the reference system so that, that is, the origin of the coordinate system is placed at the point where the body is located at the initial moment. Then

that is, the vector is equal to the vector sum of the vectors and Let's draw in the figure (see Fig.).

As in the previous case, depending on the conditions, the displacement triangle will have its own form. Such a representation makes it possible to use geometric considerations in solving, which often simplifies the solution of the problem.

To build this graph, the time of movement is plotted on the abscissa axis, and the speed (velocity projection) of the body is plotted on the ordinate axis. In uniformly accelerated motion, the speed of a body changes over time. If the body moves along the O x axis, the dependence of its speed on time is expressed by the formulas
v x \u003d v 0x +a x t and v x \u003d at (for v 0x \u003d 0).

From these formulas it can be seen that the dependence of v x on t is linear, therefore, the speed graph is a straight line. If the body moves with some initial speed, this straight line intersects the y-axis at the point v 0x . If the initial velocity of the body is zero, the velocity graph passes through the origin.

Graphs of the speed of rectilinear uniformly accelerated motion are shown in fig. 9. In this figure, graphs 1 and 2 correspond to movement with a positive acceleration projection on the O x axis (speed increases), and graph 3 corresponds to movement with a negative acceleration projection (speed decreases). Graph 2 corresponds to movement without initial speed, and graphs 1 and 3 correspond to movement with initial speed v ox . The angle of inclination a of the graph to the x-axis depends on the acceleration of the body. As can be seen from fig. 10 and formulas (1.10),

tg=(v x -v 0x)/t=a x .

According to the speed graphs, you can determine the path traveled by the body for a period of time t. To do this, we determine the area of ​​the trapezoid and the triangle shaded in Fig. eleven.

On the selected scale, one base of the trapezoid is numerically equal to the module of the projection of the initial velocity v 0x of the body, and its other base is the module of the projection of its velocity v x at time t. The height of the trapezoid is numerically equal to the duration of the time interval t. Trapezium area

S=(v0x+vx)/2t.

Using formula (1.11), after transformations, we find that the area of ​​the trapezoid

S=v 0x t+at 2 /2.

the path traveled in a rectilinear uniformly accelerated motion with an initial speed is numerically equal to the area of ​​the trapezoid limited by the speed graph, the coordinate axes and the ordinate corresponding to the value of the body's speed at time t.

In the chosen scale, the height of the triangle (Fig. 11,b) is numerically equal to the modulus of the projection of the velocity v x of the body at time t, and the base of the triangle is numerically equal to the duration of the time interval t. The area of ​​the triangle is S=v x t/2.

Using formula 1.12, after transformations, we find that the area of ​​the triangle

The right side of the last equality is an expression that defines the path traveled by the body. Hence, the path traveled in rectilinear uniformly accelerated motion without initial speed is numerically equal to the area of ​​the triangle bounded by the velocity graph, the abscissa axis and the ordinate corresponding to the body's velocity at time t.

« Physics - Grade 10 "

What is the difference between uniform motion and uniformly accelerated motion?
What is the difference between a path graph for uniformly accelerated motion and a path graph for uniform motion?
What is called the projection of a vector on any axis?

In the case of uniform rectilinear motion, you can determine the speed according to the graph of coordinates versus time.

The velocity projection is numerically equal to the tangent of the slope of the straight line x(t) to the x-axis. In this case, the greater the speed, the greater the angle of inclination.

Rectilinear uniformly accelerated motion.

Figure 1.33 shows graphs of the projection of acceleration versus time for three different values ​​of acceleration in a rectilinear uniformly accelerated motion of a point. They are straight lines parallel to the x-axis: a x = const. Graphs 1 and 2 correspond to movement when the acceleration vector is directed along the OX axis, graph 3 - when the acceleration vector is directed in the direction opposite to the OX axis.

With uniformly accelerated motion, the velocity projection depends linearly on time: υ x = υ 0x + a x t. Figure 1.34 shows the graphs of this dependence for these three cases. In this case, the initial speed of the point is the same. Let's analyze this chart.

Acceleration projection It can be seen from the graph that the greater the acceleration of the point, the greater the angle of inclination of the straight line to the t axis and, accordingly, the greater the tangent of the inclination angle, which determines the acceleration value.

For the same period of time at different accelerations, the speed changes by different values.

With a positive value of the acceleration projection for the same time interval, the velocity projection in case 2 increases 2 times faster than in case 1. With a negative value of the acceleration projection on the OX axis, the velocity projection modulo changes by the same value as in case 1, but the speed is decreasing.

For cases 1 and 3, the graphs of the dependence of the velocity modulus on time will coincide (Fig. 1.35).

Using the speed versus time graph (Figure 1.36), we find the change in the coordinate of the point. This change is numerically equal to the area of ​​the shaded trapezoid, in this case, the change in coordinate for 4 with Δx = 16 m.

We found a change in coordinates. If you need to find the coordinate of a point, then you need to add its initial value to the found number. Let at the initial moment of time x 0 = 2 m, then the value of the coordinate of the point at a given moment of time, equal to 4 s, is 18 m. In this case, the displacement module is equal to the path traveled by the point, or the change in its coordinates, i.e. 16 m .

If the movement is uniformly slowed down, then the point during the selected time interval can stop and start moving in the opposite direction to the initial one. Figure 1.37 shows the projection of velocity versus time for such a motion. We see that at the moment of time equal to 2 s, the direction of the velocity changes. The change in coordinate will be numerically equal to the algebraic sum of the areas of the shaded triangles.

Calculating these areas, we see that the change in coordinate is -6 m, which means that in the direction opposite to the OX axis, the point has traveled a greater distance than in the direction of this axis.

Square above we take the t axis with the plus sign, and the area under axis t, where the velocity projection is negative, with a minus sign.

If at the initial moment of time the speed of a certain point was equal to 2 m / s, then its coordinate at the moment of time equal to 6 s is equal to -4 m. The module of point movement in this case is also equal to 6 m - the module of coordinate change. However, the path traveled by this point is 10 m, the sum of the areas of the shaded triangles shown in Figure 1.38.

Let's plot the dependence of the x-coordinate of a point on time. According to one of the formulas (1.14), the time dependence curve - x(t) - is a parabola.

If the point moves at a speed, the time dependence of which is shown in Figure 1.36, then the branches of the parabola are directed upwards, since a x\u003e 0 (Figure 1.39). From this graph, we can determine the coordinate of the point, as well as the speed at any given time. So, at the moment of time equal to 4 s, the coordinate of the point is 18 m.

For the initial moment of time, drawing a tangent to the curve at point A, we determine the tangent of the slope α 1, which is numerically equal to the initial speed, i.e. 2 m / s.

To determine the speed at point B, we draw a tangent to the parabola at this point and determine the tangent of the angle α 2 . It is equal to 6, therefore, the speed is 6 m/s.

The path versus time graph is the same parabola, but drawn from the origin (Fig. 1.40). We see that the path is continuously increasing with time, the movement is in one direction.

If the point moves at a speed whose projection versus time graph is shown in Figure 1.37, then the branches of the parabola are directed downwards, since a x< 0 (рис. 1.41). При этом моменту времени, равному 2 с, соответствует вершина параболы. Касательная в точке В параллельна оси t, угол наклона касательной к этой оси равен нулю, и скорость также равна нулю. До этого момента времени тангенс угла наклона касательной уменьшался, но был положителен, движение точки происходило в направлении оси ОХ.

Starting from the time t = 2 s, the tangent of the slope angle becomes negative, and its module increases, which means that the point moves in the opposite direction to the initial one, while the module of the movement speed increases.

The displacement modulus is equal to the modulus of the difference between the coordinates of the point at the final and initial moments of time and is equal to 6 m.

The graph of dependence of the path traveled by a point on time, shown in Figure 1.42, differs from the graph of the dependence of displacement on time (see Figure 1.41).

No matter how the speed is directed, the path traveled by the point continuously increases.

Let us derive the dependence of the point coordinate on the velocity projection. Velocity υx = υ 0x + a x t, hence

In the case of x 0 \u003d 0 and x\u003e 0 and υ x\u003e υ 0x, the graph of the dependence of the coordinate on the speed is a parabola (Fig. 1.43).

In this case, the greater the acceleration, the less steep the branch of the parabola will be. This is easy to explain, since the greater the acceleration, the smaller the distance that the point must cover in order for the speed to increase by the same amount as when moving with less acceleration.

In case a x< 0 и υ 0x >0 speed projection will decrease. Let us rewrite equation (1.17) in the form where a = |a x |. The graph of this dependence is a parabola with branches pointing downwards (Fig. 1.44).

Accelerated movement.

According to the graphs of dependence of the projection of velocity on time, it is possible to determine the coordinate and projection of the acceleration of a point at any moment in time for any type of movement.

Let the projection of the speed of a point depend on time as shown in Figure 1.45. It is obvious that in the time interval from 0 to t 3 the movement of the point along the X axis occurred with variable acceleration. Starting from the moment of time equal to t 3 , the motion is uniform with a constant speed υ Dx . From the graph, we see that the acceleration with which the point moved was continuously decreasing (compare the angle of inclination of the tangent at points B and C).

The change in the x coordinate of a point over time t 1 is numerically equal to the area of ​​the curvilinear trapezoid OABt 1, over time t 2 - the area OACt 2, etc. As we can see from the graph of the dependence of the velocity projection on time, you can determine the change in body coordinates for any period of time.

According to the graph of the dependence of the coordinate on time, one can determine the value of the speed at any moment of time by calculating the tangent of the slope of the tangent to the curve at the point corresponding to the given moment of time. From figure 1.46 it follows that at time t 1 the velocity projection is positive. In the time interval from t 2 to t 3 the speed is zero, the body is motionless. At time t 4 the speed is also zero (the tangent to the curve at point D is parallel to the x-axis). Then the projection of the velocity becomes negative, the direction of movement of the point changes to the opposite.

If the graph of the dependence of the velocity projection on time is known, it is possible to determine the acceleration of the point, and also, knowing the initial position, determine the coordinate of the body at any time, i.e., solve the main problem of kinematics. One of the most important kinematic characteristics of movement, speed, can be determined from the graph of the dependence of coordinates on time. In addition, according to the specified graphs, you can determine the type of movement along the selected axis: uniform, with constant acceleration or movement with variable acceleration.