What is a dependency graph in physics. Graphical representation of uniform rectilinear motion - document

Lesson Topic: "Material point. Frame of reference" Objectives: to give an idea of ​​kinematics

Definition uniform rectilinear movement. - What is speed uniform movements? - Name the unit of speed movements in ... projection of the velocity vector on time movements U (O. 2. Graphic representation movements. - At point C...

Graphical representation of a uniformly accelerated rectilinear motion.

Movement with uniformly accelerated motion.

Ilevel.

Many physical quantities, describing the motions of bodies, change over time. Therefore, for greater clarity, the description of the movement is often depicted graphically.

Let us show how the dependences on time of kinematic quantities describing a rectilinear uniformly accelerated motion are graphically depicted.

Uniformly accelerated rectilinear motion- this is a movement in which the speed of the body changes in the same way for any equal time intervals, that is, this is a movement with constant acceleration in magnitude and direction.

a=const - acceleration equation. That is, a has a numerical value that does not change with time.

By definition of acceleration

From here we have already found equations for the dependence of speed on time: v = v0 + at.

Let's see how this equation can be used to graphically represent uniformly accelerated motion.

Let us graphically depict the dependences of kinematic quantities on time for three bodies

.

1 the body moves along the 0X axis, while increasing its speed (the acceleration vector a is co-directed with the velocity vector v). vx >0, ax > 0

2 the body moves along the 0X axis, while reducing its speed (the acceleration vector and not co-directed with the velocity vector v). vx >0, ax< 0

2 the body moves against the 0X axis, while reducing its speed (the acceleration vector and not co-directed with the velocity vector v). vx< 0, ах > 0

Acceleration Graph

Acceleration is by definition a constant. Then, for the presented situation, the graph of the dependence of acceleration on time a(t) will look like:

From the acceleration graph, you can determine how the speed changed - increased or decreased, and by what numerical value the speed changed and for which body the speed changed more.

Speed ​​Graph

If we compare the dependence of the coordinate on time for uniform motion and the dependence of the projection of velocity on time for uniformly accelerated motion, we can see that these dependencies are the same:

x= x0 + vx t vx = v 0 x + a X t

This means that the dependency graphs have the same form.

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.

Movement with uniformly accelerated motion.

With uniformly accelerated rectilinear motion, the speed of the body is determined by the formula

vx = v 0 x + a X t

In this formula, υ0 is the speed of the body at t = 0 (starting speed ), a= const - acceleration. On the velocity graph υ ( t), this dependence has the form of a straight line (Fig.).

The slope of the velocity graph can be used to determine the acceleration a body. The corresponding constructions are made in Figs. for graph I. The acceleration is numerically equal to the ratio of the sides of the triangle ABC:MsoNormalTable">

The greater the angle β that forms the velocity graph with the time axis, i.e. the greater the slope of the graph ( steepness), the greater the acceleration of the body.

For plot I: υ0 = –2 m/s, a= 1/2 m/s2.

For graph II: υ0 = 3 m/s, a= –1/3 m/s2.

The velocity graph also allows you to determine the displacement projection s body for a while t. Let us allocate on the time axis some small time interval Δ t. If this time interval is small enough, then the change in speed over this interval is also small, i.e., the movement during this time interval can be considered uniform with some average speed, which is equal to the instantaneous velocity υ of the body in the middle of the interval Δ t. Therefore, displacement Δ s in time Δ t will be equal to Δ s = υΔ t. This displacement is equal to the area of ​​the shaded strip (Fig.). Breaking down the time span from 0 to some point t for small intervals Δ t, we get that the displacement s for a given time t with uniformly accelerated rectilinear motion is equal to the area of ​​the trapezoid ODEF. Corresponding constructions are made for graph II in fig. 1.4.2. Time t taken equal to 5.5 s.

Since υ – υ0 = at, the final formula for moving s bodies with uniformly accelerated motion over a time interval from 0 to t will be written in the form:

To find the coordinate y body at any given time. t y t: https://pandia.ru/text/78/516/images/image008_63.gif" width="84" height="48 src=">

To find the x coordinate of a body at any time t to the starting coordinate x 0 add displacement over time t:

When analyzing a uniformly accelerated motion, sometimes the problem arises of determining the displacement of a body according to the given values ​​of the initial υ0 and final υ velocities and acceleration a. This problem can be solved using the equations written above by eliminating time from them. t. The result is written as

If the initial velocity υ0 is equal to zero, these formulas take the form MsoNormalTable">

It should once again be noted that the quantities υ0, υ, s, a, y 0 are algebraic quantities. Depending on the specific type of movement, each of these quantities can take both positive and negative values.

An example of solving the problem:

Petya moves down the mountain slope from rest with an acceleration of 0.5 m/s2 in 20 s and then moves along the horizontal section. Having traveled 40 m, he crashes into a gaping Vasya and falls into a snowdrift, reducing his speed to 0 m/s. With what acceleration did Petya move along the horizontal surface to the snowdrift? What is the length of the slope of the mountain from which Petya so unsuccessfully slid down?

Stage 1.

The equation for Petit's speed at the end of the descent from the mountain:

v 1 = v 01 + a 1t 1.

In projections on the axis X we get:

v 1x = a 1xt.

Let's write an equation relating the projections of Petya's speed, acceleration and displacement at the first stage of movement:

or because Petya was driving from the very top of the hill with the initial speed V01=0

(if I were Petya, I would be careful not to ride from such high hills)

Considering that Petya's initial speed at this 2nd stage of movement is equal to his final speed at the first stage:

v 02 x = v 1 x, v 2x = 0, where v1 is the speed with which Petya reached the bottom of the hill and started moving towards Vasya. V2x - Petya's speed in a snowdrift.

We use the equation and find the speed v1

On the horizontal section of the road, the path of Petit ramen:

BUT!!! it is more expedient to use another equation, since we do not know the time of Petya's movement to Vasya t2

The acceleration will turn out to be negative - this means that Petya tried very hard to slow down not about Vasya, but a little earlier.

Answer: a 2 = -1.25 m/s2; s 1 = 100 m.

IIlevel. Solve problems in writing.

1. Using the graphs shown in the figure, write down the equations for the dependence of speed on time. How the bodies moved at each stage of their movement (make according to the model, see example).

2. According to this acceleration graph, tell how the speed of the body changes. Write down the equations of the dependence of the speed on time, if at the moment of the beginning of the movement (t=0) the speed of the body is v0х =0. Please note that each subsequent segment of the movement, the body begins to pass at some speed (which was achieved in the previous time!).

3. A subway train leaving a station can reach a speed of 72 km/h in 20 seconds. Determine with what acceleration the bag forgotten in the subway car is moving away from you. What path will she take?

4. A cyclist moving at a speed of 3 m/s starts downhill with an acceleration of 0.8 m/s2. Find the length of the mountain if the descent took 6 s.

5. Starting braking with an acceleration of 0.5 m/s2, the train went to a stop 225 m. What was its speed before braking?

6. Starting to move, the soccer ball reached a speed of 50 m / s, traveled a distance of 50 m and crashed into the window. Determine the time it took the ball to travel this path and the acceleration with which it moved.

7. Uncle Oleg's neighbor's reaction time = 1.5 minutes, during which time he will figure out what happened to his window and have time to run out into the yard. Determine what speed young football players should develop so that the joyful owners of the window do not catch up with them if they need to run 350 m to their entrance.

8. Two cyclists are going towards each other. The first, having a speed of 36 km/h, began to climb uphill with an acceleration of 0.2 m/s2, and the second, having a speed of 9 km/h, began to descend from the mountain with an acceleration of 0.2 m/s2. After how much time and in what place will they collide because of their absent-mindedness, if the length of the mountain is 100 m?

If the trajectory of the point is known, then the dependence of the path traveled by the point on the elapsed time interval gives Full description this movement. We have seen that for uniform motion such a dependence can be given in the form of formula (9.2). The connection between and for individual points in time can also be specified in the form of a table containing the corresponding values ​​of the time interval and the distance traveled. Let us be given that the speed of some uniform motion is 2 m/s. Formula (9.2) in this case has the form . Let's make a table of the path and time of such a movement:

It is often convenient to depict the dependence of one quantity on another not by formulas or tables, but by graphs, which more clearly show the picture of changes in variable quantities and can facilitate calculations. Let's build a graph of the distance traveled versus time for the movement under consideration. To do this, take two mutually perpendicular lines - the coordinate axes; one of them (the abscissa axis) is called the time axis, and the other (the ordinate axis) is the path axis. Let's choose the scales for depicting time intervals and paths and take the point of intersection of the axes as the initial moment and as the starting point on the trajectory. Let's put on the axes the values ​​of time and the distance traveled for the considered movement (Fig. 18). To “bind” the values ​​of the distance traveled to time points, we draw perpendiculars to the axes from the corresponding points on the axes (for example, points 3 s and 6 m). The point of intersection of the perpendiculars corresponds simultaneously to both quantities: the path and the moment, - in this way the "binding" is achieved. The same construction can be performed for any other time points and corresponding paths, obtaining for each such pair of time - path values ​​one point on the graph. On fig. 18, such a construction is performed, replacing both rows of the table with one row of dots. If such a construction were performed for all moments of time, then instead of individual points, a solid line would be obtained (also shown in the figure). This line is called the path versus time graph, or, in short, the path graph.

Rice. 18. Graph of the path of uniform movement at a speed of 2 m / s

Rice. 19. To exercise 12.1

In our case, the path graph turned out to be a straight line. It can be shown that the graph of the path of uniform motion is always a straight line; and vice versa: if the path versus time graph is a straight line, then the motion is uniform.

Repeating the construction for a different speed of movement, we find that the points of the graph for a higher speed lie higher than the corresponding points of the graph for a lower speed (Fig. 20). Thus, the greater the speed of uniform movement, the steeper the straight-line graph of the path, i.e., the greater the angle it makes with the time axis.

Rice. 20. Graphs of the path of uniform movements with speeds of 2 and 3 m/s

Rice. 21. Graph of the same movement as in fig. 18, drawn to a different scale

The slope of the graph depends, of course, not only on the numerical value of the speed, but also on the choice of time and length scales. For example, the graph shown in Fig. 21 gives the path versus time for the same movement as the graph in Fig. 18, although it has a different slope. From this it is clear that it is possible to compare movements by the slope of the graphs only if they are drawn on the same scale.

With the help of path graphs, you can easily solve various problems about movement. For an example in fig. 18 dashed lines show the constructions necessary to solve the following problems for a given movement: a) find the path traveled in 3.5 s; b) find the time for which the path of 9 m has been covered. In the figure, the answers are found graphically (dashed lines): a) 7 m; b) 4.5 s.

On graphs that describe uniform rectilinear motion, you can plot the coordinate of the moving point along the y-axis instead of the path. Such a description opens up great possibilities. In particular, it makes it possible to distinguish the direction of motion with respect to the axis. In addition, taking the origin of time as zero, one can show the movement of a point at earlier times, which should be considered negative.

Rice. 22. Graphs of movements with the same speed, but with different initial positions of the moving point

Rice. 23. Graphs of several movements with negative velocities

For example, in fig. 22, straight line I is a graph of motion occurring at a positive speed of 4 m / s (i.e., in the direction of the axis), and at the initial moment the moving point was at a point with coordinate m. For comparison, the same figure shows a graph of motion that occurs with the same speed, but at which at the initial moment the moving point is at the point with the coordinate (line II). Straight. III corresponds to the case when at the moment the moving point was at the point with the coordinate m. Finally, the straight line IV describes the motion in the case when the moving point had the coordinate at the moment c.

We see that the slopes of all four graphs are the same: the slope depends only on the speed of the moving point, and not on its initial position. When changing the initial position, the entire graph is simply transferred parallel to itself along the axis up or down by the appropriate distance.

Graphs of movements occurring at negative velocities (i.e., in the direction opposite to the direction of the axis ) are shown in fig. 23. They are straight, inclined downwards. For such movements, the coordinate of a point decreases with time., had coordinates

Path graphs can also be built for cases in which the body moves uniformly for a certain period of time, then moves uniformly, but at a different speed for a different period of time, then changes speed again, etc. For example, in fig. 26 shows a motion graph in which the body moved during the first hour at a speed of 20 km/h, during the second hour at a speed of 40 km/h, and during the third hour at a speed of 15 km/h.

The task: 12.8. Construct a path graph for movement in which the body had speeds of 10, -5, 0, 2, -7 km/h for successive hourly intervals. What is the total displacement of the body?

For greater clarity, the movement can be described using graphs. The graph shows how one value changes when another value changes, on which the first depends.

To build a graph, both quantities on the selected scale are plotted along the coordinate axes. If on the horizontal axis (abscissa) we plot the time that has elapsed since the beginning of the time reference, and on the vertical axis (y-axis) - the values ​​of the body coordinates, the resulting graph will express the dependence of the body coordinate on time (it is also called a motion graph).

Let us assume that the body moves uniformly along the X axis (Fig. 29). At moments of time, etc., the body is respectively in the positions measured by the coordinates (point A), .

This means that only its coordinate changes. In order to obtain a graph of the body's motion, we will plot the values ​​along the vertical axis, and the time values ​​along the horizontal axis. The motion graph is a straight line shown in Figure 30. This means that the coordinate depends linearly on from time.

The graph of the dependence of the body's coordinate on time (Fig. 30) should not be confused with the trajectory of the body's movement - a straight line, at all points of which the body visited during its movement (see Fig. 29).

Motion graphs provide a complete solution to the problem of mechanics in the case of rectilinear motion of the body, since they allow you to find the position of the body at any time, including at times preceding the initial moment (assuming that the body was moving before the start of time). Continuing the graph shown in Figure 29, in the direction opposite to the positive direction of the time axis, we, for example, find that the body, 3 seconds before it was at point A, was at the origin of the coordinate

By the form of graphs of dependence of coordinates on time, one can also judge the speed of movement. It is clear that the greater the speed, the steeper the graph, i.e., the greater the angle between it and the time axis (the greater this angle, the greater the change in the coordinate for the same time).

Figure 31 shows several motion graphs with different speeds. Graphs 1, 2 and 3 show that the bodies are moving along the X axis in a positive direction. The body, the motion graph of which is straight line 4, moves in the direction opposite to the direction of the X axis. From the motion graphs, one can also find the displacement of a moving body for any period of time.

Figure 31 shows, for example, that body 3 moved in a positive direction in the absolute value of 2 m in the time between 1 and 5 seconds, and body 4 in the same time moved in a negative direction equal in absolute value to 4 m.

Along with motion graphs, speed graphs are often used. They are obtained by plotting along the coordinate axis the projection of the velocity

body, and the x-axis is still time. Such graphs show how the speed changes over time, i.e. how the speed depends on time. In the case of rectilinear uniform motion, this “dependence” consists in the fact that the speed does not change over time. Therefore, the speed graph is a straight line parallel to the time axis (Fig. 32). The graph in this figure refers to the case when the body is moving in the direction of the positive X-axis. Graph II refers to the case when the body is moving in the opposite direction (because the velocity projection is negative).

From the velocity graph, you can also find out the absolute value of the movement of the body for a given period of time. It is numerically equal to the area of ​​the shaded rectangle (Fig. 33): the upper one, if the body moves in the direction of the positive direction, and the lower one, in the opposite case. Indeed, the area of ​​a rectangle is equal to the product of its sides. But one of the sides is numerically equal to time and the other, to speed. And their product is just equal to the absolute value of the displacement of the body.

Exercise 6

1. What movement corresponds to the graph depicted by the dotted line in Figure 31?

2. Using the graphs (see Fig. 31), find the distance between bodies 2 and 4 at time sec.

3. From the graph shown in Figure 30, determine the magnitude and direction of the speed.

CHARTS

Determination of the type of movement according to the schedule

1. Uniformly accelerated motion corresponds to a graph of the dependence of the acceleration module on time, indicated in the figure by the letter

2. The figures show graphs of the dependence of the acceleration module on time for different types of movement. Which chart matches uniform motion?

1) 1 2) 2 3) 3 4) 4

3.
body moving along the axis Oh rectilinearly and uniformly accelerated, for some time reduced its speed by 2 times. Which of the graphs of the projection of acceleration versus time corresponds to such a movement?

1) 1 2) 2 3) 3 4) 4

4. The skydiver moves vertically down with a constant speed. Which graph - 1, 2, 3 or 4 - correctly reflects the dependence of its coordinates Y from the time of movement t relative to the surface of the earth? Ignore air resistance.

1) 1 2) 2 3) 3 4) 4

5. Which of the graphs of the dependence of the projection of velocity on time (Fig.) Corresponds to the movement of a body thrown vertically upwards with a certain speed (axis Y directed vertically upward)?

1) 1 2) 2 3) 3 4) 4

6.
A body is thrown vertically upwards with some initial velocity from the surface of the earth. Which of the graphs of the dependence of the height of the body above the earth's surface on time (Fig.) Corresponds to this movement?

1) 1 2) 2 3) 3 4) 4

Determination and comparison of characteristics of movement according to the schedule

7. The graph shows the dependence of the projection of the speed of the body on time for rectilinear motion. Determine the projection of the acceleration of the body.

1) - 10 m/s 2

2) - 8 m/s 2

3) 8 m/s 2

8.
The figure shows a graph of the dependence of the speed of movement of bodies on time. What is the acceleration of the body?

2) 2 m/s 2

9. According to the graph of the dependence of the projection of speed on time, presented in the figure, determine the acceleration of a rectilinearly moving body at the time t= 2 s.

3) 10 m/s 2

10. The figure shows the schedule of the bus from point A to point B and back. Point A is at the point x = 0, and point B at the point x = 30 km. What is the speed of the bus on the way from A to B?

11. The figure shows a bus schedule from point A to point B and back. Point A is at the point x = 0, and point B at the point x = 30 km. What is the speed of the bus on the way from B to A?

12. The car is moving along a straight street. The graph shows the dependence of the speed of the car on time. The acceleration modulus is maximum in the time interval

1) 0 s to 10 s

2) from 10 s to 20 s

3) 20s to 30s

4) 30s to 40s

13. Four bodies move along an axis Ox.The figure shows the graphs of the projections of velocities υ x from time t for these bodies. Which of the bodies is moving with the least modulo acceleration?

1) 1 2) 2 3) 3 4) 4

14. The figure shows a path dependence graph S cyclist from time to time t. Determine the time interval when the cyclist was moving at a speed of 2.5 m/s.

1) 5 s to 7 s

3 s to 5 s

3) 1s to 3s

4) 0 to 1 s

15. The figure shows a graph of the dependence of the coordinates of a body moving along the axis Oh, from time. Compare speeds v 1 , v 2 and v 3 bodies at times t1, t2, t3

1) v1 > v2 = v3

2) v1 > v2 > v3

3) v1< v 2 < v 3

4) v1 = v2 > v3

16. The figure shows a graph of the dependence of the projection of the body's velocity on time.

The projection of the acceleration of the body in the time interval from 5 to 10 s is represented by a graph

1) 1 2) 2 3) 3 4) 4

17. A material point moves in a straight line with acceleration, the time dependence of which is shown in the figure. The initial speed of the point is 0. Which point on the graph corresponds to the maximum speed material point:

Compilation of kinematic dependencies (functions of the dependence of kinematic quantities on time) according to the schedule

18. In fig. shows a graph of body coordinates versus time. Determine the kinematic law of motion of this body

1) x(t)= 2 + 2t

2) x(t)= – 2 – 2t

3) x(t)= 2 – 2t

4) x(t) = – 2 + 2t

19. According to the graph of the dependence of the speed of the body on time, determine the function of the dependence of the speed of this body on time

1) v x= – 30 + 10t

2) v x = 30 + 10t

3) v x = 30 – 10t

4) v x = – 30 + 10t

Determination of displacement and path according to the schedule

20. According to the graph of the dependence of the speed of a body on time, determine the path traveled by a rectilinearly moving body in 3 s.

21. A stone is thrown vertically upwards. The projection of its velocity on the vertical direction changes with time according to the graph in the figure. What is the distance traveled by the stone in the first 3 seconds?

22. A stone is thrown vertically upwards. The projection of its velocity on the vertical direction changes with time according to the graph in figure h.17. What is the distance traveled by the stone during the entire flight?

23. A stone is thrown vertically upwards. The projection of its velocity on the vertical direction changes with time according to the graph in figure h.17. What is the displacement of the stone in the first 3 s?

24. A stone is thrown vertically upwards. The projection of its velocity on the vertical direction changes with time according to the graph in figure h.17. What is the displacement of the stone during the entire flight?

25. The figure shows a graph of the dependence of the projection of the velocity of a body moving along the Ox axis on time. What is the path traveled by the body by the time t = 10 s?

26. The trolley starts moving from rest along the paper tape. There is a dropper on the cart, which at regular intervals leaves spots of paint on the tape.

Choose a graph of speed versus time that correctly describes the movement of the cart.

1) 1 2) 2 3) 3 4) 4

EQUATIONS

27. The movement of a trolleybus during emergency braking is given by the equation: x \u003d 30 + 15t - 2.5 t 2, m What is the initial coordinate of the trolleybus?

28. The movement of the aircraft during the run is given by the equation: x = 100 + 0.85t2, m What is the acceleration of the aircraft?

3) 1.7 m/s 2

29. The movement of a car is given by the equation: x = 150 + 30t + 0.7t2, m. What is the initial speed of the car?

30. The equation of dependence of the projection of the speed of a moving body on time: v x = 2 +3t(m/s). What is the corresponding equation for the projection of the displacement of the body?

1) S x= 2t+ 3t2 2)S x = 4t+ 3t2 3)S x = t+ 6t2 4)S x = 2t + 1,5t2

31. The dependence of the coordinate on time for some body is described by the equation x \u003d 8t - t 2. At what point in time is the body's velocity zero?

TABLES

32. The table shows the results of measurements of the free fall path of a steel ball at different times. What, most likely, was the value of the path traveled by the ball during the fall, by the time t = 2 s?

1) 7.5 m 2) 10 m 3) 20 m 4) 40 m

34. The table shows the dependence of the coordinates X body movements over time t:

With what speed did the body move from time 0 s to time 3 s?

4) 3 m/s

36. The table shows the dependence of the coordinates X body movements over time t:

With what speed did the body move from time 3 s to time 5 s?

38. The table shows the dependence of the speed of the body v from time t:

3) 17 m

40. The table shows the dependence of the speed of the body v from time t:

Determine the path traveled by the body in the interval from time 0 s to time 2 s.

42. The table shows the dependence of the speed of the body v from time t:

Determine the path traveled by the body in the interval from time 0 s to time 5 s.

4) 25 m

43. Four bodies moved along the Ox axis. The table shows the dependence of their coordinates on time.

Which of the bodies could have a constant velocity and be different from zero?

1) 1 2) 2 3) 3 4) 4

44. Four bodies moved along the Ox axis. The table shows the dependence of their coordinates on time.

Which of the bodies could have constant acceleration and be different from zero?