1. 
   What is called uniform motion?
2. 
   What is speed?
3. 
   What is the formula for speed in uniform rectilinear motion?
4. 
   In what units is speed measured?
5. 
   What is the equation of uniform rectilinear motion?
6. 
   How to find the time of motion of the body from the equation of motion?
7. 
   The equation of motion is s = 2.5t (m). What information can we extract from this record?

R
eat tasks:

W
hell 1.

According to the path graph (Fig. 3), determine the speed of the body.

Task 2.

According to the speed graph (Fig. 4), determine the path traveled by the body in 6 s. How to represent the numerical value of the path using the speed graph?

W
hell 3.

Body speed 20 m/s. Plot the velocity versus time graph. Choose your own scale units.

W
5 cm
hell 4.

D
moving straight and evenly, the car traveled 240 km in 3 hours. Plot a path versus time graph. Calculate the speed of the car and plot the speed versus time graph.

Task 5.

What is the speed of the boat whose motion graph is shown in Fig. 5?

Task 6.

According to the graphs of movements of bodies (Fig. 6), it can be argued that:

a) bodies move along hills of different slopes;

b) that the speeds of both bodies are the same;

in
) the speed of the first body is 2.5 times less than the speed of the second body;

d) the speed of the first body is 2.5 times the speed of the second body.

Build graphs of the speed of bodies.

Task 7.

Moving at constant speeds, a pedestrian traveled 5.4 km in an hour, and a cyclist traveled 200 m in 20 s. Build in one coordinate system: a) graphs of the speed of these bodies; b) graphs of the path for 20 seconds of movement.

H
Figure 7 shows the graphs of the path of three bodies versus time. How did these bodies move? Determine the speed of movement of each body, build graphs of the speed of υ 1, υ 2, υ 3 bodies versus time.

Task 9.

Figure 8 shows a path vs. time graph. How did the body move during the time of movement? Determine the path s covered by the body and the speed υ 1 , υ 2 , υ 3 bodies in all areas of movement.

Task 10.

Using the graph of the speed of the body (Fig. 9), it can be argued that the path s 1, traveled in the first three seconds, and the path s 2, traveled in the last three seconds, are related by the relation:

a) s 2 \u003d 0.5s 1;

b) s 2 \u003d 1.5s 1;

d) s 2 \u003d 3s 1.

Task 11.

A simplified graph of the speed υ of the car is shown in Figure 10. Describe the movement of the car. What are the distances traveled by the car in each section of the movement? What plot details are omitted?

Task 12.

Using the graph of the speed of the body (Fig. 11), it can be proved that half of the entire path will be covered by the body:

a) by the end of the 10th second;

b) by the end of the 13th second;

c) by the end of the 18th second;

d) by the end of the 20th second.

1. 
   12 m;
2. 
   9 m;
3. 
   6 m;
4. 
   3m.

1. 
   According to the graph (Fig. 12), determine the speed of movement at a time point of 3 s from the beginning of the movement.

1. 
   27 m/s;
2. 
   12 m/s;
3. 
   3 m/s;
4. 
   0 m/s.

Science never solves a problem without raising a dozen new ones.

George Bernard Shaw

SHEET CONTROL

Learning element	Answer	Points	Result
Problem solving	1. v= 10 m/s 2. s= 30 m , 5. v ≈ 333.3 m/min ≈5.5m/s 9. s = 54 m v 1 = 3 m/s, v 2 = 0 m/s, v 3 = 6 m/s, v 4 = 0 m/s 11. s 1 = 50 m - uniform, s 2 = 7 5 m - uniform, s 3 = 0 m - did not move, s 4 = 38 m - not uniform.	2 4 5 5 8	Give yourself a final score: 37-71 points - "excellent" 17-36 points - "good"; 6-16 points - "test"; ≤5 points - "failure". Give the checklist to the teacher. GRADE ↓
Day off the control	1. g 3. in 5. 3 times	2
Total:		71

Horace

A person who hits the target is a talent; a person who hits an invisible target is a genius.

Arthur Schopenhauer

DIFFERENTIATED HOMEWORK

Mechanical movement is represented graphically. The dependence of physical quantities is expressed using functions. designate

Graphs of uniform motion

Time dependence of acceleration. Since the acceleration is equal to zero during uniform motion, the dependence a(t) is a straight line that lies on the time axis.

Dependence of speed on time. The speed does not change with time, the graph v(t) is a straight line parallel to the time axis.

The numerical value of the displacement (path) is the area of ​​the rectangle under the speed graph.

Path versus time. Graph s(t) - sloping line.

The rule for determining the speed according to the schedule s(t): The tangent of the slope of the graph to the time axis is equal to the speed of movement.

Graphs of uniformly accelerated motion

Dependence of acceleration on time. Acceleration does not change with time, has a constant value, graph a(t) is a straight line parallel to the time axis.

Speed ​​versus time. With uniform motion, the path changes, according to a linear relationship. in coordinates. The graph is a sloping line.

The rule for determining the path according to the schedule v(t): The path of the body is the area of ​​the triangle (or trapezoid) under the velocity graph.

The rule for determining the acceleration according to the schedule v(t): The acceleration of the body is the tangent of the slope of the graph to the time axis. If the body slows down, the acceleration is negative, the angle of the graph is obtuse, so we find the tangent of the adjacent angle.

Path versus time. With uniformly accelerated movement, the path changes, according to

If the trajectory of the point is known, then the dependence of the path traveled by the point on the elapsed time interval gives a complete description of 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. We choose 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 was 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.

Exercise: 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?

« 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 angle of inclination, which determines the value of acceleration.

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 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 the 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, you can determine the value of the speed at any time by calculating the tangent of the slope of the tangent to the curve at the point corresponding to present moment 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.