The displacement vector projection for rectilinear uniformly accelerated motion is calculated by the following formula:

• Sx=V0x*t+(ax*t^2)/2.

Let us consider the case when the movement starts with zero initial speed. In this case, the above equation will take the following form:

• Sx=ax*t^2)/2.

For the modules of the vectors a and S, the following equation can be written:

• S=(a*t^2)/2.

Dependence of displacement and time

We see that with a rectilinear uniformly accelerated movement without an initial speed, the module of the displacement vector will be directly proportional to the square of the time interval during which this movement took place. That is, in other words, if we increase the time of movement by n times, then the movement will increase by n ^ 2 times.

For example, if for a certain period of time t1 from the beginning of the movement the body moved s1=(a/2)*(t1)^2,

Then for the time interval t2=2*t1, this body will move S2=(a/2)*4*(t1)^2=4*S1.

During the interval t3=3*t1, this body will move S3=9*S1, etc., for any natural n. This will, of course, be true, provided that the time must be counted from the same moment.

The following figure shows this relationship well.

• OA:OB:OC:OD:OE = 1:4:9:16:25.

With an increase in the time interval, which is counted from the beginning of the movement, by an integer number of times compared to t1, the modules of the displacement vectors will increase as a series of squares of consecutive natural numbers.

In addition to this pattern, from the above figure, one more, the following pattern can be established:

• OA:AB:BC:CD:DE = 1:3:5:7:9.

For successive equal periods of time, the modules of the vectors of displacements performed by the body will be related to each other as a series of consecutive odd numbers.

It is worth noting that such patterns will be true only in uniformly accelerated motion. That is, they are, as it were, a kind of peculiar sign of uniformly accelerated movement. If it is necessary to check whether the movement is uniformly accelerated, then these patterns can be checked, and if they are fulfilled, then the movement will be uniformly accelerated.

Consider some features of the movement of the body during rectilinear uniformly accelerated motion without initial speed. The equation that describes this motion was derived by Galileo in the 16th century. It must be remembered that in the case of rectilinear uniform or non-uniform movement without changing the direction of speed, the displacement modulus coincides in its value with the distance traveled. The formula looks like this:

where is the acceleration.

Examples of uniformly accelerated motion without initial speed

Uniformly accelerated motion with no initial velocity is an important special case of uniformly accelerated motion. Consider examples:

1. Free fall with no initial velocity. An example of such a movement can be the fall of an icicle at the end of winter (Fig. 1).

Rice. 1. Falling icicle

At the moment when the icicle breaks away from the roof, its initial speed is zero, after which it moves with uniform acceleration, because free fall is a uniformly accelerated movement.

2. Start of any movement. For example, a car starts off and accelerates (Figure 2).

Rice. 2. Start driving

When we say that the acceleration time of 100 km / h for a car of one brand or another, for example, is 6 s, most often we are talking about uniformly accelerated movement without an initial speed. Similarly, when we talk about the launch of a rocket, etc.

3. Uniformly accelerated motion is of particular relevance for weapons developers. After all departure of any projectile or bullet- this is movement without an initial velocity, and while moving in the barrel, the bullet (projectile) moves uniformly accelerated. Consider an example.

The length of the Kalashnikov assault rifle is . The bullet in the barrel of the machine gun moves with acceleration . How fast will the bullet exit the barrel?

Rice. 3. Illustration for the problem

To find the speed of a bullet leaving the barrel of an automaton, we use the expression for moving in a rectilinear uniformly accelerated movement, if the time is unknown:

The movement is carried out without an initial speed, which means that , then .

We obtain the following expression for finding the speed of a bullet leaving the barrel:

We write the solution of the problem as follows, taking into account units of measurement in SI:

Given:		Decision:
		Answer:.

Uniformly accelerated motion without an initial velocity is often found both in nature and in technology. Moreover, the ability to work with such a movement allows you to solve inverse problems when the initial speed exists, and the final one is zero.

If , then the equation above becomes the equation:

This equation makes it possible to find the distance traveled uniform movement. in this case is a projection of the displacement vector. It can be defined as the difference in coordinates: . If we substitute this expression into the formula, we get the dependence of the coordinate on time:

Let's consider a situation when - the initial speed is equal to zero. This means that movement begins from a state of rest. The body is at rest, then begins to acquire and increase speed. Movement from rest will be recorded without initial velocity:

If S (displacement projection) is denoted as the difference between the initial and final coordinates (), then the equation of motion will be obtained, which makes it possible to determine the coordinate of the body for any moment of time:

The acceleration projection can be both negative and positive, so we can talk about the coordinate of the body, which can both increase and decrease.

Graph of speed versus time

Since uniformly accelerated motion without an initial velocity is a special case of uniformly accelerated motion, consider a graph of the velocity projection versus time for such motion.

On fig. Figure 4 shows a plot of velocity projection versus time for uniformly accelerated motion without initial velocity (the graph starts at the origin).

The chart is pointing up. This means that the acceleration projection is positive.

Rice. 4. Graph of the dependence of the projection of speed on time for uniformly accelerated motion without initial speed

Using the graph, you can determine the projection of the movement of the body or the distance traveled. To do this, it is necessary to calculate the area of ​​\u200b\u200bthe figure bounded by the graph, the coordinate axes and the perpendicular lowered onto the time axis. That is, it is necessary to find the area of ​​\u200b\u200ba right triangle (half the product of the legs)

where is the final speed with uniformly accelerated motion without initial speed:

On fig. Figure 5 shows a plot of the displacement projection versus time for two bodies for uniformly accelerated motion without initial velocity.

Rice. 5 Graph of the dependence of the projection of displacement on time of two bodies for uniformly accelerated motion without initial speed

The initial velocity of both bodies is zero, since the vertex of the parabola coincides with the origin:

For the first body the projection of acceleration is positive, for the second it is negative. Moreover, the projection of the acceleration of the body is larger for the first body, since its movement is faster.

- the distance traveled (up to a sign), it is proportional to, i.e., the square of time. If we consider equal time intervals - , , , then we can notice the following relationships:

If you continue the calculations, the pattern will be preserved. The distance traveled increases in proportion to the square of the increase in time intervals.

For example, if , then the distance traveled will be proportional to . If , the distance traveled will be proportional, etc. The distance will increase in proportion to the square of these time intervals (Fig. 6).

Rice. 6. Proportionality of the path to the square of time

If we select a certain interval as a unit of time, then the total distances traveled by the body over subsequent equal periods of time will be treated as squares of integers.

In other words, the movements made by the body for each subsequent second will be treated as odd numbers:

Rice. 7. Movements per second are treated as odd numbers

The studied two very important conclusions are peculiar only to rectilinear uniformly accelerated motion without initial speed.

Task. The car starts moving from a stop, i.e., from a state of rest, and in the fourth second of its movement it travels 7 m. Determine the acceleration of the body and the instantaneous speed 6 s after the start of movement (Fig. 8).

Rice. 8. Illustration for the problem

Given:

Topic: “Body displacement during rectilinear uniformly accelerated motion. No initial speed.

Lesson Objectives:

Tutorial:

• to form the concept of displacement in a rectilinear uniformly accelerated motion, taking into account the existence of cause-and-effect relationships;
• consider a graphical representation of uniformly accelerated motion and work out the solution of problems for finding the parameters of uniformly accelerated motion using formulas;
• to form practical skills to apply knowledge in specific situations.

Developing:

• develop the ability to read and build graphs of the dependence of displacement, speed and acceleration on time with uniformly accelerated movement;
• to develop students' speech through the organization of dialogical communication in the classroom;
• to develop and maintain the attention of students through a change in learning activities.

Educational:

• to cultivate cognitive interest, curiosity, activity, accuracy in performing tasks, interest in the subject being studied.

Lesson equipment:

computer, multimedia projector, screen, presentation "Movement with uniformly accelerated rectilinear motion" (own development), printed table for reflection.

Demo equipment:

easily mobile trolleys, stopwatch, weights on the block.

Lesson plan:

1. front poll. Solving graphic problems.

1. Main part. Learning new material (20 min).Presentation of new material using a presentation with additional teacher comments, elements of a conversation, demonstration of experiments.

1. Fixing (10 min).

front poll. Problem solving.

Grading. Homework.

During the classes

1. Updating of basic knowledge (10 min).

Organizing time. Announcement of the topic and objectives of the lesson.

slide 1.2.

Front poll:

1. What types of movement do you know?
2. Define each of them.
3. What quantities characterize these types of movement?
4. What is called acceleration of uniformly accelerated motion?
5. What is uniformly accelerated motion?
6. What does the acceleration module show?
7. The train leaves the station. What is the direction of its acceleration?
8. The train starts to slow down. What is the direction of its speed and acceleration?

Demonstrations (teacher shows experiments):

1. The movement of the cart on an inclined plane with an initial zero speed.

2. The movement of two loads suspended on a thread thrown over a block.

(Students give a description of the movement of bodies in the experiments they see).

Slide 3.

Decide verbally. No. 1.

Describe the motions of material points, dependence graphs v x(t),

which 1 and 2 are shown in Figure 1. How to determine from these graphs the projection of the point displacement on the x-axis, its module and the distance traveled?

slide 4.

Decide verbally. No. 2.

Figure 2 schematically shows the graphs of the dependence of the speed of bodies on time.

What do these movements have in common, how do they differ?

Slide 5.

Decide verbally. No. 3.

Which of the sections of the graph of the dependence of speed on time (Fig. 3) corresponds to uniform motion, uniformly accelerated with increasing speed, uniformly accelerated with decreasing speed?

slide 6.

Decide verbally. No. 3.

Figure 4 schematically shows the graphs of the dependence of the speed of bodies on time. What do all movements have in common, how do they differ?

1. Main part. Learning new material (15 min).

Slide 7.

The teacher analyzes the graphs of the dependence of physical quantities during uniformly accelerated motion in the form of a dialogue with students (slides 7-11).

Graph of the projection of the velocity vector of a body moving with constant acceleration (Fig. 5).

The area under the velocity graph is numerically equal to the displacement. Therefore, the area of ​​the trapezoid is numerically equal to the displacement.

slide 8.

The equation for determining the projection of the displacement vector of the body during its rectilinear uniformly accelerated motion:

slide 9.

Movement of a body during rectilinear uniformly accelerated motion without initial speed:

slide 10.

Graph of the dependence of the projection of the displacement vector of the body on time (Fig. 6), if the body moves with constant acceleration.

Slide 11.

Graph of the dependence of the coordinate of the body on the time of the body moving with constant acceleration (Fig. 7).

1. Fixing (15 min).

slide 12.

Think and answer! #5.

What is the displacement of the body if the graph of the change in its speed over time is shown schematically in Figure 8?

slide 13.

Think and answer! #6.

Figure 9 schematically shows the plots of bodies versus time. What do all movements have in common, how do they differ?

slide 14.

Task #8 (solution by the student at the blackboard).

The kinematic law of train movement along the Ox axis has the form: x= 0.2t 2 .

Is the train accelerating or slowing down? Determine the projection of the initial velocity and acceleration.

Write down the equation for the projection of velocity on the Ox axis. Plot graphs of acceleration and velocity projections.

Task #9 (solution by the student at the blackboard).

The position of a soccer ball rolling along the x-axis along the field is given by the equation
x=10 + 5t - 0.2t 2 . Determine the projection of the initial velocity and acceleration. What is the coordinate of the ball and the projection of its speed at the end of the 5th second?

slide 15.

Think and find a match (Fig. 10). #7.

IV. Reflection. Summing up the lesson (5 min).

Slide 16, 17.

Filling in the conceptual table.

(A reflection table for each student on the table)

(Exchange of opinions, quotes from tables with reflection).

Summing up, grading.

D/Z: p. 7.8; .Check yourself.

Questions.

1. What formulas are used to calculate the projection and modulus of the displacement vector of a body during its uniformly accelerated movement from a state of rest?

2. How many times will the modulus of the displacement vector of the body increase with an increase in the time of its movement from rest by n times?

3. Write down how the modules of the displacement vectors of a body moving uniformly accelerated from a state of rest relate to each other with an increase in the time of its movement by an integer number of times compared to t 1.

4. Write down how the moduli of the vectors of displacements performed by the body in successive equal time intervals relate to each other if this body moves uniformly accelerated from a state of rest.

5. For what purpose can regularities (3) and (4) be used?

Regularities (3) and (4) are used to determine whether the movement is uniformly accelerated or not (see p.33).

Exercises.

1. The train departing from the station during the first 20 s moves in a straight line and uniformly accelerated. It is known that in the third second from the start of the movement the train traveled 2 m. Determine the module of the displacement vector made by the train in the first second and the module of the acceleration vector with which it moved.

2. A car, moving uniformly accelerated from rest, travels 6.3 m in the fifth second of acceleration. What speed has the car developed by the end of the fifth second from the start of movement?