Moving a body without initial speed. Presentation on the topic "Movement of a body during rectilinear uniformly accelerated motion without initial velocity (along the x-axis)"

Let's show how you can find the path traveled by a body using a graph of speed versus time.

Let's start with the simplest case - uniform motion. Figure 6.1 shows a graph of v(t) – speed versus time. It represents a segment of a straight line parallel to the base of time, since with uniform motion the speed is constant.

The figure enclosed under this graph is a rectangle (it is shaded in the figure). Its area is numerically equal to the product of speed v and time of movement t. On the other hand, the product vt is equal to the path l traversed by the body. So, with uniform motion

way numerically equal to area the figure enclosed under the graph of speed versus time.

Let us now show that uneven motion also has this remarkable property.

Let, for example, the graph of speed versus time look like the curve shown in Figure 6.2.

Let us mentally divide the entire time of movement into such small intervals that during each of them the movement of the body can be considered almost uniform (this division is shown by dashed lines in Figure 6.2).

Then the path traveled during each such interval is numerically equal to the area of ​​the figure under the corresponding lump of the graph. Therefore, the entire path is equal to the area of ​​the figures contained under the entire graph. (The technique we used is the basis of integral calculus, the basics of which you will study in the course “Beginnings of Mathematical Analysis.”)

2. Path and displacement during rectilinear uniformly accelerated motion

Let us now apply the method described above for finding the path to rectilinear uniformly accelerated motion.

The initial speed of the body is zero

Let's direct the x axis in the direction of body acceleration. Then a x = a, v x = v. Hence,

Figure 6.3 shows a graph of v(t).

1. Using Figure 6.3, prove that for a straight line uniformly accelerated motion without initial speed the path l is expressed in terms of the acceleration module a and the time of movement t by the formula

l = at 2 /2. (2)

Main conclusion:

In case of rectilinear uniformly accelerated motion without initial speed, the distance traveled by the body is proportional to the square of the time of movement.

In this way, uniformly accelerated motion differs significantly from uniform motion.

Figure 6.4 shows graphs of the path versus time for two bodies, one of which moves uniformly, and the other uniformly accelerates without an initial speed.

2. Look at Figure 6.4 and answer the questions.
a) What color is the graph for a body moving with uniform acceleration?
b) What is the acceleration of this body?
c) What are the speeds of the bodies at the moment when they have covered the same path?
d) At what point in time are the velocities of the bodies equal?

3. Having started, the car covered a distance of 20 m in the first 4 s. Consider the car’s motion to be linear and uniformly accelerated. Without calculating the acceleration of the car, determine how far the car will travel:
a) in 8 s? b) in 16 s? c) in 2 s?

Let us now find the dependence of the projection of displacement s x on time. IN in this case the projection of acceleration onto the x axis is positive, so s x = l, a x = a. Thus, from formula (2) it follows:

s x = a x t 2 /2. (3)

Formulas (2) and (3) are very similar, which sometimes leads to errors in solving simple tasks. The fact is that the displacement projection value can be negative. This will happen if the x axis is directed opposite to the displacement: then s x< 0. А путь отрицательным быть не может!

4. Figure 6.5 shows graphs of travel time and displacement projection for a certain body. What color is the displacement projection graph?

The initial speed of the body is not zero

Let us recall that in this case the dependence of the velocity projection on time is expressed by the formula

v x = v 0x + a x t, (4)

where v 0x is the projection of the initial velocity onto the x axis.

We will further consider the case when v 0x > 0, a x > 0. In this case, we can again take advantage of the fact that the path is numerically equal to the area of ​​the figure under the graph of speed versus time. (Consider other combinations of signs for the projection of initial velocity and acceleration yourself: the result will be the same general formula (5).

Figure 6.6 shows a graph of v x (t) for v 0x > 0, a x > 0.

5. Using Figure 6.6, prove that in case of rectilinear uniformly accelerated motion with an initial speed, the projection of displacement

s x = v 0x + a x t 2 /2. (5)

This formula allows you to find the dependence of the x coordinate of the body on time. Let us recall (see formula (6), § 2) that the coordinate x of a body is related to the projection of its displacement s x by the relation

s x = x – x 0 ,

where x 0 is the initial coordinate of the body. Hence,

x = x 0 + s x , (6)

From formulas (5), (6) we obtain:

x = x 0 + v 0x t + a x t 2 /2. (7)

6. The dependence of the coordinate on time for a certain body moving along the x axis is expressed in SI units by the formula x = 6 – 5t + t 2.
a) What is the initial coordinate of the body?
b) What is the projection of the initial velocity onto the x-axis?
c) What is the projection of acceleration on the x-axis?
d) Draw a graph of the x coordinate versus time.
e) Draw a graph of the projected velocity versus time.
f) At what moment is the speed of the body equal to zero?
g) Will the body return to the starting point? If so, at what point(s) in time?
h) Will the body pass through the origin? If so, at what point(s) in time?
i) Draw a graph of the displacement projection versus time.
j) Draw a graph of the distance versus time.

3. Relationship between path and speed

When solving problems, the relationships between path, acceleration and speed (initial v 0, final v or both) are often used. Let us derive these relations. Let's start with movement without an initial speed. From formula (1) we obtain for the time of movement:

Let's substitute this expression into formula (2) for the path:

l = at 2 /2 = a/2(v/a) 2 = v 2 /2a. (9)

Main conclusion:

in rectilinear uniformly accelerated motion without initial speed, the distance traveled by the body is proportional to the square of the final speed.

7. Having started, the car picked up a speed of 10 m/s over a distance of 40 m. Consider the car’s motion to be linear and uniformly accelerated. Without calculating the acceleration of the car, determine how far from the beginning of the movement the car traveled when its speed was equal to: a) 20 m/s? b) 40 m/s? c) 5 m/s?

Relationship (9) can also be obtained by remembering that the path is numerically equal to the area of ​​the figure enclosed under the graph of speed versus time (Fig. 6.7).

This consideration will help you easily cope with the next task.

8. Using Figure 6.8, prove that when braking with constant acceleration the body travels the distance l t = v 0 2 /2a until it comes to a complete stop, where v 0 is the initial speed of the body, a is the acceleration modulus.

In case of braking vehicle(car, train) the distance traveled to a complete stop is called the braking distance. Please note: the braking distance at the initial speed v 0 and the distance traveled during acceleration from standstill to speed v 0 with the same acceleration a are the same.

9. When emergency braking on dry asphalt, the acceleration of the car is equal in absolute value to 5 m/s 2 . What is the braking distance of a car at initial speed: a) 60 km/h (maximum permitted speed in the city); b) 120 km/h? Find the braking distance at the indicated speeds during icy conditions, when the acceleration modulus is 2 m/s 2 . Compare the braking distances you found with the length of the classroom.

10. Using Figure 6.9 and the formula expressing the area of ​​a trapezoid through its height and half the sum of the bases, prove that for rectilinear uniformly accelerated motion:
a) l = (v 2 – v 0 2)/2a, if the speed of the body increases;
b) l = (v 0 2 – v 2)/2a, if the speed of the body decreases.

11. Prove that the projections of displacement, initial and final velocity, as well as acceleration are related by the relation

s x = (v x 2 – v 0x 2)/2ax (10)

12. A car on a path of 200 m accelerated from a speed of 10 m/s to 30 m/s.
a) How fast was the car moving?
b) How long did it take the car to travel the indicated distance?
c) What is it equal to average speed car?

Additional questions and tasks

13. The last car is uncoupled from a moving train, after which the train moves uniformly, and the car moves with constant acceleration until it comes to a complete stop.
a) Draw on one drawing graphs of speed versus time for a train and a carriage.
b) How many times is the distance covered by the carriage to the stop less than the distance covered by the train in the same time?

14. Having left the station, the train traveled at a uniform acceleration for some time, then for 1 minute at a uniform speed of 60 km/h, and then again at a uniform acceleration until it stopped at the next station. The acceleration modules during acceleration and braking were different. The train covered the distance between stations in 2 minutes.
a) Draw a schematic graph of the projection of the speed of the train as a function of time.
b) Using this graph, find the distance between the stations.
c) What distance would the train travel if it accelerated on the first section of the route and slowed down on the second? What would be its maximum speed?

15. A body moves uniformly accelerated along the x axis. At the initial moment it was at the origin of coordinates, and the projection of its speed was equal to 8 m/s. After 2 s, the coordinate of the body became 12 m.
a) What is the projection of the acceleration of the body?
b) Plot a graph of v x (t).
c) Write a formula expressing the dependence x(t) in SI units.
d) Will the speed of the body be zero? If yes, at what point in time?
e) Will the body visit the point with coordinate 12 m a second time? If yes, at what point in time?
f) Will the body return to the starting point? If so, at what point in time, and what will be the distance traveled?

16. After the push, the ball rolls up an inclined plane, after which it returns to the starting point. The ball was at a distance b from the initial point twice at time intervals t 1 and t 2 after the push. The ball moved up and down along the inclined plane with the same acceleration.
a) Direct the x-axis upward along the inclined plane, select the origin at the initial position of the ball and write a formula expressing the dependence x(t), which includes the modulus of the initial velocity of the ball v0 and the modulus of the acceleration of the ball a.
b) Using this formula and the fact that the ball was at a distance b from the starting point at times t 1 and t 2, create a system of two equations with two unknowns v 0 and a.
c) Having solved this system of equations, express v 0 and a in terms of b, t 1 and t 2.
d) Express the entire path l traveled by the ball in terms of b, t 1 and t 2.
e) Find the numerical values ​​of v 0, a and l for b = 30 cm, t 1 = 1 s, t 2 = 2 s.
f) Plot graphs of v x (t), s x (t), l(t).
g) Using the graph of sx(t), determine the moment when the ball’s modulus of displacement was maximum.

Let us consider some features of the movement of a body during rectilinear uniformly accelerated motion without an initial speed. The equation that describes this movement was derived by Galileo in the 16th century. It must be remembered that with a rectilinear uniform or uneven movement without changing the direction of speed, the displacement module coincides in 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 without initial velocity is an important special case of uniformly accelerated motion. Let's look at examples:

1. Free fall without initial speed. An example of such a movement would be the fall of an icicle at the end of winter (Fig. 1).

Rice. 1. Falling icicle

At the moment when the icicle comes off the roof, its initial speed is zero, after which it moves uniformly accelerated, because free fall- This is uniformly accelerated motion.

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

Rice. 2. Start of movement

When we say that the time it takes a car of one brand or another to reach 100 km/h is, for example, 6 seconds, most often we are talking about uniformly accelerated motion without an initial speed. Likewise when we talk about a rocket launch, etc.

3. Uniformly accelerated motion is of particular relevance for weapon developers. After all departure of any projectile or bullet- this is movement without an initial speed, and while moving in the barrel, the bullet (projectile) moves uniformly accelerated. Let's look at an example.

The length of the Kalashnikov assault rifle is . A bullet in a machine gun barrel moves with acceleration. At what speed will the bullet leave the barrel?

Rice. 3. Illustration for the problem

To find the speed of a bullet leaving the barrel of a machine gun, we use the expression for displacement during rectilinear uniformly accelerated motion if the time is unknown:

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

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

We write the solution to the problem as follows, taking into account SI units:

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

If , then the equation above becomes the equation:

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

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

If S (the projection of displacement) is denoted as the difference between the initial and final coordinates (), then we obtain an equation of motion that makes it possible to determine the coordinate of the body for any moment in time:

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

Speed ​​versus time graph

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

In Fig. Figure 4 shows a graph of the velocity projection versus time for uniformly accelerated motion without an initial speed (the graph starts at the origin).

The graph is pointing upward. This suggests that the acceleration projection is positive

Rice. 4. Graph of the velocity projection versus time for uniformly accelerated motion without initial speed

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

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

In Fig. Figure 5 shows a graph of the projection of displacement versus time of two bodies for uniformly accelerated motion without an initial speed.

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

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

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

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

If we continue the calculations, the pattern will remain. The distances traveled increase 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 choose a certain interval for a unit of time, then the total distances traveled by the body over subsequent equal periods of time will be related as the 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 for each second are treated as odd numbers

The two very important conclusions studied are characteristic only of rectilinear uniformly accelerated motion without an initial speed.

Task. The car begins to move 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

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

• Sx=V0x*t+(ax*t^2)/2.

Let's consider the case when the movement begins with zero initial speed. In this case, the equation written above will take the following form:

• Sx= ax*t^2)/2.

For the magnitudes of vectors a and S, we can write the following equation:

• S=(a*t^2)/2.

Dependence of displacement and time

We see that in case of rectilinear uniformly accelerated motion without initial speed, the magnitude of the displacement vector will be directly proportional to the square of the period of time during which this displacement occurred. 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 during a certain period of time t1 from the beginning of the movement the body made a movement s1=(a/2)*(t1)^2,

Then, during 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 number n. This will of course be carried out, provided that 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 period of time, 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 figure presented above one can establish another, the following pattern:

• OA:AB:BC:CD:DE = 1:3:5:7:9.

Over successive equal periods of time, the modules of the vectors of displacements made by the body will relate 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 like some kind of peculiar sign of uniformly accelerated motion. 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.

Questions.

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

2. How many times will the module of the body’s displacement vector increase when the time of its movement from rest increases 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 when the time of its movement increases by an integer number of times compared to t 1.

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

5. For what purpose can laws (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. A train leaving the station moves rectilinearly and uniformly accelerated during the first 20 s. It is known that in the third second from the start of movement the train traveled 2 m. Determine the magnitude of the displacement vector made by the train in the first second and the magnitude of the acceleration vector with which it moved.

2. A car, moving uniformly accelerated from a state of rest, travels 6.3 m during the fifth second of acceleration. What speed did the car develop by the end of the fifth second from the start of movement?