This video lesson is devoted to the topic “Speed ​​of rectilinear uniformly accelerated motion. Speed ​​graph." During the lesson, students will have to remember such a physical quantity as acceleration. Then they will learn how to determine the velocities of uniformly accelerated linear motion. Afterwards the teacher will tell you how to correctly construct a speed graph.

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.

Every second a car increases its speed by . Is the car moving with uniform acceleration?

At first glance, it seems yes, because over equal periods of time the speed increases by equal amounts. Let's take a closer look at the movement for 1 second. It is possible that the car moved uniformly for the first 0.5 s and increased its speed by the second 0.5 s. There could have been another situation: the car accelerated at first, 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 This is a movement in which a body changes its speed by the same amount over ANY equal periods of time.

Often uniformly accelerated motion is called a motion in which a 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 acceleration, it is convenient to write a formula for calculating the instantaneous speed of any interval and for any moment in time:

The velocity equation in projections has the form:

This equation makes it possible to determine the speed at any moment of movement of a body. When working with the law of changes in speed over time, it is necessary to take into account the direction of speed in relation to the selected reference point.

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 look at the most typical examples of the direction of velocity and acceleration.

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

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

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

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

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

This type of motion is sometimes called uniformly slow motion. In this case, they say that the body is 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 upward.

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

Rice. 3. Velocity and acceleration are mutually perpendicular

Examples of such movement are the movement of the Earth around the Sun and the movement of the Moon around the Earth. In this case, the trajectory of movement 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 in velocity.

Speed ​​graph(velocity projection) is the law of change of velocity (velocity projection) over time for uniformly accelerated rectilinear motion, presented graphically.

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

Let's analyze various graphs.

First. Velocity projection equation: . As time increases, 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 begins from the point, which characterizes the initial speed.

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

The graph begins at point and continues until 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 speed equation, you will remember that in mathematics there was a similar function:

Where and are some constants, for example:

Rice. 5. Graph of a function

This is the equation of a straight line, which is confirmed by the graphs we 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 projection of acceleration is greater than zero.

Writing this equation. And the type of graph itself is quite simple (graph 1).

Rice. 6. Various cases of uniformly accelerated motion

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

The third case is a situation where the acceleration projection is less than zero and the body continuously moves in the direction opposite to the positive direction of the axis. In this case, the velocity module constantly increases, the body accelerates.

Graph of acceleration versus time

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

Let's look at the graphs:

Rice. 7. Graph of acceleration projections versus time

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

Let's look at Fig. 8. The area of ​​the figure bounded by the axes, the graph and the perpendicular to the x-axis is equal to:

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

Rice. 8. Speed ​​change

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

We used the word "numerically" because the units of area and change in velocity are not the same.

In this lesson, we became acquainted with the velocity equation and learned how to graphically represent this equation.

Bibliography

1. Kikoin I.K., Kikoin A.K. Physics: Textbook for 9th grade of high school. - M.: “Enlightenment”.
2. Peryshkin A.V., Gutnik E.M., Physics. 9th grade: 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: A reference book with examples of problem solving. - 2nd edition repartition. - X.: Vesta: Ranok Publishing House, 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. Characterize 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 graph shows the dependence of the projection of the body’s velocity on time during uniformly accelerated motion at ?

1) Analytical method.

We consider the highway to be straight. Let's write down the equation of motion of a cyclist. Since the cyclist moved uniformly, his equation of motion is:

(we place the origin of coordinates at the starting point, so the initial coordinate of the cyclist is zero).

The motorcyclist was moving at uniform acceleration. He also started moving from the starting point, so his initial coordinate is zero, the initial speed of the motorcyclist is also zero (the motorcyclist began to move from a state of rest).

Considering that the motorcyclist started moving later, the equation of motion for the motorcyclist is:

In this case, the speed of the motorcyclist changed according to the law:

At the moment when the motorcyclist caught up with the cyclist, their coordinates are equal, i.e. or:

Solving this equation for , we find the meeting time:

This is a quadratic equation. We define the discriminant:

Determining the roots:

Let's substitute numerical values ​​into the formulas and calculate:

We discard the second root as not corresponding to the physical conditions of the problem: the motorcyclist could not catch up with the cyclist 0.37 s after the cyclist started moving, since he himself left the starting point only 2 s after the cyclist started.

Thus, the time when the motorcyclist caught up with the cyclist:

Let's substitute this time value into the formula for the law of change in speed of a motorcyclist and find the value of his speed at this moment:

2) Graphic method.

On the same coordinate plane we build graphs of changes over time in the coordinates of the cyclist and motorcyclist (the graph for the cyclist’s coordinates is in red, for the motorcyclist – in green). It can be seen that the dependence of the coordinate on time for a cyclist is a linear function, and the graph of this function is a straight line (the case of uniform rectilinear motion). The motorcyclist was moving with uniform acceleration, so the dependence of the motorcyclist’s coordinates on time is a quadratic function, the graph of which is a parabola.

In this topic we will look at a very special type of irregular motion. Based on the opposition to uniform motion, uneven motion is movement at unequal speed along any trajectory. What is the peculiarity of uniformly accelerated motion? This is an uneven movement, but which "equally accelerated". We associate acceleration with increasing speed. Let's remember the word "equal", we get an equal increase in speed. How do we understand “equal increase in speed”, how can we evaluate whether the speed is increasing equally or not? To do this, we need to record time and estimate the speed over the same time interval. For example, a car starts to move, in the first two seconds it develops a speed of up to 10 m/s, in the next two seconds it reaches 20 m/s, and after another two seconds it already moves at a speed of 30 m/s. Every two seconds the speed increases and each time by 10 m/s. This is uniformly accelerated motion.

The physical quantity that characterizes how much the speed increases each time is called acceleration.

Can the movement of a cyclist be considered uniformly accelerated if, after stopping, in the first minute his speed is 7 km/h, in the second - 9 km/h, in the third - 12 km/h? It is forbidden! The cyclist accelerates, but not equally, first he accelerated by 7 km/h (7-0), then by 2 km/h (9-7), then by 3 km/h (12-9).

Typically, movement with increasing speed is called accelerated movement. Movement with decreasing speed is slow motion. But physicists call any movement with changing speed accelerated movement. Whether the car starts moving (the speed increases!) or brakes (the speed decreases!), in any case it moves with acceleration.

Uniformly accelerated motion- this is the movement of a body in which its speed for any equal intervals of time changes(can increase or decrease) the same

Body acceleration

Acceleration characterizes the rate of change in speed. This is the number by which the speed changes every second. If the acceleration of a body is large in magnitude, this means that the body quickly gains speed (when it accelerates) or quickly loses it (when braking). Acceleration is a physical vector quantity, numerically equal to the ratio of the change in speed to the period of time during which this change occurred.

Let's determine the acceleration in the next problem. At the initial moment of time, the speed of the ship was 3 m/s, at the end of the first second the speed of the ship became 5 m/s, at the end of the second - 7 m/s, at the end of the third 9 m/s, etc. Obviously, . But how did we determine? We are looking at the speed difference over one second. In the first second 5-3=2, in the second second 7-5=2, in the third 9-7=2. But what if the speeds are not given for every second? Such a problem: the initial speed of the ship is 3 m/s, at the end of the second second - 7 m/s, at the end of the fourth 11 m/s. In this case, you need 11-7 = 4, then 4/2 = 2. We divide the speed difference by the time interval.

This formula is most often used in a modified form when solving problems:

The formula is not written in vector form, so we write the “+” sign when the body is accelerating, the “-” sign when it is slowing down.

Acceleration vector direction

The direction of the acceleration vector is shown in the figures

In this figure, the car moves in a positive direction along the Ox axis, the velocity vector always coincides with the direction of movement (directed to the right). When the acceleration vector coincides with the direction of the speed, this means that the car is accelerating. Acceleration is positive.

During acceleration, the direction of acceleration coincides with the direction of speed. Acceleration is positive.

In this picture, the car is moving in the positive direction along the Ox axis, the velocity vector coincides with the direction of movement (directed to the right), the acceleration does NOT coincide with the direction of the speed, this means that the car is braking. Acceleration is negative.

When braking, the direction of acceleration is opposite to the direction of speed. Acceleration is negative.

Let's figure out why the acceleration is negative when braking. For example, in the first second the ship slowed down from 9 m/s to 7 m/s, in the second second to 5 m/s, in the third to 3 m/s. The speed changes to "-2m/s". 3-5=-2; 5-7=-2; 7-9=-2m/s. This is where the negative acceleration value comes from.

When solving problems, if the body slows down, acceleration is substituted into the formulas with a minus sign!!!

Moving during uniformly accelerated motion

An additional formula called timeless

Formula in coordinates

Medium speed communication

With uniformly accelerated motion, the average speed can be calculated as the arithmetic mean of the initial and final speeds

From this rule follows a formula that is very convenient to use when solving many problems

Path ratio

If a body moves uniformly accelerated, the initial speed is zero, then the paths traversed in successive equal intervals of time are related as a successive series of odd numbers.

The main thing to remember

1) What is uniformly accelerated motion;
2) What characterizes acceleration;
3) Acceleration is a vector. If a body accelerates, the acceleration is positive, if it slows down, the acceleration is negative;
3) Direction of the acceleration vector;
4) Formulas, units of measurement in SI

Exercises

Two trains are moving towards each other: one is heading north at an accelerated rate, the other is moving slowly to the south. How are train accelerations directed?

Equally to the north. Because the first train's acceleration coincides in direction with the movement, and the second train's acceleration is opposite to the movement (it slows down).

In the first second of uniformly accelerated motion, the body travels a distance of 1 m, and in the second - 2 m. Determine the path traveled by the body in the first three seconds of movement.

Problem No. 1.3.31 from the “Collection of problems for preparing for entrance exams in physics at USPTU”

Given:

\(S_1=1\) m, \(S_2=2\) m, \(S-?\)

The solution of the problem:

Note that the condition does not say whether the body had an initial speed or not. To solve the problem it will be necessary to determine this initial speed \(\upsilon_0\) and acceleration \(a\).

Let's work with the available data. The path in the first second is obviously equal to the path in \(t_1=1\) second. But the path for the second second must be found as the difference between the path for \(t_2=2\) seconds and \(t_1=1\) second. Let's write down what was said in mathematical language.

\[\left\( \begin(gathered)

(S_2) = \left(((\upsilon _0)(t_2) + \frac((at_2^2))(2)) \right) — \left(((\upsilon _0)(t_1) + \frac( (at_1^2))(2)) \right) \hfill \\
\end(gathered) \right.\]

Or, which is the same:

\[\left\( \begin(gathered)
(S_1) = (\upsilon _0)(t_1) + \frac((at_1^2))(2) \hfill \\
(S_2) = (\upsilon _0)\left(((t_2) — (t_1)) \right) + \frac((a\left((t_2^2 — t_1^2) \right)))(2) \hfill\\
\end(gathered) \right.\]

This system has two equations and two unknowns, which means it (the system) can be solved. We will not try to solve it in general form, so we will substitute the numerical data known to us.

\[\left\( \begin(gathered)
1 = (\upsilon _0) + 0.5a \hfill \\
2 = (\upsilon _0) + 1.5a \hfill \\
\end(gathered) \right.\]

Subtracting the first from the second equation, we get:

If we substitute the resulting acceleration value into the first equation we get:

\[(\upsilon _0) = 0.5\; m/s\]

Now, in order to find out the path traveled by a body in three seconds, it is necessary to write down the equation of motion of the body.

As a result, the answer is:

Answer: 6 m.

If you do not understand the solution and you have any questions or you have found an error, then feel free to leave a comment below.