This video tutorial is devoted to the topic “The speed of a 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 speeds of a rectilinear uniformly accelerated motion. After that, the teacher will tell you how to build a speed graph correctly.

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 during the time during which this change occurred.

Once again about what is uniformly accelerated motion

Let's consider the problem.

For each second, the car increases its speed by. Is the vehicle moving at uniform acceleration?

At first glance, it seems, yes, because for equal periods of time, the speed increases by equal amounts. Let's take a closer look at the movement for 1 s. It is possible that the first 0.5 s the car moved uniformly and increased its speed by the second 0.5 s. There could be another situation: the car accelerated on the first, and the rest moved evenly. Such movement will not be uniformly accelerated.

By analogy with uniform motion, we introduce the correct formulation of uniformly accelerated motion.

Equally accelerated is called a movement in which the body for ANY equal intervals of time changes its speed by the same amount.

Often, uniformly accelerated motion is called such a motion in which the body moves with constant acceleration. The simplest example of uniformly accelerated motion is the free fall of a body (the body falls under the action of gravity).

Using the equation that determines the acceleration, it is convenient to write down the formula for calculating the instantaneous velocity of any interval and for any moment in time:

The velocity equation in projections is:

This equation makes it possible to determine the speed at any moment of movement of the body. When working with the law of speed variation from time to time, it is necessary to take into account the direction of the speed in relation to the selected CO.

On the question of the direction of speed and acceleration

In uniform movement, the direction of speed and movement is always the same. In the case of uniformly accelerated motion, the direction of speed 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 consider the most typical examples of direction of speed and acceleration.

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

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

In this case, the body accelerates. Examples of such movement include free fall, starting and accelerating a bus, launching and accelerating a rocket.

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

Rice. 2. Speed ​​and acceleration are directed in different directions along one straight line

This movement is sometimes called equal slow motion. In this case, the body is said to slow down. Ultimately, it will either stop or start moving in the opposite direction. An example of such a movement is a stone thrown vertically upward.

3. Speed ​​and acceleration are mutually perpendicular (Fig. 3).

Rice. 3. Speed ​​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 will be a circle.

Thus, the direction of acceleration does not always coincide with the direction of speed, but always coincides with the direction of change in speed.

Speed ​​graph(speed projection) is the law of speed change (speed projection) versus time for uniformly accelerated rectilinear motion, represented graphically.

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

Let's analyze various graphs.

First. Velocity projection equation:. With increasing time, the speed also increases. Note that in a graph where one of the axes is time and the other is speed, there will be a straight line. This line starts from a point that characterizes the initial speed.

The second is the dependence at a negative value of the projection of acceleration, when the movement is slowed down, that is, the speed modulo first decreases. In this case, the equation looks like this:

The graph starts at a point and continues to the point where the time axis is crossed. At this point, the body's speed becomes zero. This means that the body has stopped.

If you look closely at the equation of speed, you will recall that there was a similar function in mathematics:

Where and are some constants, for example:

Rice. 5. Function graph

This is a straight line equation, which is confirmed by the graphs we have considered.

To finally understand the speed graph, consider special cases. In the first graph, the dependence of speed on time is associated with the fact that the initial speed,, is equal to zero, the projection of acceleration is greater than zero.

Writing this equation. And the very form of the graph is quite simple (graph 1).

Rice. 6. Various cases of uniformly accelerated motion

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

The third case is a situation when the projection of the acceleration is less than zero and the body is continuously moving in the direction opposite to the positive direction of the axis. In this case, the speed module is constantly increasing, the body is accelerating.

Acceleration versus time graph

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

Consider the graphs:

Rice. 7. Graph of dependence of projections of acceleration on 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. Note that line I lies above the straight line abscissa (the projection of the acceleration is positive), and line II is below (the projection of the acceleration is negative). If the movement was uniform, then the projection of the acceleration would coincide with the abscissa axis.

Consider fig. 8. The area of ​​the figure, bounded by the axes, the graph and the perpendicular to the abscissa axis, is equal to:

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

Rice. 8. Change of speed

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

We used the word "numerically" because the units for area and speed change do not match.

In this lesson, we got acquainted with the equation of speed and learned how to graphically represent this equation.

Bibliography

1. Kikoin I.K., Kikoin A.K. Physics: Textbook for grade 9 high school. - M .: "Education".
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 Handbook with Examples of Problem Solving. - 2nd edition redistribution. - X .: Vesta: Ranok Publishing House, 2005. - 464 p.

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Homework

1. What is uniformly accelerated motion?

2. Describe the movement of the body and determine the traversed path of the body according to the graph within 2 s from the beginning of the movement:

3. Which of the graphs shows the dependence of the projection of the body's velocity on time for uniformly accelerated motion at?

In this thread we will look at a very special kind of uneven motion. Based on the opposition to uniform movement, uneven movement is movement with an unequal speed, along any trajectory. What is the peculiarity of uniformly accelerated motion? This is an uneven movement, but which "accelerates equally"... Acceleration is associated with an increase in speed. Let's remember the word "equal", we get an equal increase in speed. And how to understand "equal increase in speed", how to estimate the speed is equal to the increase or not? To do this, we need to measure the time, estimate the speed at the same time interval. For example, a car starts moving, in the first two seconds it develops a speed of up to 10 m / s, in the next two seconds 20 m / s, after another two seconds it is already moving at a speed of 30 m / s. Every two seconds the speed increases and every 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 a cyclist's movement 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 in the same way, first accelerated by 7 km / h (7-0), then by 2 km / h (9-7), then by 3 km / h (12-9).

Usually movement with increasing modulus of speed is called accelerated movement. Movement with a decreasing speed is a slow motion. But physicists call any movement with changing speed accelerated movement. Whether the car starts to move (the speed increases!), Or brakes (the speed decreases!), In any case it moves with acceleration.

Equally accelerated movement- this is the movement of the body, at which its speed for any equal intervals of time changes(can increase or decrease) the same

Body acceleration

Acceleration is the rate at which the speed changes. This is the number by which the speed changes every second. If the acceleration of a body is large in modulus, it means that the body quickly picks up 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 time interval during which this change occurred.

Let's define the acceleration in the next problem. At the initial moment of time the speed of the motor ship was 3 m / s, at the end of the first second the speed of the motor 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 consider the difference in speeds in 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 task: the initial speed of the motor 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, 11-7 = 4, then 4/2 = 2. We divide the speed difference by the time interval.

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

The formula is not written in vector form, so the "+" sign is written when the body is accelerating, the "-" sign is when it slows down.

Acceleration vector direction

The direction of the acceleration vector is shown in the figures

In this figure, the car is moving 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 speed, it means that the car is accelerating. The acceleration is positive.

When accelerating, the direction of acceleration coincides with the direction of speed. The acceleration is positive.

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

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

Let's see why acceleration is negative during braking. For example, a motor ship in the first second dropped the speed from 9m / s to 7m / s, in the second second to 5m / s, in the third to 3m / s. The speed changes by "-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, the acceleration is substituted into the formulas with a minus sign !!!

Moving with uniformly accelerated motion

An additional formula called timeless

Formula in coordinates

Medium speed communication

With uniformly accelerated movement, 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 the body moves uniformly accelerated, the initial velocity is zero, then the paths traversed in successive equal intervals of time are referred to as a sequential row of odd numbers.

The main thing to remember

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

Exercises

Two trains are going towards each other: one - accelerating to the north, the other - slowly to the south. How are train accelerations directed?

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

In the first second of uniformly accelerated movement, the body travels a path of 1 m, and in the second - 2 m. Determine the path traversed 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 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 velocity 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 data we have. The path in the first second is obviously equal to the path in \ (t_1 = 1 \) second. But the path in the second second must be found as the difference between the path in \ (t_2 = 2 \) seconds and \ (t_1 = 1 \) second. Let's write 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 substitute the known numerical data.

\ [\ 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 traversed by the 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 a question or you have found an error, then feel free to leave a comment below.

1) Analytical method.

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

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

The motorcyclist was moving at uniform acceleration. He also began to move 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 rest).

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

At the same time, 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. Determine the discriminant:

We define the roots:

Substitute numerical values ​​into the formulas and calculate:

We discard the second root as inappropriate to the physical conditions of the problem: the motorcyclist could not catch up with the cyclist in 0.37 s after the cyclist started to move, 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:

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

2) Graphic method.

On one coordinate plane, we build graphs of the change over time of the coordinates of the cyclist and the motorcyclist (the graph for the coordinate of the cyclist 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 movement). The motorcyclist was moving uniformly, so the dependence of the motorcyclist's coordinate on time is a quadratic function, the graph of which is a parabola.