This video lesson is devoted to the topic “Speed of rectilinear uniformly accelerated motion. Speed Graph. During the lesson, students will need to remember such a physical quantity as acceleration. Then they will learn how to determine the speeds of a uniformly accelerated rectilinear motion. After the teacher will tell you how to build a speed graph correctly.
Let's remember what acceleration is.
Definition
Acceleration- this physical quantity, which 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.
The car increases its speed by . Is the car moving with uniform acceleration?
At first glance, it seems so, 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 car moved uniformly for the first 0.5 s and increased its speed by 0.5 s in the second. There could be another situation: the car accelerated to the first yes, 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 called such a movement in which the body for ANY equal intervals of time changes its speed by the same amount.
Often called uniformly accelerated is such a movement in which the body moves with constant acceleration. by the most simple example uniformly accelerated motion is the free fall of the 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 of time:
The velocity equation in projections is:
This equation makes it possible to determine the speed at any moment of the movement of the body. When working with the law of change of speed from time, it is necessary to take into account the direction of speed in relation to the selected CO.
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 consider the most typical examples of the direction of velocity and acceleration.
1. Velocity and acceleration are directed in the same direction along one straight line (Fig. 1).
Rice. 1. Velocity and acceleration are directed in the same direction along one straight line
In this case, the body accelerates. Examples of such movement are free fall, the start and acceleration of a bus, the launch and acceleration of a rocket.
2. Speed and acceleration are directed in different sides along one straight line (Fig. 2).
Rice. 2. Speed and acceleration are directed in different directions along the same straight line
Such a movement is sometimes called uniformly slow. In this case, the body is said to be 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 upwards.
3. Velocity and acceleration are mutually perpendicular (Fig. 3).
Rice. 3. Velocity and acceleration are mutually perpendicular
Examples of such motion are the motion of the Earth around the Sun and the motion of the Moon around the Earth. In this case, the trajectory of motion 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 of velocity.
Speed Graph(projection of speed) is the law of change of speed (projection of speed) from time for uniformly accelerated rectilinear motion, presented graphically.
Rice. 4. Graphs of the dependence of the projection of speed on time for uniformly accelerated rectilinear motion
Let's analyze different charts.
First. Velocity projection equation: . As the time increases, the 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 starts from the point , which characterizes the initial speed.
The second is the dependence at a negative value of the acceleration projection, when the movement is slow, that is, the modulo speed first decreases. In this case, the equation looks like this:
The graph starts at the point and continues until the 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 velocity equation, you will remember that there was a similar function in mathematics:
Where and are some constants, for example:
Rice. 5. Function Graph
This is the equation of a straight line, which is confirmed by the graphs we have 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 acceleration projection is greater than zero.
Write this equation. And the type of chart itself is quite simple (chart 1).
Rice. 6. Various cases of uniformly accelerated motion
Two more cases uniformly accelerated motion are shown in the following 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 the situation where the acceleration projection is less than zero and the body is continuously moving in the direction opposite to the positive axis direction. At the same time, the modulus of speed is constantly increasing, the body is accelerating.
Graph of acceleration versus time
Uniformly accelerated motion is a movement in which the acceleration of the body does not change.
Let's look at the charts:
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 x-axis. Lines I and II - direct movements for two different bodies. Note that line I lies above the abscissa line (positive acceleration projection), and line II lies below (negative acceleration projection). If the motion were uniform, then the acceleration projection would coincide with the abscissa axis.
Consider Fig. 8. The area of \u200b\u200bthe figure bounded by the axes, the graph and the perpendicular to the x-axis is:
The product of acceleration and time is the change in speed over a given time.
Rice. 8. Speed change
The area of the figure bounded by the axes, dependence and perpendicular to the abscissa axis is numerically equal to the change in the speed of the body.
We used the word "number" because the units for area and change in speed are not the same.
In this lesson, we got acquainted with the equation of speed and learned how to graphically represent this equation.
Bibliography
- Kikoin I.K., Kikoin A.K. Physics: Textbook for Grade 9 high school. - M.: "Enlightenment".
- Peryshkin A.V., Gutnik E.M., Physics. Grade 9: textbook for general education. institutions / A.V. Peryshkin, E.M. Gutnik. - 14th ed., stereotype. - M.: Bustard, 2009. - 300 p.
- Sokolovich Yu.A., Bogdanova G.S. Physics: Handbook with examples of problem solving. - 2nd edition redistribution. - X .: Vesta: Publishing house "Ranok", 2005. - 464 p.
- Internet portal "class-fizika.narod.ru" ()
- Internet portal "youtube.com" ()
- Internet portal "fizmat.by" ()
- Internet portal "sverh-zadacha.ucoz.ru" ()
Homework
1. What is uniformly accelerated motion?
2. Describe 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 of the graphs shows the dependence of the projection of the body's velocity on time during uniformly accelerated motion at ?
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 motion.
Task No. 1.3.31 from the "Collection of tasks for preparing for entrance exams in physics of 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 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 it down 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, so it (the system) can be solved. Let's not try to solve it general view, so we 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 equation from the second equation, we get:
If we substitute the obtained acceleration value into the first equation, we get:
\[(\upsilon _0) = 0.5\; m/s\]
Now, in order to find out the path traveled 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 some question or you find an error, then feel free to leave a comment below.
1) Analytical method.We consider the highway to be straight. Let's write down the equation of motion of a cyclist. Since the cyclist was moving uniformly, his equation of motion is:
(the origin of coordinates is placed at the starting point, so the initial coordinate of the cyclist is zero).
The motorcyclist was moving at a uniform speed. He also started moving from the starting point, so his initial coordinate is zero, the initial speed of the motorcyclist is also equal to zero (the motorcyclist began to move from a state of rest).
Considering that the motorcyclist started moving a little later, the motorcyclist's equation of motion 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 with respect to , we find the meeting time:
This quadratic equation. We define the discriminant:
Define roots:
Substitute in formulas numerical values 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 began 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) Graphical way.
On one coordinate plane we build graphs of changes over time in the coordinates of the cyclist and motorcyclist (the graph for the coordinates 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 motion). The motorcyclist was moving with uniform acceleration, so the dependence of the motorcyclist’s coordinates on time is quadratic function, whose graph is a parabola.
In this topic, we will consider a very special kind of non-uniform motion. Based on the opposition to uniform movement, uneven movement is movement at an unequal speed, along any trajectory. What is the characteristic of uniformly accelerated motion? This is an uneven movement, but which "equally accelerating". Acceleration is associated with an increase in speed. Remember the word "equal", we get an equal increase in speed. And how to understand "an equal increase in speed", how to evaluate the speed is equally increasing or not? To do this, we need to detect the time, estimate the speed through 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 each time by 10 m/s. This is uniformly accelerated motion.
The physical quantity that characterizes how much each time the speed increases is called acceleration.
Can a cyclist's movement be considered uniformly accelerated if, after stopping, his speed is 7 km/h in the first minute, 9 km/h in the second, and 12 km/h in the third? It is forbidden! The cyclist accelerates, but not equally, first accelerating by 7 km/h (7-0), then by 2 km/h (9-7), then by 3 km/h (12-9).
Usually, the movement with increasing speed is called accelerated movement. Movement with decreasing speed - slow motion. But physicists call any motion with a changing speed accelerated motion. Whether the car starts off (speed increases!), or slows down (speed decreases!), in any case, it moves with acceleration.
Uniformly accelerated motion- this is such a movement of a body in which its speed for any equal intervals of time changes(may increase or decrease) equally
body acceleration
Acceleration characterizes the rate of change of speed. This is the number by which the speed changes every second. If the modulo acceleration of the body is large, this means that the body quickly picks up speed (when it accelerates) or quickly loses it (when decelerating). Acceleration is a physical vector quantity, numerically equal to the ratio change in speed to the time interval during which this change occurred.
Let's determine the acceleration in the following 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 do we determine? We consider the speed difference 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 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 we write the "+" sign when the body accelerates, the "-" sign - when it slows down.
Direction of the acceleration vector
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, this means that the car is accelerating. The acceleration is positive.
During acceleration, the direction of acceleration coincides with the direction of speed. The acceleration is positive.
In this picture, the car is moving in the positive direction on the Ox axis, the velocity vector is the same as the direction of motion (rightward), the acceleration is NOT the same as the direction of the speed, which means that the car is decelerating. The acceleration is negative.
When braking, the direction of acceleration is opposite to the direction of speed. The acceleration is negative.
Let's figure out why the acceleration is negative when braking. For example, the 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 to "-2m/s". 3-5=-2; 5-7=-2; 7-9=-2m/s. That's where it comes from negative meaning acceleration.
When solving problems, if the body slows down, the acceleration in the formulas is substituted with a minus sign!!!
Moving with uniformly accelerated motion
An additional formula called untimely
Formula in coordinates
Communication with medium speed
With uniformly accelerated movement, the average speed can be calculated as the arithmetic mean of the initial and final speed
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 speed is zero, then the paths traveled in successive equal time intervals are related as a 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 the 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 go towards each other: one - accelerated to the north, the other - slowly to the south. How are train accelerations directed?
Same to the north. Because the first train has the same acceleration in the direction of movement, and the second has the opposite movement (it slows down).
