Types of rectilinear motion. Uniformly accelerated motion
1.1.3 Kinematics of rectilinear motion
Uniform rectilinear motion. Uniform rectilinear called such a movement that occurs along a rectilinear trajectory, and when for any equal intervals of time the body makes the same movement. speed uniform rectilinear motion is called a vector quantity equal to the ratio of the movement of the body to the time interval during which this movement was made: v = r / t
The direction of speed in rectilinear motion coincides with the direction of movement, so the module of movement is equal to the path of movement: / r/ = S. Since in uniform rectilinear motion for any equal time intervals the body makes equal displacements, the speed of such motion is a constant value ( v = const):
This movement can be graphically displayed in different coordinates. In system v(t), uniform rectilinear movement, the speed will be a straight line parallel to the abscissa axis, and the path will be the area of a quadrangle with sides equal to the constant speed and time during which the movement took place (Figure - 1.8). In coordinates S(t), the path is reflected by an inclined straight line, and the speed can be judged by the tangent of the angle of inclination of this straight line (Figure - 1.9) Let the axis Oh coordinate system associated with the reference body coincides with the straight line along which the body moves, and x 0 is the coordinate of the starting point of the body motion.
According to this formula, knowing the coordinate X 0 starting point of body movement and body speed v (her projection v x per axle Oh), at any time, you can determine the position of a moving body. The right side of the formula is an algebraic sum, since and X 0 , and v x can be both positive and negative (its graphical representation is given in figure 1.10).
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Figure - 1.9 |
Figure - 1.10 |
Rectilinear uniformly variable motion is motion with constant acceleration ( a = const). In rectilinear uniformly accelerated motion, the vectors v 0 , v and a directed in a straight line. Therefore, the modules of their projections onto this line are equal to the modules of these vectors themselves.
Let us find the kinematic law of rectilinear uniformly accelerated motion. After the transformation, we obtain the equation for the speed of uniformly accelerated motion:
If the body was initially at rest (v0 ==0),
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v=√ 2aS |
Graphs of the speed of rectilinear uniformly accelerated motion are shown in figure - 1.11. In this figure, graphics 1 and 2 correspond to the movement with a positive acceleration projection on the axis Oh(speed increases) and the graph 3 corresponds to movement with a negative projection of acceleration (speed decreases). Schedule 2 corresponds to motion without initial velocity, and the graphs 1 and 3 - movement with initial speed v 0x. The angle of inclination of the graph to the abscissa axis depends on the acceleration of the body. To build the dependence of the coordinate on time (motion graph), the time of movement is plotted on the abscissa axis, and the coordinate of the moving body is plotted on the ordinate axis.
Let the body move with uniform acceleration in the positive direction Oh selected coordinate system. Then the equation of motion of the body has the form:
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x = x 0 + v ox t |
The graph of this dependence is a parabola, the branches of which are directed upwards, if a>0, or down if a<0. Чтобы построить графикпути, на оси абсцисс откладывают время, а на оси ординат - длину пути, пройденного телом. В равноускоренном прямолинейном движении зависимость пути от времени выражается формулами, которые отражают квадратичную зависимость. Следовательно, графиком пути прямолинейного равнопеременного движения является ветвь параболы (рисунок - 1.12).
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Figure - 1.11 |
Figure - 1.12 |
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Uniform movement- mechanical movement, in which the body travels the same distance in any equal time intervals. (v=const) Uniform movement of a material point is a movement in which the value of the speed of the point remains unchanged. The distance traveled by the point in time t (\displaystyle t) is given in this case by the formula l = v t (\displaystyle l=vt) .
Types of uniform motion
Uniform circular motion is the simplest example of curvilinear motion.
When a point moves uniformly along a circle, its trajectory is an arc. The point moves at a constant angular velocity ω (\displaystyle \omega ) , and the dependence of the point's rotation angle on time is linear:
φ = φ 0 + ω t (\displaystyle \varphi =\varphi _(0)+\omega t) ,
where φ 0 (\displaystyle \varphi _(0)) is the initial value of the rotation angle.
The same formula determines the angle of rotation of an absolutely rigid body during its uniform rotation around a fixed axis, that is, during rotation with a constant angular velocity ω → (\displaystyle (\vec (\omega ))) .
An important characteristic of this type of motion is the linear velocity of the material point v → (\displaystyle (\vec (v)))
It must be remembered that uniform motion in a circle is uniformly accelerated motion. Although the linear velocity modulus does not change, the direction of the linear velocity vector does change (due to normal acceleration).
Literature
- Physical encyclopedia. T.4. M .: "Great Russian Encyclopedia", 1994. test in physics
Links
Play media file Uniform and uneven motion1.1.3 Kinematics of rectilinear motion
Uniform rectilinear motion. Uniform rectilinear called such a movement that occurs along a rectilinear trajectory, and when for any equal intervals of time the body makes the same movement. speed uniform rectilinear motion is called a vector quantity equal to the ratio of the movement of the body to the time interval during which this movement was made: v = r / t
The direction of speed in rectilinear motion coincides with the direction of movement, so the module of movement is equal to the path of movement: / r/ = S. Since in uniform rectilinear motion for any equal time intervals the body makes equal displacements, the speed of such motion is a constant value ( v = const):
This movement can be graphically displayed in different coordinates. In system v(t), uniform rectilinear movement, the speed will be a straight line parallel to the abscissa axis, and the path will be the area of a quadrangle with sides equal to the constant speed and time during which the movement took place (Figure - 1.8). In coordinates S(t), the path is reflected by an inclined straight line, and the speed can be judged by the tangent of the angle of inclination of this straight line (Figure - 1.9) Let the axis Oh coordinate system associated with the reference body coincides with the straight line along which the body moves, and x 0 is the coordinate of the starting point of the body motion.
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Figure - 1.7 |
Drawing - 1.8 |
Both the displacement S and the velocity v of the moving body are directed along the Ox axis. Now you can establish the kinematic law of uniform rectilinear motion, i.e., find an expression for the coordinates of a moving body at any time.
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x= x 0 + v x t |
According to this formula, knowing the coordinate X 0 starting point of body movement and body speed v(her projection v x per axle Oh), at any time, you can determine the position of a moving body. The right side of the formula is an algebraic sum, since and X 0 , and v x can be both positive and negative (its graphical representation is given in figure 1.10).
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Figure - 1.9 |
Figure - 1.10 |
Rectilinear motion, in which the speed of the body for any equal intervals of time changes in the same way, is called uniform rectilinear motion. The rate of change of speed is characterized by a value denoted a and called acceleration. Acceleration is called a vector quantity equal to the ratio of the change in the speed of the body (v- v 0 ) to the time span t, during which this change occurred: a =(v - v 0 )/ t. Here v 0 - the initial speed of the body, v is the instantaneous velocity of the body at the given moment of time.
Rectilinear uniformly variable motion is motion with constant acceleration ( a = const). In rectilinear uniformly accelerated motion, the vectors v 0 , v and a directed in a straight line. Therefore, the modules of their projections onto this line are equal to the modules of these vectors themselves.
Let us find the kinematic law of rectilinear uniformly accelerated motion. After the transformation, we obtain the equation for the speed of uniformly accelerated motion:
If the body was initially at rest (v0 ==0),
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v=√ 2aS |
Graphs of the speed of rectilinear uniformly accelerated motion are shown in figure - 1.11. In this figure, graphics 1 and 2 correspond to the movement with a positive acceleration projection on the axis Oh(speed increases) and the graph 3 corresponds to movement with a negative projection of acceleration (speed decreases). Schedule 2 corresponds to motion without initial velocity, and the graphs 1 and 3 - movement with initial speed v 0x. The angle of inclination of the graph to the abscissa axis depends on the acceleration of the body. To build the dependence of the coordinate on time (motion graph), the time of movement is plotted on the abscissa axis, and the coordinate of the moving body is plotted on the ordinate axis.
Let the body move with uniform acceleration in the positive direction Oh selected coordinate system. Then the equation of motion of the body has the form:
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x = x 0 + v ox t |
The graph of this dependence is a parabola, the branches of which are directed upwards, if a>0, or down if a
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Figure - 1.11 |
Uniform movement. The formula for uniform motion.Acquaintance with the classical course of physics begins with the simplest laws that bodies moving in space obey. Rectilinear uniform motion is the simplest form of changing the position of a body in space. Such motion is studied in the section of kinematics. Opponent of AristotleGalileo Galilei remains in the annals of history as one of the greatest naturalists of the late Renaissance. He dared to check the statements of Aristotle - an unheard-of heresy at that time, because the teaching of this ancient sage was supported in every possible way by the church. The idea of uniform motion was not considered then - the body either moved "in general", or was at rest. Numerous experiments were needed to explain the nature of motion. Galileo's experimentsA classic example of the study of movement was the famous experiment of Galileo, when he threw various weights from the famous Leaning Tower of Pisa. As a result of this experiment, it turned out that bodies having different masses fall at the same speed. Later, the experiment was continued in the horizontal plane. Galileo suggested that any ball in the absence of friction will roll downhill for an arbitrarily long time, while its speed will also be constant. So, experimentally, Galileo Galilei discovered the essence of Newton's first law - in the absence of external forces, the body moves in a straight line at a constant speed. Rectilinear uniform motion is the expression of Newton's first law. Currently, a special branch of physics, kinematics, deals with various types of motion. Translated from Greek, this name means - the doctrine of movement. New coordinate systemThe analysis of uniform motion would be impossible without the creation of a new principle for determining the position of bodies in space. Now we call it a rectilinear coordinate system. Its author is the famous philosopher and mathematician Rene Descartes, thanks to whom we call the coordinate system Cartesian. In this form, it is very convenient to represent the trajectory of the body in three-dimensional space and analyze such movements by tying the position of the body to the coordinate axes. A rectangular coordinate system consists of two straight lines intersecting at a right angle. The point of intersection is usually taken as the origin of measurements. The horizontal line is called the abscissa, the vertical line is called the ordinate. Since we live in three-dimensional space, a third axis is added to the planar coordinate system - it is called the applicate. Speed detectionSpeed cannot be measured the way we measure distance and time. This is always a derivative value, which is written as a ratio. In its most general form, the speed of a body is equal to the ratio of the distance traveled to the elapsed time. The formula for speed is: Where d is the distance traveled, t is the elapsed time. The direction directly affects the vector designation of the speed (the value that determines the time is a scalar, that is, it has no direction). The concept of uniform motionIn uniform motion, a body moves along a straight line at a constant speed. Since speed is a vector quantity, its properties are described not only by a number, but also by a direction. Therefore, it is better to clarify the definition, and say that the speed of uniform rectilinear motion is constant in magnitude and direction. To describe rectilinear uniform motion, it is sufficient to use the Cartesian coordinate system. In this case, the OX axis will be conveniently laid in the direction of travel. With uniform displacement, the position of the body in any period of time is determined by only one coordinate - x. The direction of body movement and the velocity vector are directed along the x axis, while the beginning of the movement can be counted from the zero mark. Therefore, the analysis of the movement of a body in space can be reduced to the projection of the trajectory of motion onto the axis ОХ and the process can be described by algebraic equations. Uniform motion from the point of view of algebraSuppose that at a certain time t 1 the body is at a point on the x-axis, the coordinate of which is equal to x 1 . After a certain period of time, the body will change its location. Now the coordinate of its location in space will be equal to x 2. Reducing the consideration of the movement of the body to its location on the coordinate axis, we can determine that the path that the body has traveled is equal to the difference between the initial and final coordinates. Algebraically, this is written as follows: Δs \u003d x 2 - x 1. Travel amountThe value that determines the movement of the body can be both greater and less than 0. It all depends on the direction in which the body moved relative to the direction of the axis. In physics, you can write down both negative and positive displacement - it all depends on the coordinate system chosen for the reference. Rectilinear uniform motion occurs at a speed that is described by the formula: In this case, the speed will be greater than zero if the body moves along the OX axis from zero; less than zero - if the movement goes from right to left along the x-axis. Such a brief record reflects the essence of uniform rectilinear motion - whatever the changes in coordinates, the speed of movement remains unchanged. We owe Galileo another brilliant idea. Analyzing the movement of a body in a world devoid of friction, the scientist insisted that forces and speeds do not depend on each other. This brilliant conjecture is reflected in all existing laws of motion. Thus, the forces acting on the body do not depend on each other and act as if there were no others. Applying this rule to the analysis of the motion of a body, Galileo realized that the entire mechanics of the process can be decomposed into forces that add up geometrically (vector) or linearly if they act in one direction. Approximately it will look like this:
What is uniform motion here? Everything is very simple. At very short distances, the speed of the body can be considered uniform, with a rectilinear trajectory. Thus, a brilliant opportunity arose to study more complex movements, reducing them to simple ones. Thus, the uniform motion of a body along a circle was studied. Uniform circular motionUniform and uniformly accelerated motion can be observed in the movement of the planets in their orbits. In this case, the planet participates in two types of independent motions: it moves uniformly in a circle and at the same time moves uniformly accelerated towards the Sun. Such a complex movement is explained by the forces acting on the planets. The scheme of the impact of planetary forces is shown in the figure: As you can see, the planet is involved in two different movements. The geometric addition of velocities will give us the speed of the planet on a given segment of the path. Uniform motion is the basis for further study of kinematics and physics in general. This is an elementary process to which much more complex movements can be reduced. But in physics, as elsewhere, the great begins with the small, and when launching spaceships into airless space, driving submarines, one should not forget about those simple experiments on which Galileo once tested his discoveries. Write, please, a hundred formulas for uniform. rectilinear movement - coordinate, speed, etc.Alyonochka Uniform rectilinear motion is such a rectilinear motion in which a material point (body) moves in a straight line and makes the same movements at any equal time intervals. |
It consists in the fact that, considering this or that body, it should be taken into account that all its points move in the same direction with exactly the same speed. That is why it is not necessary to characterize the motion of the entire given body; one can confine oneself to only one of its points.
The main characteristics of any movement are its trajectory, movement and speed. A trajectory is just a line that exists only in the imagination, along which a given material point moves in space. The displacement is a vector directed from the start point to the end point. Finally, speed is a general indicator of the movement of a point, which characterizes not only its direction, but also the speed of movement relative to any body taken as a reference point.
Uniform rectilinear motion is largely an imaginary concept, which is characterized by two main factors - uniformity and straightness.
Uniformity of movement means that it is carried out at a constant speed without any acceleration. Straightness of motion implies that it occurs along a straight line, that is, its trajectory is an absolutely straight line.
Based on all of the above, we can conclude that uniform rectilinear motion is a special type of motion, as a result of which the body performs the same movement in absolutely equal time intervals. So, by dividing a certain interval into equal intervals (for example, one second each), it will be possible to see that with the movement indicated above, the body will cover the same distance for each of these segments.
The speed of uniform rectilinear motion is which in numerical terms is equal to the ratio of the path traveled by the body in a given period of time to the numerical value of this interval. This value does not depend on time in any way, moreover, it is worth noting that the speed of uniform rectilinear motion at any point of the trajectory absolutely coincides with the movement of the body. In this case, the quantitative value for an arbitrarily taken period of time is equal to
Uniform rectilinear motion is characterized by a special approach to the path that a body travels in a certain period of time. The distance traveled with this is nothing more than a displacement module. The movement, in turn, is the product of the speed with which the body was moving and the time during which this movement was carried out.
It is quite natural that if the displacement vector coincides with the positive direction of the abscissa axis, then the projection of the calculated velocity will not only be positive, but also coincide with the velocity value.
Uniform rectilinear motion can be represented, among other things, in the form of an equation, which will reflect the relationship between the coordinates of the body and time.
Many problems in physics are based on the consideration of rectilinear uniform and uniformly accelerated motion. They are the simplest and most idealized cases of moving bodies in space. We describe them in more detail in this article.
Before considering the uniform and it is useful to understand the concept itself.
Movement is a process of changing the coordinates of a material point in space over a certain period of time. According to this definition, we single out the following signs by which we can immediately tell whether we are talking about movement or not:
- There must be a change in spatial coordinates. Otherwise, the body can be considered at rest.
- The process must evolve over time.
Let us also pay attention to the concept of "material point". The fact is that when studying questions of mechanical motion (including uniform and uniformly accelerated rectilinear motion), the structure of the body and its dimensions are not taken into account. This approximation is connected with the fact that the magnitude of the change in coordinates in space far exceeds the physical dimensions of a moving object, therefore it is considered a material point (the word "material" implies taking into account its mass, since its knowledge is necessary when solving the problems under consideration).
The main physical quantities characterizing the movement
These include speed, acceleration, distance traveled, and the concept of trajectory. Let's analyze each value in order.
The speed of rectilinear uniform and uniformly accelerated motion (vector value) reflects the rate of change of body coordinates in time. For example, if it moved 100 meters in 10 seconds (typical values for sprinters in sports competitions), then one speaks of a speed of 10 meters per second (100/10 = 10 m/s). This value is denoted by the Latin letter "v" and is measured in units of distance divided by time, for example, kilometers per hour (km / h), meters per minute (m / min.), Miles per hour (mil / h) and so on. Further.
Acceleration - physical, which is denoted by the letter "a", and is characterized by the speed of change of the speed itself. Returning to the example of sprinters, it is known that at the beginning of the race they start at a low speed, as they move, it increases, reaching maximum values. The dimension of acceleration is obtained by dividing that for speed by time, for example, (m/s)/s or m/s 2 .
The distance traveled (a scalar value) reflects the distance traveled (traveled, flown, swum) by a moving object. This value is uniquely determined only by the initial and final position of the object. It is measured in units of distance (meters, kilometers, millimeters, and others) and is denoted by the letter "s" (sometimes "d" or "l").
The trajectory, unlike the path, characterizes the curved line along which the body moved. Since in this article only uniformly accelerated and uniform rectilinear motion is considered, then the trajectory for it will be a straight line.
The question of the relativity of motion
Many people have noticed that while on the bus, you can see that the car moving in the next lane seems to be at rest. This example clearly confirms the relativity of motion (uniformly accelerated, uniform rectilinear motion and its other types).
Taking into account the named feature, when considering problems with moving objects, a frame of reference is always introduced, with respect to which the problem is solved. So, if the passenger in the bus in the example above is taken as the reporting system, then the speed of the car relative to him will be equal to zero. If we consider the movement relative to a person standing at a stop, then the car moves relative to him with a certain speed v.
In the case of rectilinear motion, when two objects move along one line, then the speed of one of them relative to the other is determined by the formula: v ¯ = v ¯ 1 + v ¯ 2, here v ¯ 1 and v ¯ 2 are the speeds of each object (the line means , which add up vector quantities).
The easiest way to move
Of course, this is the movement of an object in a straight line at a constant speed (uniform rectilinear). An example of this type of movement is the flight of an aircraft through clouds or the walking of a pedestrian. In both cases, the trajectory of the object remains straight, and each of them moves at a specific speed.
Formulas describing this type of object movement are as follows:
- s = v*t;
- v = s/t.
Here t is the period of time during which the movement is considered.
Uniformly accelerated rectilinear movement
It is understood as such a type of rectilinear movement of an object, in which its speed changes according to the formula v \u003d a * t, where a is a constant acceleration. The change in speed occurs due to the action of external forces of a different nature. For example, the same aircraft, before reaching cruising speed, must gain it from a state of rest. Another example is the braking of a car when the speed changes from a certain value to zero. This type of motion is called uniformly decelerated, since the acceleration has a negative sign in it (directed against the velocity vector).
The distance traveled s for this type of movement can be calculated by integrating the speed over time, resulting in the formula: s = a*t 2 /2, where t is the acceleration (deceleration) time.
Mixed type of movement
In some cases, rectilinear movement of objects in space occurs both at a constant speed and with acceleration, so it is useful to give formulas for this mixed type of movement.
The speed and acceleration of uniform and uniformly accelerated rectilinear motion are related to each other by the following expression: v \u003d v 0 + a * t, where v 0 is the value of the initial speed. It is easy to understand this formula: at first, the object moved at a constant speed v 0 , for example, a car on the road, but then it began to accelerate, that is, for each time interval t, it began to increase the speed of its movement by a * t. Since the speed is an additive value, the sum of its initial value with the change value will lead to the marked expression.
Integrating this formula over time, we obtain another equation for rectilinear uniform and uniformly accelerated motion, which allows us to calculate the distance traveled: s = v 0 *t + a*t 2 /2. As you can see, this expression is equal to the sum of similar formulas for simpler types of movement discussed in the previous paragraphs.
Problem solution example
Let's solve a simple problem that will demonstrate the use of the above formulas. The condition of the problem is as follows: the car, moving at a speed of 60 km/h, began to brake and after 10 seconds it completely stopped. What distance did he travel while braking?
In this case, we are dealing with rectilinear equally slow motion. The initial speed v 0 = 60 km / h, the final value of this value is v = 0 (the car has stopped). To determine the deceleration acceleration, we use the formula: v = v 0 - a * t (the "-" sign says that the body slows down). Let's convert km/h to m/s (60 km/h = 16.667 m/s), and taking into account that the braking time t = 10 s, we get: a = (v 0 - v)/t = 16.667/10 = 1.667 m /s 2 . We have determined the braking acceleration of the car.
To calculate the distance traveled, we also use the equation for the mixed type of movement, taking into account the sign of the acceleration: s = v 0 *t - a*t 2 /2. Substituting the known values, we get: s \u003d 16.667 * 10 - 1.667 * 10 2 / 2 \u003d 83.33 meters.
Note that the distance traveled could be found using the formula for uniformly accelerated motion (s = a * t 2 /2), since during braking the car will cover exactly the same distance as during acceleration from rest to reaching speed v 0 .
Curve driving
It is important to note that the considered expressions for the path traveled are applicable not only for the case of rectilinear motion, but also for any movement of an object along a curvilinear trajectory.
For example, to calculate the distance that our planet will fly around the Sun (circular motion) for a certain period of time, you can successfully apply the expression s = v * t. This can be done because it uses the speed modulus, which is a constant value, while the speed vector changes. When applying the formula for a path along a curved path, keep in mind that the resulting value will reflect the length of this path, and not the difference between the end and start coordinates of the object.
I. PHYSICAL FOUNDATIONS OF MECHANICS
TOPIC 1.1. "KINEMATICS OF RECTILINEAR AND CURVILINEAR MOVEMENT"
KINEMATICS OF RECTILINEAR MOVEMENT
In this chapter, we will study the simplest type of movement - LINEAR MOVEMENT.
Rectilinear is a movement that is carried out along a straight line. Scientifically speaking, it is a movement whose trajectory is a straight line.
Any physical phenomenon is described by means of mathematical formulas in which physical quantities appear. Therefore, it is necessary to stipulate these very physical quantities that characterize motion, including rectilinear motion. These are:
Table 1.1
Note that Table 1.1 deliberately omits the definition of time, as it is more philosophical than physical. And for the study of this section of physics, the everyday idea of time is quite enough.
Thus, with the help of these four quantities, all types of rectilinear motion are described. And there are only three of them:
- UNIFORM RECTILINEAR MOVEMENT
- EQUI-VARIABLE RECTILINEAR MOVEMENT
- UNEQUIVARIABLE RECTIOLINEAR MOVEMENT
Let's consider each of them. And let's start with the simplest - uniform rectilinear motion.
1. Uniform rectilinear motion is motion at a constant speed. If the speed of the body does not change, then it simply does not have acceleration. The mathematical signs of this movement are written as follows:
υ=const, a=0.
Let's try to imagine this movement: the body moves at a speed, for example,
5 m/s, and since the motion is uniform, its speed does not change. This means that in every second it travels a distance of 5 meters. How to determine how far this body will travel in time t= 20 seconds? To do this, you need to multiply 5 m / s by 20 s - we get the distance S= 100 m. Thus, we can write the formula for uniform rectilinear motion:
S = υt
From here it is easy to derive the velocity formula: (1.1)
2. Uniform motion is motion with constant acceleration. In this case, the speed changes all the time, but changes uniformly: for every second by the same amount. This value is equal to the acceleration of the body. For example: a body is moving with constant acceleration a \u003d 2 m / s 2. If at a certain point in time the speed of the body is, for example, 10 m/s, then in the next second it will increase by 2 m/s and will be equal to 12 m/s, in another second it will increase by another 2 m/s and become equal to already
14 m / s - so every second. It turns out uniformly accelerated traffic.
But the body can move in such a way that its speed will not increase, but rather decrease. And in this case, the body also has acceleration. But, if in the previous example it was greater than zero ( a > 0 ), i.e. positive, then as the speed decreases, the acceleration is less than zero ( a< 0 ), i.e. considered negative. For example: a body is moving with constant acceleration a \u003d - 2 m / s 2. If at a certain point in time the speed of the body is, for example, 10 m/s, then in the next second it will decrease by 2 m/s and will be equal to 8 m/s, in another second it will decrease by another 2 m/s and become equal to already 6 m / s - and, in the end, after 3 seconds the body will stop. It turns out equally slow traffic. True, the word “uniformly slowed down” is not accepted, therefore such a movement is considered uniformly accelerated, but with negative acceleration. And, in general, motion with constant acceleration is called uniformly variable.
Signs of uniform motion can be written as follows:
υ ≠ const, a = const(a≠0).
Mathematically, uniformly variable motion is described by two equations -
the path equation and the velocity equation forming the system:
(1.2),
where υ 0 is the initial speed of the body (i.e., the speed at the beginning of the movement).
3. Non-uniform motion is motion with varying acceleration . In the case of this movement, not only the speed, but also the acceleration changes all the time. Moreover, they can change completely arbitrarily: they can increase all the time or decrease all the time, or they can either increase or decrease. But, as in the previous case, if the speed increases, then the acceleration at this time is positive and co-directed with the speed. And, if the speed decreases, then the acceleration is negative and is directed opposite to the speed (see Fig.1.1 and 1.2).
Rice. 1.1 Fig. 1.2 
a > 0 a< 0
Signs of uneven movement can be written as follows:
υ ≠ const, a ≠ const.
As you can see, of all the rectilinear movements, this type is the most difficult. But, nevertheless, for him there are formulas that allow you to calculate all the characteristics of the movement. There are also two of them: the velocity equation and the acceleration equation.
The symbol " " means that you need to perform action of differentiation by time. Formally, differentiation is performed in the same way as taking a derivative, only written in a different form.
Note that formulas (1.1) and (1.4) differ only in the presence of the differentiation symbol. And it is not surprising, because they describe varieties of rectilinear motion. And formulas (1.4) and (1.5) are general formulas for all three cases of rectilinear motion.
The question arises: how can one calculate, for example, S, guided by these formulas? - To do this, you need to perform the action, the opposite of differentiation. And that is integration. Let's do it.
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