Mechanical movement Change the change in the position of the body relative to other bodies
System reference Call the body of reference associated with it the coordinate system and clock.
Body reference Call the body relative to which the position of other bodies consider.
Material point Call the body, the sizes of which in this task can be neglected.
Trajectory They call a mental line, which when its movement describes the material point.
In the form of the trajectory, the movement is divided into:
but) straightforward - The trajectory is a straight line;
b) curvilinear - The trajectory is a segment of the curve.
Way - This is the length of the trajectory, which the material point describes during this period of time. This is a scalar value.
Move - This is a vector connecting the initial position of the material point with its end position (see Fig.).
It is very important to understand what the path differs from moving. The most important difference is that moving is a vector with the beginning at the point of departure and with the end at the destination point (at the same time it absolutely no matter how the route this movement was performed). And the path is, the set, the scalar value reflecting the length of the trajectory passed.
Uniform straight movement Call traffic at which the material point for any equal intervals of time makes the same movement
Speed \u200b\u200bof uniform rectilinear movement call the ratio of movement to the time for which this move happened:
For uneven movements, we use the concept mid speed. Often enter the average speed as a scalar value. This is the speed of such a uniform movement, in which the body passes the same path for the same time as under uneven movement:
Instant speed Call the body rate at this point of the trajectory or at the moment.
Equal asked straight movement - This is a straightforward movement in which instantaneous speed in any equal periods varies to the same magnitude.
The dependence of the body coordinates from time to a uniform straight line movement is: x \u003d x 0 + V x Twhere X 0 is the initial coordinate of the body, V x is the speed of movement.
Frequent fall Called an equilibrium movement with constant acceleration g \u003d 9.8 m / s 2not dependent on the mass of the falling body. It occurs only under the action of gravity.
The speed at a free fall is calculated by the formula:
Vertical movement is calculated by the formula:
One of the types of motion of the material point is the movement around the circumference. With this movement, the body speed is aimed at a tangent, conducted to the circle at that point where the body is located (linear speed). It is possible to describe the position of the body on the circle using a radius conducted from the center of the circumference to the body. The movement of the body when driving around the circle is described by the rotation of the circle radius connecting the center of the circle with the body. The ratio of the angle of rotation of the radius by the interval during which this turn occurred, characterizes the speed of moving the body around the circumference and is called angular velocity ω.:
Angle speed is associated with linear speed by the ratio
where R is the radius of the circle.
The time for which the body describes the full turn is called a period of circulation. The value, reverse period - the frequency of circulation - ν
Since with a uniform movement around the circle, the speed module does not change, but the direction of speed changes, with such a movement there is an acceleration. He's called centripetal accelerationIt is directed along the radius to the center of the circle:
Basic concepts and laws of speakers
Part of the mechanics studying the reasons that caused the acceleration of bodies is called dynamics
The first law of Newton:
There are such reference systems relative to which the body retains its speed constant or resting if other bodies or the effect of other bodies are compensated for it.
Body property to maintain the state of rest or uniform rectilinear movement with balanced external forces acting on it, called inertia. The phenomenon of preserving the velocity of the body under balanced external forces is called inertia. Inertial reference systems Called systems in which the first Newton law is performed.
The principle of reliability of Galilee:
in all inertial reference systems under the same initial conditions, all mechanical phenomena proceed equally, i.e. obey the same laws
Weight - this is a measure of inertness of the body
Force - This is a quantitative measure of interaction tel.
The second law of Newton:
The force acting on the body is equal to the product of body weight on acceleration, reported by this force:
$ F↖ (→) \u003d M⋅A↖ (→) $
The addition of forces is to find the resultant multiple forces, which produces the same action as several simultaneous forces.
The third Newton law:
Forces with which two bodies act on each other are located on one straight line, are equal to the module and are opposite to the direction:
$ F_1↖ (→) \u003d -F_2↖ (→) $
III Newton law emphasizes that the effect of bodies to each other is the nature of interaction. If the body A acts on the body B, then the body B acts on the body A (see Fig.).
Or in short, the strength of action is equal to the power of opposition. Often the question arises: why the horse pulls Sani, if these bodies interact with equal forces? This is only possible at the expense of interaction with the third body - the earth. The force with which the hooves rests in the ground should be greater than the power of friction of Sanya about the Earth. Otherwise, the hooves will slip, and the horse will not move from the place.
If the body is subject to deformation, then there are forces that prevent this deformation. Such forces are called forces of elasticity.
Law Guka. Record in the form
where k is the rigidity of the spring, X is the deformation of the body. The sign "-" indicates that power and deformation are directed in different directions.
When the bodies are moved relative to each other, there are forces that impede movement. These forces are called friction forces. Disassemble the friction of peace and friction slip. Slip friction force Calculated by the formula
where n is the reaction force of the support, μ is the friction coefficient.
This force does not depend on the area of \u200b\u200brubbing tel. The friction coefficient depends on the material from which the bodies and the quality of the processing of their surface are made.
Friction of rest It occurs if the bodies do not move relative to each other. The friction force of rest can change from zero to some maximum value.
Gravitational forces Call the forces with which any two bodies are attracted to each other.
The law of world gravity:Any two bodies are attracted to each other with force, directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Here R is the distance between the bodies. The law of global gravity in this form is true or for material points or for the bodies of a spherical shape.
Weight body Call the strength with which the body presses on the horizontal support or stretches the suspension.
Gravity - This is the force with which all the bodies are attracted to the ground:
With a fixed support, the body weight is equal to the module of gravity:
If the body moves vertically with acceleration, then its weight will change.
When the body moves with acceleration directed up, its weight
It can be seen that the weight of the body is greater than the weight of the resting body.
When the body moves with acceleration directed down, its weight
In this case, the body weight is less than the weight of the resting body.
Uncomfortable This is the movement of the body, in which its acceleration is equal to accelerating the free fall, i.e. a \u003d g. This is possible if only one force is valid on the body - the strength of gravity.
Artificial satellite land - This is a body having a V1 speed sufficient to move around the circumference around the Earth
Only one force is valid for the Earth's satellite - the strength of gravity directed towards the center of the Earth
First cosmic speed - This is the speed that the body must be informed so that it addresses around the planet on a circular orbit.
Where R is the distance from the center of the planet to the satellite.
For the Earth, near her surface, the first cosmic speed is equal
1.3. Basic concepts and laws of statics and hydrostatics
The body (material point) is in a state of equilibrium, if the vector sum of the forces acting on it is zero. Distinguish 3 types of equilibrium: sustainable, unstable and indifferent. If when the body is removed from the equilibrium position, there are forces seeking to return this body back, it sustainable equilibrium. If there are forces seeking to lead the body even further from the equilibrium position, this unstable position; If no strength occurs - indifferent (See Fig. 3).
When it comes not about the material point, but about the body that can have an axis of rotation, then to achieve an equilibrium position, in addition to zero, the sums of the forces acting on the body are necessary that the algebraic sum of the moments of all forces acting on the body have been zero.
Here D is a strength. Shoulder power D Call the distance from the axis of rotation to the line of action.
Lever equilibrium condition:
the algebraic sum of the moments of all the torments of the body is zero.
Pressure They call the physical value equal to the ratio of force acting on the platform perpendicular to this force, to the area of \u200b\u200bthe site:
For liquids and gases fair pascal Law:
Pressure applies in all directions unchanged.
If the liquid or gas is in the gravity field, then each advanced layer presses on the following and as the inside of the liquid or gas is immersed. For liquids
where ρ is the density of the fluid, H is the depth of penetration into the liquid.
The homogeneous liquid in the reporting vessels is set at one level. If in the knee of the reporting vessels, pour liquid with different densities, the liquid with a greater density is set at a smaller height. In this case
The height of the pillars of the liquid is inversely proportional to the densities:
Hydraulic Press It is a vessel filled with oil or other liquid, in which two holes are cut, closed with pistons. Pistons have a different area. If one piston is attached to some power, the force attached to the second piston turns out to be different.
Thus, the hydraulic press serves to transform the amount of force. Since the pressure under pistons should be the same, then 
Then A1 \u003d A2.
On the body, immersed in liquid or gas, from the side of this liquid or gas acting upwards the ejecting force called force of Archimedes
The magnitude of the pushing force sets archimedes Act: The body, immersed in liquid or gas, acts the ejecting force, directed vertically upwards and equal weight of the liquid or gas displaced by the body:
where ρ liquid is the density of the liquid into which the body is immersed; V Pogot - the volume of the submersible part of the body.
Body swimming condition - The body floats in a liquid or gas when the pushing force acting on the body is equal to the strength of gravity acting on the body.
1.4. Conservation laws
Pulse Body Call physical quantity equal to the product of body weight at its speed:Pulse - vector magnitude. [p] \u003d kg · m / s. Along with the body pulse often enjoy pulse force. This is a work of force at her time
Changing the body pulse is equal to the impulse of the body acting on this. For an isolated body system (system, the body of which only interacts with each other) is performed law of preserving impulse: The sum of the impulses of the body of an isolated system to interaction is equal to the sum of the pulses of the same bodies after interaction.
Mechanical work They call the physical value, which is equal to the product of force acting on the body, to move the body and on the cosine of the angle between the direction of force and movement:
Power - This is a job performed per unit of time:
Body ability to perform work characterize the value called energy. Mechanical energy is divided by kinetic and potential. If the body can work at the expense of its movement, they say it possesses kinetic energy. The kinetic energy of the progressive motion of the material point is calculated by the formula
If the body can work due to changes in its position relative to other bodies or by changing the position of body parts, it possesses potential energy. An example of potential energy: the body raised above the earth, its energy is calculated by the formula
where h is the height of the lifting
Spring energy:
where k is the coefficient of the rigidity of the spring, X is the absolute deformation of the spring.
The amount of potential and kinetic energy is mechanical energy. For an isolated system bodies in the mechanics fair mechanical energy conservation law: If there are no friction forces between the bodies of the insulated system (or other forces leading to the scattering of energy), then the amount of mechanical energy of the bodies of this system does not change (the law of conservation of energy in the mechanics). If the friction force between the bodies of the isolated system is, then when the interaction is part of the mechanical energy, the bodies turn into internal energy.
1.5. Mechanical oscillations and waves
Oscillations Called movements with one or another degree of time recipes. The oscillations are called periodic if the values \u200b\u200bof the physical quantities changed during the oscillation process are repeated at equal intervals.Harmonic oscillations These oscillations are called in which the fluctuating physical value of X varies according to the law of sine or cosine, i.e.
The value A equal to the greatest absolute value of the fluctuating physical value of X is called amplitude oscillations. The expression α \u003d ωt + φ determines the value of x at a given time and is called the oscillation phase. T. The time is called for which the oscillating body makes one complete oscillation. Frequency of periodic oscillations Call the number of full oscillations performed per unit of time:
Frequency is measured in C -1. This unit is called Hertz (Hz).
Mathematical pendulum It is called a material point M, suspended on a weightless unpretentious thread and performing oscillations in the vertical plane.
If one end of the spring is fixed motionless, and to another end it is to attach some body mass m, then when removing the body from the position of the equilibrium spring, the body fluctuations on the spring in the horizontal or vertical plane will occur. Such a pendulum is called spring.
Period of oscillations of the mathematical pendulum Determined by the formula 
Where L is the length of the pendulum.
Period of cargo oscillations on the spring Determined by the formula 
where k is the rigidity of the spring, M is the weight of the cargo.
Distribution of oscillations in elastic media.
The medium is called elastic, if there are interaction forces between its particles. The waves are called the process of distribution of oscillations in elastic media.
The wave is called transverseIf the medium particles fluctuate in directions perpendicular to the direction of the wave propagation. The wave is called longitianIf the oscillations of the medium particles occur in the direction of the spread of the wave.
Wavelength It is called the distance between the two closest dots, fluctuating in the same phase:
where V is the speed of wave propagation.
Sound waves Called waves, fluctuations in which occur with frequencies from 20 to 20,000 Hz.
Sound speed is different in different environments. Sound speed in air is 340 m / c.
Ultrasonic waves Called waves, the frequency of oscillations in which exceeds 20,000 Hz. Ultrasonic waves are not perceived by human ear.
With rectilinear equilibrium movement body
- moves along the conditional straight line,
- its speed gradually increases or decreases,
- during equal intervals, the speed changes to an equal value.
For example, a car from the state of rest is starting to move on a straight road, and to speed, say, 72 km / h, it moves equatively. When the specified speed is achieved, then the car is moving without changing the speed, i.e. evenly. With an equilibrium movement, its speed increased from 0 to 72 km / h. And let each second movement speed increased by 3.6 km / h. Then the time of the equivalent movement of the car will be equal to 20 seconds. Since the acceleration in SI is measured in meters per second in a square, it is necessary to accelerate 3.6 km / h per second to translate into the appropriate units of measurement. It will be equal to (3.6 * 1000 m) / (3600 C * 1 s) \u003d 1 m / s 2.
Suppose, after a while driving with a constant speed, the car began to slow down to stop. Movement during braking was also equal to the zoom (for equal intervals, the speed decreased to the same value). In this case, the speed of the acceleration will be opposite to the velocity vector. It can be said that acceleration is negative.
So, if the initial body speed is zero, its speed through time in T seconds will be equal to the work of acceleration at this time:
When the body drops, the acceleration of the free fall, and the body rate at the very surface of the Earth will be determined by the formula:
If the current body speed and the time that required is known to develop such a speed from the rest state, it is possible to determine the acceleration (i.e., how quickly the speed changed), separating the speed at the time:
However, the body could start an equilibrium movement not from the state of rest, but already possessing some speed (or the initial speed was given). Suppose you throw a stone from the tower vertically down with the power app. There is an acceleration of free drop on such a body, equal to 9.8 m / s 2. However, your power gave a stone even speeds. Thus, the final speed (at the time of the clue of the Earth) will be folded from the speed that has developed as a result of acceleration and initial speed. Thus, the final speed will be in the formula:
However, if the stone threw up. That initial speed is directed upward, and the acceleration of the free fall down. That is, velocity vector directed in opposite sides. In this case (as well as in braking), the product of acceleration for a while must be deducted from the initial speed:
We obtain from these formulas of the acceleration formula. In case of acceleration:
aT \u003d V - V 0
a \u003d (V - V 0) / T
In case of braking:
aT \u003d V 0 - V
a \u003d (v 0 - V) / T
In the case when the body is equally stopped, then at the time of stopping its speed is 0. Then the formula is reduced to this type:
Knowing the initial body speed and acceleration of braking, the time is determined through which the body will stop:
Now bring out formulas for the path that the body passes with rectilinear equative movement. A graph of the speed dependence on time with a straight-line uniform movement is a segment parallel to the axis of time (the X axis is usually taken). The path is calculated as the area of \u200b\u200bthe rectangle under the segment. That is the multiplication of speed at the time (S \u003d VT). With rectilinear equative movement, the schedule is straight, but not the parallel axis of the time. This direct either increases in the case of acceleration, or decreases in the case of braking. However, the path is also defined as the area of \u200b\u200bthe figure under the schedule.
With straight equalized movement, this figure is a trapezium. Its bases are a segment on the Y axis (speed) and a segment connecting the point of the end of the graph with its projection on the X axis. Side parties are the chart of the speed dependence on time and its projection on the X axis (time axis). The projection on the X axis is not only a side side, but also the height of the trapezium, since it is perpendicular to its grounds.
As is known, the area of \u200b\u200bthe trapezium is equal to the height of the base. The length of the first base is equal to the initial speed (V 0), the length of the second base is equal to the final speed (V), the height is equal to time. Thus we get:
s \u003d ½ * (V 0 + V) * T
Above the formula for the dependence of the final speed from the initial and acceleration (V \u003d V 0 + AT) was given. Therefore, in the path formula, we can replace V:
s \u003d ½ * (V 0 + V 0 + AT) * T \u003d ½ * (2V 0 + AT) * T \u003d ½ * T * 2V 0 + ½ * T * AT \u003d V 0 T + 1 / 2AT 2
So, the path passed is determined by the formula:
s \u003d V 0 T + AT 2/2
(It is possible to come to this formula, considering not the square of the trapezoid, but by the summing area of \u200b\u200bthe rectangle and the rectangular triangle, which is broken.)
If the body starts moving equal to the state of rest (V 0 \u003d 0), then the path formula is simplified to S \u003d AT 2/2.
If the acceleration vector was opposite to the speed, the AT 2/2 product must be deducted. It is clear that the difference V 0 T and AT 2/2 should not be negative. When it becomes equal to zero, the body will stop. Braking will be found. Above the formula of time until a complete stop (T \u003d V 0 / A). If we substitute the value of T in the path formula, then the deceleration path is given to this formula.
In this topic, we will look at a very special kind of uneven movement. Based on the opposition to uniform motion, uneven movement is a movement with a different speed, along any trajectory. What is the feature of the equilibrium movement? This is uneven movement, but which "EXCALLY accelerated". Acceleration is associated with increasing speed. Recall the word "equal", we obtain an equal increase in speed. And how to understand the "equal increase in speed", how to evaluate the speed equals or not? To do this, we will need to retrieve time, evaluate the speed through the same time interval. For example, the car begins to move, in the first two seconds it develops speed up to 10 m / s, over 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 10 m / s. This is an equilibrium movement.
The physical quantity characterizing how many times the speed increases is called acceleration.
Is it possible to consider the cyclist equivalent if after stopping in the first minute its speed 7km / h, in the second - 9km / h, in the third 12km / h? It is impossible! The cyclist accelerates, but not the same, first accelerated at 7km / h (7-0), then 2 km / h (9-7), then 3 km / h (12-9).
Usually, the movement with an increasing velocity is called an accelerated movement. Movement with a decreasing speed - slow motion. But physics, any movement with a changing rate is called accelerated movement. Does the car begins from the scene (the speed grows!), Or slows down (speed decreases!), In any case, it moves with acceleration.
Equal asked movement - this is a body movement, in which its speed in any equal periods of time changes (may increase or decrease) the same
Acceleration of body
Acceleration characterizes speed change. This is the number that changes the speed per second. If the acceleration of the body in the module is large, it means that the body quickly gains speed (when it is accelerated) or quickly loses it (when braking). Acceleration - This is a physical vector magnitude, numerically equal to the ratio of the speed change by the period of time during which this change occurred.
We define acceleration in the next task. In the initial moment of time, the height rate was 3 m / s, at the end of the first second the speed of the ship was 5 m / s, at the end of the second - 7m / s, at the end of the third 9 m / s, etc. Obviously. But how did we define? We consider speed difference in one second. In the first second, 5-3 \u003d 2, in the second second 7-5 \u003d 2, in the third 9-7 \u003d 2. And what if there are no speeds for every second? Such a task: the initial height rate of 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, it is necessary 11-7 \u003d 4, then 4/2 \u003d 2. The speed difference we divide for a period of time. 
This formula is most often used in solving tasks in a modified form:
The formula is not written in a vector form, so the sign "+" write when the body is accelerated, the sign "-" - when slows down.
The direction of the acceleration vector
The direction of the acceleration vector is shown in the drawings.
In this picture, the machine moves in the positive direction along the OX axis, the speed vector always coincides with the direction of movement (directed to the right). When the vector acceleration coincides with the direction of speed, this means that the car is accelerated. Acceleration is positive.
When acceleration, the acceleration direction coincides with the direction of speed. Acceleration is positive.
In this figure, the machine moves in the 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 machine is slow. Acceleration is negative.
When braking, the direction of acceleration is opposite to the direction of speed. Acceleration is negative.
We'll figure it out why when braking acceleration is negative. For example, the motor ship for the first second dropped the speed from 9m / s to 7m / s, for the second second to 5m / s, for the third to 3m / s. Speed \u200b\u200bchanges to "-2m / s". 3-5 \u003d -2; 5-7 \u003d -2; 7-9 \u003d -2m / s. This is where the negative acceleration value appears.
When solving tasks, If the body slows down, the acceleration into the formula is substituted with the "minus" sign !!!
Move with equivalent movement
An additional formula called strong
Formula in coordinates
Connection with average speed
With an equilibrium movement, the average speed can be calculated as a medium-ray initial and final speed.
This rule follows a formula that is very convenient to use when solving many tasks
Ratio path
If the body moves equally, the initial speed is zero, the paths passing into sequential equal intervals are related as a sequential number of odd numbers.
The main thing is to remember
1) what is an equidalized movement;
2) what characterizes acceleration;
3) Acceleration - vector. If the body accelerates the acceleration is positive, if slows down - the acceleration is negative;
3) the 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 is slowed to the south. How are trains accelerate?
Equally north. Because at the first train, the acceleration coincides towards the movement, and the second - the opposite movement (it slows down).
In general equal asked movement This movement is called, in which the acceleration vector remains unchanged by module and direction. An example of such a movement is the movement of a stone thrown at some angle to the horizon (excluding air resistance). At any point of the trajectory, the acceleration of the stone is equal to the acceleration of the free fall. For the kinematic description of the movement of the stone, the coordinate system is convenient to choose so that one of the axes, such as the axis Oy.was directed parallel to the acceleration vector. Then the curvilinear movement of the stone can be represented as the sum of two movements - rectilinear equative movement Along the axis Oy. and uniform rectilinear movement in the perpendicular direction, i.e. along the axis OX. (Fig. 1.4.1).
Thus, the study of an equilibrium movement is reduced to the study of a straight-line equivalent movement. In the case of a straightforward movement, velocity and acceleration vectors are directed along a direct movement. Therefore, the speed υ and acceleration a. In projections on the direction of movement, it can be considered as algebraic values.
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Figure 1.4.1. Projection of velocity vectors and acceleration on coordinate axes. a.x. = 0, a.y. = -g. |
With an equilibrium straight line movement, the body speed is determined by the formula
(*)
In this formula υ 0 - body speed t. = 0 (starting speed ), a. \u003d Const - acceleration. On the speed chart υ ( t.) This dependence has the form of a straight line (Fig. 1.4.2).
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Figure 1.4.2. Speed \u200b\u200bgraphics of equivalent movement |
At the inclination of the speed schedule can be determined a. Body. The corresponding constructions are made in Fig. 1.4.2 For graph I. Acceleration is numerically equal to the attitude of the side of the triangle ABC:
The greater the angle of β, which forms a speed chart with the axis of time, that is, the more slope of the graph ( steep), the greater the acceleration of the body.
For graph I: υ 0 \u003d -2 m / s, a. \u003d 1/2 m / s 2.
For graph II: υ 0 \u003d 3 m / s, a. \u003d -1/3 m / s 2
Speed \u200b\u200bgraph also allows you to determine the projection of movement s. bodies for a while t.. We highlight on the axis of time some small period of time Δ t.. If this period of time is sufficiently small, then the change in the speed during this gap is small, i.e., the movement during this period can be considered uniform at a certain average rate, which is equal to the instantaneous velocity υ of the body in the middle of the gap Δ t.. Consequently, moving Δ s. During Δ. t. It will be equal to Δ. s. = υΔ t.. This movement is equal to the area of \u200b\u200bthe shaded strip (Fig. 1.4.2). After breaking the time interval from 0 to some moment t. for small gaps Δ t., I get that moving s. For a given time t. with an equilibrium straight movement equal to the area of \u200b\u200bthe trapez ODEF.. The corresponding constructions are made for graph II in Fig. 1.4.2. Time t. It is taken equal to 5.5 s.
Since υ - υ 0 \u003d aT., final formula for moving s. bodies with evenly accelerated motion during the period of time from 0 to t. Wrong in the form:
(**)
To find the coordinate y. Body at any time t. Need to the initial coordinate y. 0 Add Moving During t.:
(***)
This expression is called the law of equal to the movement .
When analyzing the equilibrium movement, it sometimes occurs a problem of determining the movement of the body according to the specified values \u200b\u200bof the initial υ 0 and the final υ speeds and acceleration a.. This task can be solved using equations written above by excluding time t.. The result is written in the form
From this formula, it is possible to obtain an expression to determine the final velocity of the body, if the initial speed υ 0 is known, acceleration a. and moving s.:
If the initial velocity υ 0 is zero, these formulas take the form
It should once again pay attention to the fact that those included in the formula of the equilibrium straight line of the magnitude of υ 0, υ, s., a., y. 0 are algebraic values. Depending on the specific type of movement, each of these quantities can take both positive and negative values.
1. A real mechanical movement is a movement with a changing rate. Movement whose speed is changed, called uneven movement.
With uneven motion, the coordinate of the tool can already be determined but the formula \\ (x \u003d x_0 + v_xt \\), since the value of the speed is not constant. Therefore, to characterize the speed of changing the position of the body over time, with uneven movement, the value called the value called average speed.
Average speed \\ (\\ VEC (V) _ (cf) \\) uneven movement is called a physical value equal to the ratio of the movement \\ (\\ VEC (S) \\) by the time \\ (t \\) for which it happened : \\ (\\ VEC (V) _ (CP) \u003d \\ FRAC (S) (T) \\).
The recorded formula defines the average speed as a vector value. For practical purposes, this formula can be used to determine the average speed module only in the case when the body moves along a straight one way. If you need to determine the average velocity of the car from Moscow to St. Petersburg and back to calculate the flow of gasoline, then this formula cannot be applied, since the movement in this case is zero and the average speed is also zero. Therefore, in practice, when determining the average speed, the value of equal the path ratio \\ (L \\) by the time \\ (t \\), for which this path is passed: \\ (v_ (cp) \u003d \\ FRAC (L) (T) \\). This speed is usually called an average way rate.
2. It is important that, knowing the average rate of uneven movement on any site of the trajectory, it is impossible to determine the position of the body on this trajectory at any time. For example, if the average vehicle speed is 2 hours 50 km / h, then we cannot say where it was 0.5 hours from the beginning of the movement, after 1 hour, 1.5 hours, etc., since he could The first half an hour move at a speed of 80 km / h, then stand for a while, and for a while go to the traffic jam at a speed of 20 km / h.
3. Moving along the trajectory, the body passes consistently all its points. At each point of the trajectory it is at certain points of time and has some speed.
Instant speed is called the body speed at the moment of time at this point of the trajectory.
Suppose some body makes an uneven rectilinear movement (Fig. 17), its speed at the point o can be determined as follows: I will select a section of AB on the trajectory, inside the point O. Moving the body in this section - \\ (\\ VEC (S) _1 \\) During the time \\ (T_1 \\). The average speed of movement in this area - \\ (\\ VEC (V) _ (CP.1) \u003d \\ FRAC (S_1) (T_1) \\). Reduce the movement of the body. Suppose that it is equal to \\ (\\ Vec (S) _2 \\), and the movement time - \\ (t_2 \\). Then the average speed during this time: \\ (\\ VEC (V) _ (CP.2) \u003d \\ FRAC (S_2) (T_2) \\). Even reducing movement, average speed on this area: \\ (\\ VEC (V) _ (CP.3) \u003d \\ FRAC (S_3) (T_3) \\).
With a further reduction in the movement and, accordingly, the time of the body movement, they will become such small that the device, such as a speedometer, will cease to fix the change in the speed, and the movement during this low time interval can be considered uniform. The average speed in this area is the instantaneous body speed in T.O.
In this way, instant speed is called a vector physical value equal to the ratio of small displacement (\\ (\\ deelta (\\ vec (s)) \\)) to a small period of time \\ (\\ Delta (T) \\), for which this movement happened: \\ (\\ VEC (V) \u003d \\ FRAC (\\ Delta (S)) (\\ Delta (T)) \\).
4. One of the types of uneven movement is equivalent movement. Equally asked motion is called a movement at which the body's speed in any equal periods varies to the same value.
The words "any equal periods of time" mean that there would be no equal intervals of the time (2 s, 1 s, fraction of seconds, etc.), the speed will always change the same. In this case, its module can both increase and decrease.
5. The characteristic of the equilibrium movement, in addition to speed and movement, is acceleration.
Let in the initial moment of time \\ (t_0 \u003d 0 \\) the body speed is equal to \\ (\\ VEC (V) _0 \\). At some point in time \\ (t \\) it became equal to \\ (\\ VEC (V) \\). Changing the velocity over time \\ (T-T_0 \u003d T \\) is equal to \\ (\\ VEC (V) - \\ VEC (V) _0 \\) (Fig.18). Changing the speed per unit of time is: \\ (\\ FRAC (\\ VEC (V) - \\ VEC (V) _0) (t) \\). This value is the acceleration of the body, it characterizes the speed of change of speed \\ (\\ VEC (A) \u003d \\ FRAC (\\ VEC (V) - \\ VEC (V) _0) (T) \\).
Acceleration of body With an equilibrium movement - a vector physical value equal to the ratio of a change in body velocity by a period of time, for which this change occurred.
Acceleration unit \\ ([a] \u003d [v] / [t] \\); \\ ([A] \\) \u003d 1 m / s / 1 s \u003d 1 m / s 2. 1 m / s 2 is such an acceleration at which the body's speed varies in 1 s per 1 m / s.
The acceleration direction coincides with the direction of movement rate, if the speed module increases, the acceleration is directed oppositely the speed of movement if the speed module decreases.
6. Converting the acceleration formula, you can obtain an expression for body velocity with an equalized movement: \\ (\\ VEC (V) \u003d \\ VEC (V) _0 + \\ VEC (A) T \\). If the initial body rate \\ (v_0 \u003d 0 \\), then \\ (\\ VEC (V) \u003d \\ VEC (A) T \\).
To determine the value of the speed of an equivalent movement at any time, the equation should be recorded for the speed projection on the axis oh. It has the form: \\ (v_x \u003d v_ (0x) + A_XT \\); If \\ (v_ (0x) \u003d 0 \\), then \\ (v_x \u003d a_xt \\).
7. As can be seen from the speed formula of an equilibrium movement, it depends linearly on time. A graph of the dependence of the speed module is direct, which is some angle with the axis of the abscissa (axis of time). Figure 19 shows the charts of the velocity module dependence.
Chart 1 corresponds to the movement without initial speed with acceleration, directed the same as the speed; Schedule 2 - movement with the initial speed \\ (v_ (02) \\) and with acceleration directed in the same way as the speed; Schedule 3 - motion with the initial speed \\ (v_ (03) \\) and with acceleration directed towards the opposite direction of speed.
8. The figure shows the graphs of the dependence of the speed of the speed of equivalent movement from time to time (Fig. 20).
Chart 1 corresponds to the movement without initial speed with acceleration directed along the positive direction of the X axis; Schedule 2 - movement with the initial speed \\ (v_ (02) \\), with acceleration and speed directed along the positive direction of the X axis; Schedule 3 - motion at the initial speed \\ (v_ (03) \\): until the time \\ (T_0 \\), the direction of the speed coincides with the positive direction of the X axis, the acceleration is directed in the opposite direction. At the time of time \\ (t_0 \\), the speed is zero, and then the speed, and the acceleration is directed to the side opposite to the positive direction of the X axis.
9. Figure 21 shows the graphs of the dependence of the projection of the acceleration of the equilibrium movement.
Chart 1 corresponds to the movement, the projection of the acceleration of which is positive, a graph 2 - movement, the projection of the acceleration of which is negative.
10. The formula for moving the body with an equilibrium motion can be obtained using a graph of the projection of the velocity of this movement from time to time (Fig. 22).
We highlight the small section \\ (ab \\) on the chart and omit the perpendicular from the points \\ (a \\) and \\ (b \\) to the abscissa axis. If the time interval \\ (\\ deelta (t) \\), the corresponding area \\ (CD \\) on the abscissa axis is small, then we can assume that the speed during this period of time does not change and the body moves evenly. In this case, the figure \\ (Cabd \\) differs little from the rectangle and its area is numerically equal to the projection of the body movement during the corresponding segment \\ (CD \\).
On such strips you can smash the whole figure of the OAUB, and its area is equal to the sum of the squares of all strips. Consequently, the projection of the body movement during the \\ (t \\) is numerically equal to the area of \u200b\u200bthe OAAVS trapezium. The area of \u200b\u200bthe trapezium is equal to the work of its bases to the height: \\ (S_X \u003d \\ FRAC (1) (2) (OA + BC) OC \\).
As can be seen from the figure, \\ (oa \u003d v_ (0x), Bc \u003d V_x, OC \u003d T \\). It follows that the projection of movement is expressed by the formula \\ (S_X \u003d \\ FRAC (1) (2) (V_ (0x) + v_x) T \\). Since \\ (v_x \u003d v_ (0x) + A_ (XT) \\), then \\ (S_X \u003d \\ FRAC (1) (2) (2v_ (0x) + A_XT) T \\)From here \\ (S_x \u003d v_ (0x) T + \\ FRAC (A_XT ^ 2) (2) \\). If the initial speed is zero, the formula has the form \\ (S_x \u003d \\ FRAC (AT ^ 2) (2) \\). The projection of the movement is equal to the coordinate difference \\ (s_x \u003d x-x_0 \\), so: \\ (x-x_0 \u003d v_ (0x) T + \\ FRAC (AT ^ 2) (2) \\), or \\ (x \u003d x_ (0x) + v_ (0x) T + \\ FRAC (AT ^ 2) (2) \\).
The resulting formula allows determining the position (coordinate) of the body at any time, if the initial speed, the initial coordinate and acceleration are known.
11. In practice, the formula or \\ (V ^ 2_x-V ^ 2_ (0x) \u003d 2A_xs_x \\) is used, or \\ (V ^ 2-V ^ 2_ (0) \u003d 2As \\).
If the initial body velocity is zero, then: \\ (V ^ 2_x \u003d 2a_xs_x \\).
The resulting formula allows you to calculate the braking path of vehicles, i.e. The path that drives, for example, a car to a complete stop. With some acceleration of motion, which depends on the mass of the car and the power of the engine thrust, the braking path is the greater, the greater the initial speed of the car.
Part 1
1. The figure shows the tracks of the dependence of the path and body velocity. What chart corresponds to an equilibrium movement?
2. The car, starting to move from the state of rest but a straight road, for 10 ° C acquired a speed of 20 m / s. What is the acceleration of the car?
1) 200 m / s 2
2) 20 m / s 2
3) 2 m / s 2
4) 0.5 m / s 2
3. The figures are presented graphs of the dependence of the coordinate from time to four bodies moving along the axis \\ (ox \\). In what of the bodies at the time of time \\ (t_1 \\) the speed of movement is zero?
4. The figure presents a graph of the dependence of the projection of acceleration from time for the body, moving straightly along the axis \\ (ox \\).
Equivalent movement corresponds to the site
1) only OA
2) only av
3) only OA and Sun
4) only CD
5. When studying the equilibrium movement, the path passed by the body from the state of rest for successive equal intervals (in the first second, for the second second, etc.) was measured. The data obtained are shown in the table.
What is the path traveled by the body for the third second?
1) 4 m
2) 4.5 m
3) 5 m
4) 9 m
6. The figure shows the graphs of the dependence of the speed of movement from time for four bodies. Bodies are moving in a straight line.
For which (s) from the tel - 1, 2, 3 or 4 - the speed of the acceleration is directed oppositely the velocity vector?
1) only 1
2) only 2
3) only 4
4) 3 and 4
7. Using a graph of the velocity of the body of the body from time to time, determine its acceleration.