mechanical movement. Trajectory
Mechanical movement called a change in the position of a body relative to other bodies
Reference system call the body of reference, the coordinate system associated with it and the clock.
Reference body called the body, relative to which the position of other bodies is considered.
material point is called a body whose dimensions in this problem can be neglected.
trajectory called a mental line, which, during its movement, describes a material point.
According to the shape of the trajectory, the movement is divided into:
A) rectilinear- the trajectory is a straight line segment;
b) curvilinear- the trajectory is a segment of the curve.
Path- this is the length of the trajectory that the material point describes for a given period of time. This is a scalar value.
moving is a vector connecting the initial position of a material point with its final position (see Fig.).
It is very important to understand how path differs from movement. The most important difference is that the movement is a vector with the beginning at the point of departure and with the end at the destination (it does not matter at all which route this movement took). And the path is, on the contrary, a scalar value that reflects the length of the trajectory traveled.
Uniform rectilinear movement called a movement in which a material point makes the same movements for any equal intervals of time
The speed of uniform rectilinear motion called the ratio of the movement to the time for which this movement occurred:
For non-uniform motion use the concept average speed. Often the average speed is entered as a scalar value. This is the speed of such uniform motion, in which the body travels the same path in the same time as with uneven motion:
instantaneous speed called the speed of the body at a given point in the trajectory or at a given time.
Uniformly accelerated rectilinear motion- this is a rectilinear movement in which the instantaneous speed for any equal intervals of time changes by the same amount
The dependence of the body coordinate on time in uniform rectilinear motion has the form: x = x 0 + V x t, where x 0 is the initial coordinate of the body, V x is the speed of movement.
free fall called uniformly accelerated motion with constant acceleration g \u003d 9.8 m / s 2 independent of the mass of the falling body. It occurs only under the influence of gravity.
The speed in free fall is calculated by the formula:
Vertical displacement is calculated by the formula:
One of the types of movement of a material point is movement in a circle. With such a movement, the speed of the body is directed along a tangent drawn to the circle at the point where the body is located (linear speed). The position of a body on a circle can be described using a radius drawn from the center of the circle to the body. The movement of a body when moving along a circle is described by turning the radius of the circle connecting the center of the circle with the body. The ratio of the angle of rotation of the radius to the time interval during which this rotation occurred characterizes the speed of movement of the body around the circle and is called angular velocity ω:
The angular velocity is related to the linear velocity by the relation
where r is the radius of the circle.
The time it takes for a body to complete one revolution is called circulation period. The reciprocal of the period - the frequency of circulation - ν
Since with uniform motion along a circle, the velocity module does not change, but the direction of the velocity changes, with such motion there is an acceleration. He is called centripetal acceleration, it is directed along the radius to the center of the circle:
Basic concepts and laws of dynamics
The part of mechanics that studies the causes that caused the acceleration of bodies is called dynamics
Newton's first law:
There are such frames of reference with respect to which the body keeps its speed constant or is at rest if no other bodies act on it or the action of other bodies is compensated.
The property of a body to maintain a state of rest or uniform rectilinear motion with balanced external forces acting on it is called inertia. The phenomenon of maintaining the speed of a body with balanced external forces is called inertia. inertial reference systems called systems in which Newton's first law is satisfied.
Galileo's principle of relativity:
in all inertial reference systems under the same initial conditions, all mechanical phenomena proceed in the same way, i.e. obey the same laws
Weight is a measure of the body's inertia
Force is a quantitative measure of the interaction of bodies.
Newton's second law:
The force acting on a body is equal to the product of the mass of the body and the acceleration imparted by this force:
$F↖(→) = m⋅a↖(→)$
The addition of forces is to find the resultant of several forces, which produces the same effect as several simultaneously acting forces.
Newton's third law:
The forces with which two bodies act on each other are located on the same straight line, are equal in magnitude and opposite in direction:
$F_1↖(→) = -F_2↖(→) $
Newton's III law emphasizes that the action of bodies on each other has the character of interaction. If body A acts on body B, then body B also acts on body A (see figure).
Or in short, the force of action is equal to the force of reaction. The question often arises: why does a horse pull a sled if these bodies interact with equal forces? This is possible only through interaction with the third body - the Earth. The force with which the hooves rest on the ground must be greater than the friction force of the sled on the ground. Otherwise, the hooves will slip and the horse will not budge.
If the body is subjected to deformation, then forces arise that prevent this deformation. Such forces are called elastic forces.
Hooke's law written in the form
where k is the stiffness of the spring, x is the deformation of the body. The "−" sign indicates that the force and deformation are directed in different directions.
When bodies move relative to each other, forces arise that impede movement. These forces are called friction forces. Distinguish between static friction and sliding friction. sliding friction force calculated according to the formula
where N is the reaction force of the support, µ is the coefficient of friction.
This force does not depend on the area of the rubbing bodies. The coefficient of friction depends on the material from which the bodies are made and the quality of their surface treatment.
Friction of rest occurs when the bodies do not move relative to each other. The static friction force can vary from zero to some maximum value
Gravitational forces called the forces with which any two bodies are attracted to each other.
Law of gravity:any two bodies are attracted to each other with a force that is 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 universal gravitation in this form is valid either for material points or for spherical bodies.
body weight called the force with which the body presses on a horizontal support or stretches the suspension.
Gravity is the force with which all bodies are attracted to the Earth:
With a fixed support, the weight of the body is equal in absolute value to the force of gravity:
If a body moves vertically with acceleration, then its weight will change.
When a body moves with an upward acceleration, its weight
It can be seen that the weight of the body is greater than the weight of the resting body.
When a body moves with downward acceleration, its weight
In this case, the weight of the body is less than the weight of the resting body.
weightlessness is called such a movement of the body, in which its acceleration is equal to the acceleration of free fall, i.e. a = g. This is possible if only one force acts on the body - the force of gravity.
artificial earth satellite is a body with a speed V1 sufficient to move in a circle around the Earth
Only one force acts on the Earth's satellite - gravity, directed towards the center of the Earth
first cosmic speed- this is the speed that must be reported to the body so that it revolves around the planet in a circular orbit.
where R is the distance from the center of the planet to the satellite.
For the Earth, near its surface, the first escape velocity is
1.3. Basic concepts and laws of statics and hydrostatics
A body (material point) is in a state of equilibrium if the vector sum of the forces acting on it is equal to zero. There are 3 types of balance: stable, unstable and indifferent. If, when a body is taken out of equilibrium, forces arise that tend to bring this body back, this stable balance. If forces arise that tend to take the body even further away from the equilibrium position, this precarious position; if no forces arise - indifferent(See Fig. 3).
When we are talking not about a material point, but about a body that can have an axis of rotation, then in order to achieve an equilibrium position, in addition to equality to zero of the sum of forces acting on the body, it is necessary that the algebraic sum of the moments of all forces acting on the body be equal to zero.
Here d is the arm of the force. Shoulder of strength d is the distance from the axis of rotation to the line of action of the force.
Lever equilibrium condition:
the algebraic sum of the moments of all forces rotating the body is equal to zero.
By pressure they call a physical quantity equal to the ratio of the force acting on the site perpendicular to this force to the area of the site:
For liquids and gases is valid Pascal's law:
pressure is distributed in all directions without change.
If a liquid or gas is in the field of gravity, then each higher layer presses on the lower ones, and as the liquid or gas is immersed inside, the pressure increases. For liquids
where ρ is the density of the liquid, h is the depth of penetration into the liquid.
Homogeneous liquid in communicating vessels is set at the same level. If liquid with different densities is poured into the knees of communicating vessels, then the liquid with a higher density is installed at a lower height. In this case
The heights of the liquid columns are inversely proportional to the densities:
Hydraulic Press is a vessel filled with oil or other liquid, in which two holes are cut, closed by pistons. Pistons have different sizes. If a certain force is applied to one piston, then the force applied to the second piston turns out to be different.
Thus, the hydraulic press serves to convert the magnitude of the force. Since the pressure under the pistons must be the same, then 
Then A1 = A2.
A body immersed in a liquid or gas is subjected to an upward buoyant force from the side of this liquid or gas, which is called the power of Archimedes
The value of the buoyant force is set law of Archimedes: a buoyant force acts on a body immersed in a liquid or gas, directed vertically upwards and equal to the weight of the liquid or gas displaced by the body:
where ρ liquid is the density of the liquid in which the body is immersed; V submerged - the volume of the submerged part of the body.
Body floating condition- a body floats in a liquid or gas when the buoyant force acting on the body is equal to the force of gravity acting on the body.
1.4. Conservation laws
body momentum called a physical quantity equal to the product of the mass of the body and its speed:Momentum is a vector quantity. [p] = kg m/s. Along with the momentum of the body, they often use force impulse. It is the product of force times its duration.
The change in momentum of a body is equal to the momentum of the force acting on that body. For an isolated system of bodies (a system whose bodies interact only with each other), law of conservation of momentum: the sum of the impulses of the bodies of an isolated system before the interaction is equal to the sum of the impulses of the same bodies after the interaction.
mechanical work they call a physical quantity that is equal to the product of the force acting on the body, the displacement of the body and the cosine of the angle between the direction of the force and the displacement:
Power is the work done per unit of time.
The ability of a body to do work is characterized by a quantity called energy. Mechanical energy is divided into kinetic and potential. If a body can do work due to its motion, it is said to have kinetic energy. The kinetic energy of the translational motion of a material point is calculated by the formula
If a body can do work by changing its position relative to other bodies or by changing the position of parts of the body, it has potential energy. An example of potential energy: a body raised above the ground, its energy is calculated by the formula
where h is the height of the lift
Compressed spring energy:
where k is the spring constant, x is the absolute deformation of the spring.
The sum of potential and kinetic energy is mechanical energy. For an isolated system of bodies in mechanics, law of conservation of mechanical energy: if friction forces (or other forces leading to energy dissipation) do not act between the bodies of an isolated system, then the sum of the mechanical energies of the bodies of this system does not change (the law of conservation of energy in mechanics). If there are friction forces between the bodies of an isolated system, then during the interaction part of the mechanical energy of the bodies is transferred into internal energy.
1.5. Mechanical vibrations and waves
fluctuations are called movements that have one or another degree of repetition in time. Oscillations are called periodic if the values of physical quantities that change in the process of oscillations are repeated at regular intervals.Harmonic vibrations such oscillations are called in which the oscillating physical quantity x changes according to the law of sine or cosine, i.e.
The value A, equal to the largest absolute value of the oscillating physical quantity x, is called oscillation amplitude. The expression α = ωt + ϕ determines the value of x at a given time and is called the oscillation phase. Period T The time it takes for an oscillating body to make one complete oscillation is called. The frequency of periodic oscillations called the number of complete oscillations per unit of time:
Frequency is measured in s -1 . This unit is called hertz (Hz).
Mathematical pendulum is a material point of mass m suspended on a weightless inextensible thread and oscillating in a vertical plane.
If one end of the spring is fixed motionless, and some body of mass m is attached to its other end, then when the body is taken out of equilibrium, the spring will stretch and the body will oscillate on the spring in a horizontal or vertical plane. Such a pendulum is called a spring pendulum.
The period of oscillation of a mathematical pendulum is determined by the formula 
where l is the length of the pendulum.
The period of oscillation of the load on the spring is determined by the formula 
where k is the stiffness of the spring, m is the mass of the load.
Propagation of oscillations in elastic media.
A medium is called elastic if there are interaction forces between its particles. Waves is the process of propagation of oscillations in elastic media.
The wave is called transverse, if the particles of the medium oscillate in directions perpendicular to the direction of wave propagation. The wave is called longitudinal, if the oscillations of the particles of the medium occur in the direction of wave propagation.
Wavelength the distance between two nearest points oscillating in the same phase is called:
where v is the speed of wave propagation.
sound waves called waves, oscillations in which occur with frequencies from 20 to 20,000 Hz.
The speed of sound is different in different environments. The speed of sound in air is 340 m/s.
ultrasonic waves called waves, the oscillation frequency of which exceeds 20,000 Hz. Ultrasonic waves are not perceived by the human ear.
Mechanics is a branch of physics that studies the laws of motion and interaction of bodies.Kinematics is a branch of mechanics that does not study the causes of the motion of bodies.
mechanical movement- change in the position of the body in space relative to other bodies over time.
A material point is a body whose dimensions under given conditions can be neglected.
Translational called a movement in which all points of the body move in the same way. Translational is a movement in which any straight line drawn through the body remains parallel to itself.
Kinematic characteristics of movement
Trajectory – line of motion. S-way – path length.
S- moving– vector, connecting the initial and final position of the body.
Relativity of motion. Reference system - a set of a reference body, a coordinate system and a device for measuring time (hours)
coordinate systemRectilinear uniform motion called such a movement in which the body for any equal intervals of time makes the same movement.Speed - a physical quantity equal to the ratio of the displacement vector to the time interval during which this displacement occurred.The speed of uniform rectilinear motion is numerically equal to the displacement per unit of time.
Average speed of uneven movement
The main task of mechanics (OZM) is to determine the position of a body in space at any time. Instantaneous speed - speed - of the body at a given moment in time.
The classical law of addition of velocities
The speed of the body in the moving CO is equal to the vector sum of the speed of the body in the stationary CO and the speed of the moving CO
Outline of the lesson on the topic "Generalization and systematization of knowledge on the topic" »
date :
Subject: "Generalization and systematization of knowledge on the topic"Uniform and uneven movement. Addition of speeds»
Goals:
educational : the formation of practical skills in solving problems on the topic “Uneven movement. Addition of speeds";
Educational : improve intellectual skills (observe, compare, reflect, apply knowledge, draw conclusions), develop cognitive interest;
Educational : to instill a culture of mental work, accuracy, to teach to see the practical benefits of knowledge, to continue the formation of communication skills, to cultivate attentiveness, observation.
Lesson type: generalization and systematization of knowledge
Equipment and sources of information:
Isachenkova, L. A. Physics: textbook. for 9 cells. institutions of general avg. education with Russian lang. education / L. A. Isachenkova, G. V. Palchik, A. A. Sokolsky; ed. A. A. Sokolsky. Minsk: Narodnaya Aveta, 2015
Lesson structure:
Organizational moment (5 min)
Updating of basic knowledge (5 min)
Consolidation of knowledge (30 min)
Lesson summary (5 min)
Lesson content
Organizing time
Hello, have a seat! (Checking those present).Today in the lesson we must consolidate the knowledge gained by solving And this means thatLesson topic : « Generalization and systematization of knowledge on the topic " Uniform and uneven movement. Addition of speeds »
Updating of basic knowledge
What is called uniform motion?
What movement is called uneven? Can it be argued that the body moves uniformly if the paths traversed by the body in every hour. the same?
What is the average travel speed? Average travel speed? How are they calculated?
What is the meaning of Galileo's law of addition of velocities?
Consolidation of knowledge
And now let's move on to solving problems:
№ 1
If two bodies move along the same straight line in the same direction with velocities whose modules are and, then the module of the relative velocity of the bodies is always equal to:
a) ; V) ;
b);d);
№ 2
What is the distance traveled by a pedestrian moving at an average ground speed< > = 4.8 per time interval Δt= 0.5 h?
№ 3
The skater ran the first part of the distance in the timeΔ \u003d 20 s with a speed whose module is \u003d 7.6, and the second - in timeΔ t 2 = 36 s at a speed whose modulusv 2 = 9.0. Determinethe average speed of the skater over the entire distance.
№ 4
A car moving along a straight section of a highway at a speed whose modulus is= 82 , overtakes a motorcyclist. What is the modulus of the speed of the motorcyclist, if after a period of time Δt = 2.8 minutes from the moment of overtaking, the distance between the car and the motorcyclist becameL\u003d 1.4 km?
№ 5
The car traveled the first half of the way at an average speedv 1 = 60 km/h , and the second - at an average speedv 2 = 40 km/h Determine the average speed of the car for the entire journey.
Consolidation of knowledge
The speed of uneven movement on a section of the trajectory is characterized by an average speed, and at a given point of the trajectory - by instantaneous speed.
The instantaneous speed is approximately equal to the average speed determined over a short period of time. The shorter this period of time, the smaller the difference between the average speed and the instantaneous one.
The instantaneous velocity is directed tangentially to the motion trajectory.
If the modulus of instantaneous velocity increases, then the movement of the body is called accelerated, if it decreases, it is called slow.
With uniform rectilinear motion, the instantaneous speed is the same at any point of the trajectory.
The displacement of a body relative to a fixed frame of reference is equal to the vector sum of its displacement relative to the moving frame and the displacement of the moving frame relative to the stationary one.
The speed of a body in a fixed frame of reference is equal to the vector sum of its speed relative to the moving frame and the speed of the moving frame relative to the fixed one.
Lesson summary
So, let's sum up. What did you learn in class today?
Homework organization
§6-10, ex. 3 No. 5, ex. 6 no 11.
Reflection.
Continue the phrases:
Today in class I learned...
It was interesting…
The knowledge that I received in the lesson will come in handy
Lesson #3
Subject. Uniform rectilinear motion. Speed. The law of addition of speeds. Movement charts.
Target: the formation of knowledge about rectilinear motion, speed as a physical quantity, the classical law of adding speeds, the solution of the main problem of mechanics for rectilinear uniform motion; consideration of graphs of the dependence of speed, coordinates of rectilinear uniform motion on time.
Lesson type: combined lesson.
Organizational stage
^ Checking homework.
front poll.
What is a reference system?
What is a trajectory? What types of fissile motion depending on the trajectory?
What is called a path? moving?
What is the difference between path and movement?
What is the essence of the concept of relativity of motion?
Reporting the topic, purpose and tasks of the lesson
Uniform rectilinear motion.
The speed of uniform rectilinear motion as a physical quantity.
The law of adding speeds.
Moving rectilinear uniform motion. Solution of the main problem of mechanics for rectilinear uniform motion.
Movement charts.
Learning new material
The simplest type of motion is uniform rectilinear motion.
Uniform rectilinear movement called such a movement of the body, in which the body for any equal time intervals performs the same movement and the trajectory of its movement is a straight line.
Question for students:
Give examples of uniform rectilinear motion.
What do you think, do we often encounter cases of rectilinear uniform motion?
Why study this type of movement, be able to describe its patterns?
One of the characteristics of uniform rectilinear motion is its speed. The teacher offers students to characterize the speed as a physical quantity according to the generalized plan of the characteristics of the physical quantity.
Generalized plan for the characteristics of a physical quantity:
A phenomenon that characterizes a value.
Definition, designation.
Formulas that relate a given quantity to other quantities.
Units.
Measurement methods.
direct measurements (using a speedometer, radar);
indirect measurements (by formula)
- velocity vector;
υ x , υ y - projections of the velocity vector on the coordinate axes Ox, Oy;
υ - speed modulus.
Question:
Can the velocity projection be negative? (The velocity projection can be either positive or negative depending on how the body is moving (Figure 1).)
^ The law of addition of speeds
If the movement of the same material point is considered with respect to two reference systems associated with a fixed body and a moving one (for example, a person who is standing on the bank of the river along which this boat is floating, and a person who himself is watching the movement of a person along the deck of a boat) the same time is on the boat), then we can formulate the classical law of adding speeds.
The law of adding speeds: the speed of the body relative to the fixed frame of reference is equal to the vector sum of the speed of the body relative to the moving frame and the actual speed of the moving frame relative to the fixed one:
where and are the velocities of the body relative to the fixed and moving frames of reference, respectively, and is the speed of the moving frame of reference relative to the fixed one (Fig. 2).
^ Moving rectilinear uniform motion. Solution of the main problem of mechanics for rectilinear uniform motion
you can determine the displacement modulus for rectilinear uniform motion: 
.If a material point, moving along the OX axis, has moved from a point with coordinate x
0
to a point with coordinate X
, then for the time t
she moved: 
(Fig. 3).
Since the main task of mechanics is to determine the position of a body at a given moment of time according to known initial conditions, the equation 
and is a solution to the main problem of mechanics.
This equation is also called the basic law of uniform rectilinear motion.
Movement charts
Velocity vs. Time Plot
is a straight line parallel to the time axis t
(Fig. 4, a). If 
> 0, then this line passes above the time axis t
, and if t.
The area of the figure bounded by the graph and the axis t , is numerically equal to the displacement modulus (Fig. 4, b).
Graph of displacement projection versus time
is a straight line passing through the origin. If > 0, then s
x
increases with time, if s x
decreases with time (Fig. 5, a). The slope of the graph is greater, the greater the velocity modulus (Fig. 5, b). 
If we are talking about a path graph, then it should be remembered that the path is the length of the trajectory, therefore it cannot decrease, but can only grow with time, therefore, this graph cannot approach the time axis (Fig. 5, c).
^ Plot of coordinate versus time
different from chart 
only by shifting x
0
along the coordinate axis. The point of intersection of graphs 1 and 2 corresponds to the moment when the coordinates of the bodies are equal, that is, this point determines the moment in time and the coordinate of the meeting of two bodies (Fig. 6).
Application of acquired knowledge
Moving objects are given in random order: pedestrian; sound waves in the air; oxygen molecule at 0 °C; weak wind; electromagnetic waves in vacuum; storm wind.
Answer:
electromagnetic waves in vacuum (300,000 km/s);
oxygen molecule at 0 °C (425 m/s);
sound waves in the air (330 m/s);
storm wind (21 m/s);
light wind (4 m/s);
pedestrian (1.3 m/s).
Summing up the lesson and reporting homework
Homework
Learn the theoretical material from the textbook.
Solve problems.
Find the correct answer.
Which of the following examples of motion can be considered uniform?
Car is braking
The passenger goes down the subway escalator
Airplane takes off
Rectilinear uniform movement is called, in which:
the modulus of the body's velocity remains unchanged
the speed of the body changes by the same value in any equal intervals of time
the body performs the same movements for any time intervals
A passenger train, moving uniformly, covered a distance of 30 km in 20 minutes. Find the speed of the train.
A motorcycle is moving at a speed of 36 km/h. How far will it travel in 20 seconds?
On fig. Figure 7 shows a graph of the path of uniform motion versus time. What is the speed of the body?
On fig. Figure 8 shows a graph of the speed of uniform motion versus time. What is the distance traveled by the body in 3 s?
Do you think you are moving or not when you read this text? Almost every one of you will immediately answer: no, I'm not moving. And it will be wrong. Some might say I'm moving. And they are wrong too. Because in physics, some things are not quite what they seem at first glance.
For example, the concept of mechanical motion in physics always depends on the reference point (or body). So a person flying in an airplane moves relative to the relatives left at home, but is at rest relative to a friend sitting next to him. So, bored relatives or a friend sleeping on his shoulder are, in this case, reference bodies for determining whether our aforementioned person is moving or not.
Definition of mechanical movement
In physics, the definition of mechanical motion studied in the seventh grade is as follows: a change in the position of a body relative to other bodies over time is called mechanical motion. Examples of mechanical movement in everyday life would be the movement of cars, people and ships. Comets and cats. Air bubbles in a boiling kettle and textbooks in a schoolboy's heavy backpack. And every time a statement about the movement or rest of one of these objects (bodies) will be meaningless without indicating the body of reference. Therefore, in life we most often, when we talk about movement, we mean movement relative to the Earth or static objects - houses, roads, and so on.
Trajectory of mechanical movement
It is also impossible not to mention such a characteristic of mechanical movement as a trajectory. A trajectory is a line along which a body moves. For example, footprints in the snow, the footprint of an airplane in the sky, and the footprint of a tear on a cheek are all trajectories. They can be straight, curved or broken. But the length of the trajectory, or the sum of the lengths, is the path traveled by the body. The path is marked with the letter s. And it is measured in meters, centimeters and kilometers, or in inches, yards and feet, depending on what units of measurement are accepted in this country.
Types of mechanical movement: uniform and uneven movement
What are the types of mechanical movement? For example, during a trip by car, the driver moves at different speeds when driving around the city and at almost the same speed when entering the highway outside the city. That is, it moves either unevenly or evenly. So the movement, depending on the distance traveled for equal periods of time, is called uniform or uneven.
Examples of uniform and non-uniform motion
There are very few examples of uniform motion in nature. The Earth moves almost evenly around the Sun, raindrops drip, bubbles pop up in soda. Even a bullet fired from a pistol moves in a straight line and evenly only at first glance. From friction against the air and the attraction of the Earth, its flight gradually becomes slower, and the trajectory decreases. Here in space, a bullet can move really straight and evenly until it collides with some other body. And with uneven movement, things are much better - there are many examples. The flight of a football during a football game, the movement of a lion hunting its prey, the travel of a chewing gum in a seventh grader's mouth, and a butterfly fluttering over a flower are all examples of uneven mechanical movement of bodies.
