Trajectory. The displacement is called the vector connecting the start and end points of the trajectory The vector connecting the beginning and end of the path is called
Kinematic description of motion mat. points
(Mat. point, reference system, movement, trajectory, path, speed, acceleration.)
Kinematic equations of uniformly variable motion
Kinematics deals with the description of motion, abstracting from its causes. To describe the movement, you can choose different reference systems. In different frames of reference, the motion of the same body looks different. In kinematics, when choosing a frame of reference, they are guided only by considerations of expediency, determined by specific conditions. So, when considering the motion of bodies on the Earth, it is natural to associate the reference frame with the Earth, which we will do. When considering the motion of the Earth itself, it is more convenient to associate the frame of reference with the Sun, etc. It is impossible to indicate any fundamental advantages of one frame of reference over another in kinematics. All frames of reference are kinematically equivalent. Only in dynamics, which studies motion in connection with the forces acting on moving bodies, are the fundamental advantages of a certain frame of reference or, more precisely, a certain class of frames of reference revealed. So,
A material point is a macroscopic body, the dimensions of which are so small that in the considered movement they can be ignored and it can be assumed that all the substance of the body is, as it were, concentrated in one geometric point.
Material points do not exist in nature. The material point is an abstraction, an idealized image of really existing bodies. It is possible or not possible to take this or that body in the study of any movement as a material point - this depends not so much on the body itself, but on the nature of the movement, as well as on the content of the questions that we want to get an answer to. The absolute dimensions of the body do not play a role in this. Relative dimensions are important, i.e., the ratio of the dimensions of the body to certain distances characteristic of the movement under consideration. For example, the Earth, when considering its orbital motion around the Sun, can be taken with great accuracy as a material point. The characteristic length here is the radius of the earth's orbit R ~ 1.5 108 km. It is very large in comparison with the radius of the globe r zhl:6.4 103 km. Due to this, during orbital motion, all points of the Earth move almost the same way. Therefore, it suffices to consider the movement of only one point, for example, the center of the Earth, and to assume that all the matter of the Earth is, as it were, concentrated in this geometric point. Such an idealization greatly simplifies the problem of the Earth's orbital motion, retaining, however, all the essential features of this motion. But this idealization is not suitable when considering the rotation of the Earth around its own axis, because it is meaningless to talk about the rotation
geometric point about an axis passing through this point.
The reference body is the position of a material point in space at a given moment in time, determined in relation to some other body. Contacts him
Reference system - a set of coordinate systems and clocks associated with the body, in relation to which the movement of some other material points is being studied.
A displacement is a vector connecting the start and end points of the trajectory.
The trajectory of the movement of a material point is the line described by this point in space. Depending on the shape of the trajectory, the movement can be rectilinear and curvilinear.
Basic concepts of kinematics
Kinematics
Chapter 1. Mechanics
Any physical phenomenon or process in the material world around us is a natural series of changes occurring in time and space. Mechanical motion, that is, a change in the position of a given body (or its parts) relative to other bodies, is the simplest type of physical process. The mechanical motion of bodies is studied in the branch of physics called mechanics. The main task of mechanics is determine the position of the body at any time.
One of the main parts of mechanics, which is called kinematics, considers the movement of bodies without clarifying the causes of this movement. Kinematics answers the question: how does a body move? Another important part of mechanics is dynamics, which considers the action of some bodies on others as the cause of motion. Dynamics answers the question: why does the body move in this way and not otherwise?
Mechanics is one of the most ancient sciences. Certain knowledge in this area was known long before the new era (Aristotle (IV century BC), Archimedes (III century BC)). However, the qualitative formulation of the laws of mechanics began only in the 17th century AD. e., when G. Galileo discovered the kinematic law of addition of velocities and established the laws of free fall of bodies. A few decades after Galileo, the great I. Newton (1643–1727) formulated the basic laws of dynamics.
In Newtonian mechanics, the motion of bodies is considered at speeds much less than the speed of light in a vacuum. They call her classical or Newtonian mechanics, in contrast to relativistic mechanics, created at the beginning of the 20th century, mainly due to the work of A. Einstein (1879–1956).
In relativistic mechanics, the motion of bodies is considered at speeds close to the speed of light. Classical Newtonian mechanics is the limiting case of relativistic for υ<< c.
kinematics called a branch of mechanics in which the movement of bodies is considered without clarifying the causes that cause it.
Mechanical movement body is called the change in its position in space relative to other bodies over time.
mechanical movement relatively. The motion of the same body relative to different bodies turns out to be different. To describe the movement of a body, it is necessary to indicate in relation to which body the movement is considered. This body is called reference body.
The coordinate system associated with the reference body and the clock for timing form reference system , which allows determining the position of a moving body at any time.
In the International System of Units (SI), the unit of length is meter, and per unit of time - second.
Every body has a certain size. Different parts of the body are in different places in space. However, in many problems of mechanics there is no need to indicate the positions of individual parts of the body. If the dimensions of the body are small compared to the distances to other bodies, then this body can be considered its material point. This can be done, for example, when studying the motion of planets around the Sun.
If all parts of the body move in the same way, then such a movement is called progressive . For example, cabins in the Ferris Wheel attraction, a car on a straight section of the track, etc. move forward. When the body moves forward, it can also be considered as a material point.
A body whose dimensions can be neglected under given conditions is called material point .
The concept of a material point plays an important role in mechanics.
Moving over time from one point to another, the body (material point) describes a certain line, which is called trajectory of the body .
The position of a material point in space at any time ( law of motion ) can be determined either using the dependence of coordinates on time x = x (t), y = y (t), z = z (t) (coordinate method), or using the time dependence of the radius vector (vector method) drawn from the origin to a given point (Fig. 1.1.1).
Question 1. Radius vector. Displacement vector.
- radius vector is the vector drawn from the origin O to the considered point M.
- moving(or changing the radius vector) is the vector connecting the start and end of the path.
radius vector in rectangular Cartesian coordinates:
Where - call point coordinates.
Question 2. The speed of movement. Average and instantaneous speeds.
Travel speed(vector) - shows how the displacement changes per unit of time.
Average: Instant:
The instantaneous velocity is always directed tangentially to the trajectory,
and the middle one coincides with the displacement vector.
Projection: Module:
Question 3. Path. Its connection with the speed module.
S– path is the length of the trajectory (scalar value, > 0).
S is the area of the figure bounded by the curve v(t) and the straight lines t 1 and t 2 .
Question 4. Acceleration. Acceleration module.
Acceleration - in meaning - shows how the speed changes per unit of time.
Projection: Module: Mean:
Question 5. Uneven movement of a point along a curved path.
If the point moves along a curved path, then it is advisable to decompose the acceleration into components, one of which is tangentially directed and is called tangential or tangential acceleration, and the other is directed along the normal to the tangent, i.e. along the radius of curvature, to the center of curvature and is called normal acceleration.
It characterizes the change in speed in direction, - in magnitude.
Where r - radius of curvature.
A point moving along a curved path always has normal acceleration, and tangential acceleration only when the speed changes in magnitude.
(2, 3) Topic 2. KINEMATIC EQUATIONS OF MOTION.
Question 1. Get the kinematic equations of motion r(t) and v(t).
Two differential and related two integral vector equations:
→ and - kinematic equations of an equally variable points at .
Question 2. Get the kinematic equations of motion x(t), y(t), v x (t) and v y (t), for a thrown body.
Question 3. Get a cinematography. equations of motion x(t), y(t), v x (t) and v y (t), for a body thrown at an angle.
Question 4. Get the equation of motion for a body thrown at an angle.
Topic 3. ROTATION KINEMATICS.
Question 1. Kinematic characteristics of rotational motion.
angular displacement- angle of rotation of the radius vector.
angular velocity- shows how the angle of rotation of the radius vector changes.
angular acceleration- shows how the angular velocity changes per unit of time.
Question 2. The relationship between the linear and angular characteristics of the movement of a point
Question 3. Get the kinematic equationsw (t) and f(t).
Then the kinematic equations after integration will take a simpler form: - kin. equations of uniform acceleration (+) and uniform deceleration (-) of rotational motion.
(4, 5, 6) Topic 4. KINEMATICS ATT.
Question 1. Definition of ATT. Translational and rotational movements of ATT.
ATT A body whose deformations can be neglected under the conditions of a given problem is called.
All ATT movements can be decomposed into translational and rotational, relative to some instantaneous axis. Progressive movement - this is a movement in which a straight line drawn through any two points of the body moves parallel to itself. In translational motion, all points of the body make the same movement. rotational movement- this is a movement in which all points of the body move along circles, the centers of which lie on the same straight line, called the axis of rotation.
As a kinematic equation for the rotational motion of the ATT, it is sufficient to know the equation j(t) for the angle of rotation of the radius vector drawn from the axis of rotation to some point of the body (if the axis is fixed). That is, fundamentally, the kinematic equations of motion for a point and ATT do not differ.
Topic 5. NEWTON'S LAWS.
Topic 6. LAW OF CONSERVATION OF MOMENTUM.
Topic 7. WORK. POWER. ENERGY.
Question 7. Conservation laws as applied to an absolutely elastic impact of two balls.
Absolutely elastic impact is the impact at which the kinetic energy of the entire system is conserved.
Topic 10. FORCE FIELDS
Question 3. Reducing the length.
l 0 is the length of the rod in the system relative to which it is at rest (in our case, in To),l - the length of this segment in the system relative to which it moves ( K¢). because and find a connection between l and l 0: .
Thus, it follows from SRT that the dimensions of moving bodies should decrease in the direction of their movement, but there is no real reduction, because All ISOs are equal.
Question 2. Ideal gas
The simplest model of real gases is ideal gas. FROM m a cro from the scopic point of view, it is a gas for which the gas laws are satisfied ( pV = const, p/T = const, V/T = const). FROM m and cro From the scopic point of view, it is a gas for which one can neglect: 1) the interaction of molecules with each other and 2) the own volume of gas molecules compared to the volume of the vessel in which the gas is located.
The equation that relates the state parameters to each other is called equation of state gas. One of the simplest equations of state is
( ; ; ) the Mendeleev-Clapeyron equation.
(n- concentration, k- Boltzmann constant) - the ideal gas equation of state in a different form.
Topic 15. BASIC CONCEPTS OF THERMODYNAMICS
Question 1. Basic concepts. Reversible and irreversible processes.
Reversible process - it is such a process of transition of the system from the state BUT into a state AT, at which the reverse transition from AT to BUT through the same intermediate states and at the same time no changes occur in the surrounding bodies. The system is called isolated if it does not exchange energy with the environment. On the graph, states are indicated by dots, and processes by lines.
Quantities that depend only on the state of the system and do not depend on the processes by which the system came to this state are called state functions. Quantities whose values in a given state depend on previous processes are called process functions - it's warmth Q and work A, their change is often denoted as dQ, dA or . ( d- Greek letter - delta)
Work and heat- these are two forms of energy transfer from one body to another. When work is performed, the relative position of bodies or body parts changes. The transfer of energy in the form of heat is carried out at the contact of bodies - due to the thermal movement of molecules.
To internal energy include: 1) the kinetic energy of the thermal motion of molecules (but not the kinetic energy of the entire system as a whole), 2) the potential energy of the interaction of molecules with each other, 3) the kinetic and potential energy of the vibrational motion of atoms in a molecule, 4) the binding energy of electrons with the nucleus in an atom , 5) the energy of interaction of protons and neutrons inside the nucleus of an atom. These energies are very different in magnitude from each other, for example, the energy of thermal motion of molecules at 300 K is ~ 0.04 eV, the binding energy of an electron in an atom is ~ 20-50 eV, and the energy of interaction of nucleons in the nucleus is ~ 10 MeV. Therefore, these interactions are considered separately.
Internal energy of an ideal gas is the kinetic energy of the thermal motion of its molecules. It depends only on the temperature of the gas. Its change has the same expression for all processes in ideal gases and depends only on the initial and final temperatures of the gas. is the internal energy of an ideal gas.
Topic 16.
Question 1. Entropy
The second law of thermodynamics, like the first law, is a generalization of a large number of experimental facts and has several formulations.
Let us first introduce the concept of "entropy", which plays a key role in thermodynamics. E ntropium - S- one of the most important thermodynamic functions that characterizes the state or possible changes in the state of matter - this is a multifaceted concept.
1)Entropy is a state function. The introduction of such quantities is valuable in that the change in the state function is the same for any processes, so a complex real process can be replaced by "fictional" simple processes. For example, the real process of the transition of the system from state A to state B (see Fig.) can be replaced by two processes: isochoric A®C and isobaric C®B.
Entropy is defined as follows.
For reversible processes in ideal gases, one can obtain formulas for calculating the entropy in various processes. Express dQ from the I beginning and substitute into the expression for dS .
general expression for entropy change in reversible processes.
Integrating, we obtain expressions for the change in entropy in various isoprocesses in ideal gases.
Question 2,3,4. isobaric, isochoric, isothermal
In all entropy calculations, only the difference between the entropies of the final and initial states of the system matters
2)Entropy is a measure of energy dissipation.
we write down the I law of thermodynamics for a reversible isothermal process, taking into account that dQ=T×dS and express the work dA | |
the thermodynamic function is called the free energy the quantity is called the bound energy | |
From the formulas, we can conclude that not all of the internal energy of the system can be converted into work U. Part of the energy TS cannot be translated into work, it dissipates in the environment. And this "bound" energy is the greater, the greater the entropy of the system. Therefore, entropy can be called a measure of energy dissipation. |
3)Entropy is a measure of the disorder of a system
Let us introduce the concept of thermodynamic probability. Let us have a box divided into n compartments. Moves freely in all compartments in the box N molecules. In the first compartment will be N 1 molecules, in the second compartment N 2 molecules...
in n-th compartment - N n molecules. Number of ways w that can be distributed N molecules according to n states (compartments) is called thermodynamic probability. In other words, thermodynamic probability shows how many micro distributions, you can get this macro distribution It is calculated by the formula:
For example calculation w consider a system consisting of three molecules 1, 2 and 3, which move freely in a box with three compartments.
In this example N=3(three molecules) and n=3(three compartments), the molecules are considered distinguishable.
In the first case, macrodistribution is a uniform distribution of molecules in compartments; it can be carried out by 6 microdistributions. The probability of such a distribution is the highest. An even distribution can be called "disorder" (by analogy with scattered things in a room). In the latter case, when the molecules are collected in only one compartment, the probability is the smallest. Simply put, we know from everyday observations that air molecules are more or less evenly distributed in a room, and it is almost completely unbelievable that all the molecules would gather in one corner of the room. However, theoretically, such a possibility exists.
Boltzmann postulated that entropy is directly proportional to the natural logarithm of thermodynamic probability:
Therefore, entropy can be called a measure of the disorder of a system.
Question 6. Now we can formulate the II law of thermodynamics.
1) For any processes occurring in a thermally insulated system, the entropy of the system cannot decrease: |
The sign "=" refers to reversible processes, the sign ">" - to irreversible (real) processes. In non-closed systems, entropy can change in any way. |
In other words, in closed real systems, only those processes are possible in which the entropy increases. Entropy is related to thermodynamic probability, therefore, its increase in closed systems means an increase in the “disorder” of the system, i.e. molecules tend to come to the same energy state and over time all molecules should have the same energy. Hence the conclusion was made about the tendency of our Universe to heat death. "The entropy of the world tends to a maximum" (Clausius). Since the laws of thermodynamics are derived on the basis of human experience on the scale of the Earth, the question of their applicability on the scale of the Universe remains open. |
3) “It is impossible to build a perpetual motion machine of the second kind, i.e. such a periodically operating machine, the action of which would consist only in lifting the load and cooling the thermal reservoir ”(Thomson, Planck) |
There must also be a body to which it “will have to” give up part of the heat. Simply taking heat away from some body and converting it into work is impossible because such a process is accompanied by a decrease in the entropy of the heater. Therefore, one more body is needed - a refrigerator, the entropy of which will increase in order to D.S. = 0. Those. heat is taken from the heater, due to this, work can be done, but part of the heat is "lost", i.e. transferred to the refrigerator. |
Question 7. CIRCULAR PROCESSES (CYCLES)
Circular process or cycle is a process in which the system, after going through a series of states, returns to its original state. If the process is carried out clockwise, it is called direct, counterclock-wise - reverse. Because internal energy is a state function, then in a circular process
A device that expends heat and produces work is called thermal machine. All heat engines operate in a direct cycle consisting of various processes. A device that works in reverse is called refrigeration machine. Work is expended in the refrigeration machine, and as a result, heat is taken away from the cold body, i.e. there is additional cooling of this body.
Consider Carnot cycle for an ideal heat engine. It is assumed that the working fluid is an ideal gas, there is no friction. This cycle, consisting of two isotherms and two adiabats, is not really feasible, but it played a huge role in the development of thermodynamics and heat engineering and made it possible to analyze the coefficient of performance (COP) of heat engines.
1-2 isothermal expansion | the heat supplied is used to work the gas | ||
2-3 adiabatic expansion | gas does work due to internal energy | ||
3-4 isothermal compression | external forces compress the gas, transferring heat to the environment | ||
4-1 adiabatic compression | work is done on the gas, its internal energy increases | ||
(- from the adiabatic equations) | total work per cycle; full on chart BUT equal to the area covered by the curve 1-2-3-4-1 | ||
Thus, per cycle, the gas was reported Q1 heat transferred to the refrigerator Q2 warmth and work received BUT.
It follows from the resulting expression that: 1) the efficiency is always less than unity,
2) the efficiency does not depend on the type of working fluid, but only on the temperature of the heater and refrigerator, 3) to increase the efficiency, it is necessary to increase the temperature of the heater and reduce the temperature of the refrigerator. In modern engines, combustible mixtures are used as a heater - gasoline, kerosene, diesel fuel, etc., which have certain combustion temperatures. The refrigerator is most often the environment. Consequently, it is possible to really increase the efficiency only by reducing friction in various parts of the engine and machine.
Topic 18. Question 1. AGGREGATE STATES OF SUBSTANCE
Molecules are complex systems of electrically charged particles. The bulk of the molecule and all of its positive charge are concentrated in the nuclei, their dimensions are about 10 - 15 - 10 - 14 m, and the size of the molecule itself, including the electron shell, is about 10 - 10 m. In general, the molecule is electrically neutral. The electric field of its charges is mainly concentrated inside the molecule and decreases sharply outside it. When two molecules interact, both attractive and repulsive forces are simultaneously manifested, they depend differently on the distance between the molecules (see Fig. - dotted lines). The simultaneous action of intermolecular forces gives the dependence of the force F from distance r between molecules, which is also characteristic of two molecules, and atoms, and ions (solid curve). At large distances, the molecules practically do not interact, at very small distances, repulsive forces predominate. At distances equal to several molecular diameters, attractive forces act. Distance r o between the centers of two molecules, on which F=0, is the position of balance. Since force is related to potential energy F=-dE sweat /dr, then integration will give the dependence of the potential energy on r(potential curve) . The equilibrium position corresponds to the minimum potential energy - Umin. For different molecules, the shape of the potential curve is similar, but the numerical values r o and Umin are different and determined by the nature of these molecules.
In addition to potential, the molecule also has kinetic energy. Each type of molecule has its own minimum potential energy, and the kinetic energy depends on the temperature of the substance ( Ye kin~ kT). Depending on the ratio between these energies, a given substance can be in a particular state of aggregation. For example, water can be in a solid state (ice), liquid and vapor.
For inert gases Umin small, so they go into a liquid state at very low temperatures. Metals are large Umin therefore, they are in a solid state up to the melting point - it can be hundreds and thousands of degrees.
Question 3.
Wetting leads to the fact that on the walls of the vessel the liquid, as it were, "creeps" along the wall, and its surface is curved. In a wide vessel, this curvature is almost imperceptible. In narrow tubes capillaries– this effect can be observed visually. Due to the forces of surface tension, an additional (compared to atmospheric) pressure is created Dr directed towards the center of curvature of the liquid surface.
Additional pressure near a curved liquid surface D p leads to the rise (when wetted) or lowering (when not wetted) of the liquid in the capillaries.
At equilibrium, the additional pressure is equal to the hydrostatic pressure of the liquid column. From the Laplace formula for a circular capillary D p = 2s /R, hydrostatic pressure R = r g h. Equating Dr = R, find h.
It can be seen from the formula that the smaller the radius of the capillary, the higher the rise (or fall) of the liquid.
The phenomenon of capillarity is extremely common in nature and technology. For example, the penetration of moisture from the soil into plants is carried out by lifting it through capillary channels. Capillary phenomena also include such a phenomenon as the movement of moisture along the walls of the room, leading to dampness. Capillarity plays a very important role in oil production. The pore sizes in the rock containing oil are extremely small. If the extracted oil turns out to be non-wetting in relation to the rock, then it will clog the tubules, and it will be very difficult to extract it. By adding certain substances to a liquid, even in a very small amount, its surface tension can be significantly changed. Such substances are called surfactants. radius vector in rectangular Cartesian coordinates:
Where - call point coordinates.
Trajectory(from late Latin trajectories - referring to movement) - this is the line along which the body moves (material point). The trajectory of movement can be straight (the body moves in one direction) and curvilinear, that is, mechanical movement can be rectilinear and curvilinear.
Rectilinear trajectory in this coordinate system is a straight line. For example, we can assume that the trajectory of a car on a flat road without turns is a straight line.
Curvilinear motion- this is the movement of bodies in a circle, ellipse, parabola or hyperbola. An example of curvilinear motion is the movement of a point on the wheel of a moving car, or the movement of a car in a turn.
Movement can be tricky. For example, the trajectory of the movement of the body at the beginning of the path can be rectilinear, then curvilinear. For example, a car at the beginning of the journey moves along a straight road, and then the road begins to "wind" and the car begins to curve.
Path
Path is the length of the path. The path is a scalar quantity and in the international system of units SI is measured in meters (m). Path calculation is performed in many problems in physics. Some examples will be discussed later in this tutorial.
Displacement vector
Displacement vector(or simply moving) is a directed line segment connecting the initial position of the body with its subsequent position (Fig. 1.1). Displacement is a vector quantity. The displacement vector is directed from the starting point of the movement to the end point.
Displacement vector modulus(that is, the length of the segment that connects the start and end points of the movement) can be equal to the distance traveled or less than the distance traveled. But never the module of the displacement vector can be greater than the distance traveled.
The modulus of the displacement vector is equal to the distance traveled when the path coincides with the trajectory (see sections Trajectory and Path), for example, if the car moves from point A to point B along a straight road. The module of the displacement vector is less than the distance traveled when the material point moves along a curved path (Fig. 1.1).
Rice. 1.1. The displacement vector and the distance traveled.
On fig. 1.1:
Another example. If the car passes in a circle once, then it turns out that the start point of the movement will coincide with the end point of the movement, and then the displacement vector will be equal to zero, and the distance traveled will be equal to the circumference. Thus, the path and movement are two different concepts.
Vector addition rule
The displacement vectors are added geometrically according to the vector addition rule (the triangle rule or the parallelogram rule, see Fig. 1.2).
Rice. 1.2. Addition of displacement vectors.
Figure 1.2 shows the rules for adding vectors S1 and S2:
a) Addition according to the rule of a triangle
b) Addition according to the parallelogram rule
Displacement vector projections
When solving problems in physics, projections of the displacement vector onto coordinate axes are often used. The projections of the displacement vector onto the coordinate axes can be expressed in terms of the difference between the coordinates of its end and beginning. For example, if a material point has moved from point A to point B, then the displacement vector (Fig. 1.3).
We choose the OX axis so that the vector lies with this axis in the same plane. Let's lower the perpendiculars from points A and B (from the start and end points of the displacement vector) to the intersection with the OX axis. Thus, we get the projections of points A and B on the X axis. Let us denote the projections of points A and B, respectively, A x and B x. The length of the segment A x B x on the OX axis - this is displacement vector projection on the x-axis, that is
S x = A x B x
IMPORTANT!
A reminder for those who do not know mathematics very well: do not confuse a vector with the projection of a vector on any axis (for example, S x). A vector is always denoted by a letter or several letters with an arrow above it. In some electronic documents, the arrow is not put, as this can cause difficulties when creating an electronic document. In such cases, be guided by the content of the article, where the word “vector” can be written next to the letter or in some other way indicate to you that this is a vector, and not just a segment.
Rice. 1.3. Projection of the displacement vector.
The projection of the displacement vector onto the OX axis is equal to the difference between the coordinates of the end and beginning of the vector, that is
S x = x – x 0 Similarly, the displacement vector projections on the OY and OZ axes are determined and written: S y = y – y 0 S z = z – z 0
Here x 0 , y 0 , z 0 - initial coordinates, or coordinates of the initial position of the body (material point); x, y, z - final coordinates, or coordinates of the subsequent position of the body (material point).
The projection of the displacement vector is considered positive if the direction of the vector and the direction of the coordinate axis coincide (as in Figure 1.3). If the direction of the vector and the direction of the coordinate axis do not coincide (opposite), then the projection of the vector is negative (Fig. 1.4).
If the displacement vector is parallel to the axis, then the module of its projection is equal to the module of the Vector itself. If the displacement vector is perpendicular to the axis, then the module of its projection is zero (Fig. 1.4).
Rice. 1.4. Modules of displacement vector projection.
The difference between the subsequent and initial values of a quantity is called the change in that quantity. That is, the projection of the displacement vector onto the coordinate axis is equal to the change in the corresponding coordinate. For example, for the case when the body moves perpendicular to the X axis (Fig. 1.4), it turns out that the body DOES NOT MOVEMENT relative to the X axis. That is, the displacement of the body along the X axis is zero.
Consider an example of the motion of a body on a plane. The initial position of the body is point A with coordinates x 0 and y 0, that is, A (x 0, y 0). The final position of the body is point B with coordinates x and y, that is, B (x, y). Find the modulus of displacement of the body.
From points A and B we lower the perpendiculars on the coordinate axes OX and OY (Fig. 1.5).
Rice. 1.5. Movement of a body on a plane.
Let's define the projections of the displacement vector on the axes OX and OY:
S x = x – x 0 S y = y – y 0
On fig. 1.5 it can be seen that the triangle ABC is a right triangle. It follows from this that when solving the problem, one can use Pythagorean theorem, with which you can find the modulus of the displacement vector, since
AC = s x CB = s y
According to the Pythagorean theorem
S 2 \u003d S x 2 + S y 2
Where can you find the modulus of the displacement vector, that is, the length of the body's path from point A to point B:
And finally, I suggest you consolidate your knowledge and calculate a few examples at your discretion. To do this, enter any numbers in the coordinate fields and click the CALCULATE button. Your browser must support the execution of scripts (scripts) JavaScript and the execution of scripts must be allowed in your browser settings, otherwise the calculation will not be performed. In real numbers, the integer and fractional parts must be separated by a period, for example, 10.5.
The projection is considered positive if (a x > 0) from the projection of the beginning of the vector to the projection of its end, you need to go in the direction of the axis. Otherwise, the projection of the vector (a x 0) from the projection of the beginning of the vector to the projection of its end must go in the direction of the axis. Otherwise, the projection of the vector (a x 0) from the projection of the beginning of the vector to the projection of its end must go in the direction of the axis. Otherwise, the projection of the vector (a x 0) from the projection of the beginning of the vector to the projection of its end must go in the direction of the axis. Otherwise, the projection of the vector (a x 0) from the projection of the beginning of the vector to the projection of its end must go in the direction of the axis. Otherwise, the projection of the vector (a x
Do we pay for the journey or transportation when traveling in a taxi? The ball fell from a height of 3 m, bounced off the floor and was caught at a height of 1 m. Find the path and move the ball. The cyclist moves in a circle with a radius of 30 m. What is the path and displacement of the cyclist for half a turn? For a full turn?
§ § 2.3 answer the questions at the end of the paragraph. Ex. 3, p.15 the trajectory ABSD of the movement of a point from A to D is shown. Find the coordinates of the points of the beginning and end of the movement, the distance traveled, the displacement, the projection of the displacement on the coordinate axes. Solve the problem (optional): The boat went to the northeast for 2 km, and then to the north for another 1 km. Find the displacement (S) and its modulus (S) by geometric construction.