EntranceWhy do we need a material point? Dimensionless material point and different reference systems
What is a material point? Which physical quantities are connected with it, why is the concept of a material point introduced at all? In this article we will discuss these issues, give examples of problems that are related to the concept under discussion, and also talk about the formulas used to solve them.
Definition
So, what is a material point? Different sources give the definition in slightly different literary styles. The same applies to teachers in universities, colleges and educational institutions. However, according to the standard, a material point is a body whose dimensions (in comparison with the dimensions of the reference system) can be neglected.
Connection with real objects
It would seem, how can one take a person, a cyclist, a car, a ship and even an airplane, which in most cases are discussed in problems in physics when it comes to the mechanics of a moving body, as a material point? Let's look deeper! To determine the coordinates of a moving body at any time, you need to know several parameters. This is the initial coordinate, and the speed of movement, and acceleration (if it occurs, of course), and time.
What is needed to solve problems with material points?
A coordinate relationship can only be found by reference to a coordinate system. Our planet becomes such a unique coordinate system for a car and another body. And in comparison with its size, the size of the body can really be neglected. Accordingly, if we take a body to be a material point, its coordinate in two-dimensional (three-dimensional) space can and should be found as the coordinate of a geometric point.
Movement of a material point. Tasks
Depending on the complexity, tasks may acquire certain conditions. Accordingly, based on the conditions given to us, we can use certain formulas. Sometimes, even having the entire arsenal of formulas, it is still not possible to solve the problem, as they say, “head-on”. Therefore, it is extremely important not only to know the kinematics formulas related to a material point, but also to be able to use them. That is, express the desired quantity, and equate the systems of equations. Here are the basic formulas that we will use when solving problems:
Task No. 1
A car standing on the starting line suddenly starts moving from a stationary position. Find out how long it will take him to accelerate to 20 meters per second if his acceleration is 2 meters per second squared.
I would like to say right away that this task is practically the simplest thing that a student can expect. The word “practically” is there for a reason. The thing is that it can only be simpler to substitute direct values into the formulas. We must first express time and then make calculations. To solve the problem you will need a formula for determining instantaneous speed(instantaneous speed is the speed of a body at a certain moment in time). It looks like this:
As we can see, on the left side of the equation we have instantaneous speed. We absolutely don't need her there. Therefore, we do simple mathematical operations: we leave the product of acceleration and time on the right side, and transfer the initial speed to the left. In this case, you should carefully monitor the signs, since one incorrectly left sign can radically change the answer to the problem. Next, we complicate the expression a little, getting rid of the acceleration on the right side: divide by it. As a result, we should have pure time on the right, and a two-level expression on the left. We just swap this whole thing around to make it look more familiar. All that remains is to substitute the values. So, it turns out that the car will accelerate in 10 seconds. Important: we solved the problem assuming that the car in it is a material point.
Problem No. 2
Material point starts emergency braking. Determine what the initial speed was at the moment of emergency braking, if 15 seconds passed before the body came to a complete stop. Take the acceleration to be 2 meters per second squared.
The task is, in principle, quite similar to the previous one. But there are a couple of nuances here. First, we need to determine the speed, which we usually call the initial speed. That is, at a certain moment, the countdown of time and distance traveled by the body begins. Speed will indeed fall within this definition. The second nuance is a sign of acceleration. Recall that acceleration is a vector quantity. Consequently, depending on the direction it will change its sign. Positive acceleration is observed if the direction of the body's velocity coincides with its direction. Simply put, when a body accelerates. Otherwise (that is, in our braking situation), the acceleration will be negative. And these two factors must be taken into account to solve this problem:
As last time, let’s first express the quantity we need. To avoid fussing with signs, let's leave the initial speed where it is. With the opposite sign, we transfer the product of acceleration and time to the other side of the equation. Since the braking was complete, the final speed is 0 meters per second. Substituting these and other values, we easily find the initial speed. It will be equal to 30 meters per second. It is easy to see that, knowing the formulas, coping with the simplest tasks is not so difficult.
Problem No. 3
At a certain point in time, dispatchers begin monitoring the movement of an air object. Its speed at this moment is 180 kilometers per hour. After a period of time equal to 10 seconds, its speed increases to 360 kilometers per hour. Determine the distance traveled by the plane during the flight if the flight time was 2 hours.
In fact, in a broad sense, this task has many nuances. For example, aircraft acceleration. It is clear that in principle our body could not move along a straight path. That is, it needs to take off, pick up speed, and then, at a certain altitude, move in a straight line for some distance. Deviations and the deceleration of the aircraft during landing are not taken into account. But that's none of our business in this case. Therefore, we will solve the problem within the framework of school knowledge, general information about kinematic motion. To solve the problem, we need the following formula:
But here we have a snag that we talked about earlier. Knowing the formulas is not enough - you need to be able to use them. That is, derive one value using alternative formulas, find it and substitute it. When viewing the initial information that is available in the problem, it immediately becomes clear that it will not be possible to solve it simply. Nothing is said about acceleration, but there is information about how the speed has changed over a certain period of time. This means that we can find the acceleration ourselves. We take the formula for finding instantaneous speed. She looks like
We leave acceleration and time in one part, and transfer the initial speed to another. Then, by dividing both parts by time, we free the right side. Here you can immediately calculate the acceleration by substituting direct data. But it is much more appropriate to express it further. We substitute the formula obtained for acceleration into the main one. There you can reduce the variables a little: in the numerator time is given squared, and in the denominator - to the first power. Therefore, we can get rid of this denominator. Well, then it’s a simple substitution, since nothing else needs to be expressed. The answer should be the following: 440 kilometers. The answer will be different if you convert the quantities to another dimension.
Conclusion
So, what did we find out during this article?
1) A material point is a body whose dimensions, compared to the dimensions of the reference system, can be neglected.
2) To solve problems related to a material point, there are several formulas (given in the article).
3) The sign of acceleration in these formulas depends on the parameter of body motion (acceleration or braking).
Definition
A material point is a macroscopic body whose dimensions, shape, rotation and internal structure can be neglected when describing its motion.
The question of whether a given body can be considered as a material point depends not on the size of this body, but on the conditions of the problem being solved. For example, the radius of the Earth is much smaller than the distance from the Earth to the Sun, and its orbital motion can be well described as the movement of a material point with a mass equal to the mass of the Earth and located at its center. However, when considering the daily motion of the Earth around its own axis, replacing it with a material point does not make sense. The applicability of the material point model to a specific body depends not so much on the size of the body itself, but on the conditions of its motion. In particular, in accordance with the theorem on the motion of the center of mass of a system during translational motion, any rigid body can be considered a material point whose position coincides with the center of mass of the body.
Mass, position, speed and some others physical properties of a material point at any given moment in time completely determine its behavior.
The position of a material point in space is defined as the position of a geometric point. In classical mechanics, the mass of a material point is assumed to be constant in time and independent of any features of its movement and interaction with other bodies. In the axiomatic approach to the construction of classical mechanics, the following is accepted as one of the axioms:
Axiom
A material point is a geometric point that is associated with a scalar called mass: $(r,m)$, where $r$ is a vector in Euclidean space related to some Cartesian coordinate system. The mass is assumed to be constant, independent of the position of the point in space and time.
Mechanical energy can be stored by a material point only in the form kinetic energy its movement in space and (or) potential energy of interaction with the field. This automatically means that a material point is incapable of deformation (only an absolutely rigid body can be called a material point) and rotation around its own axis and changes in the direction of this axis in space. At the same time, the model of the motion of a body described by a material point, which consists in changing its distance from some instantaneous center of rotation and two Euler angles, which specify the direction of the line connecting this point with the center, is extremely widely used in many branches of mechanics.
The method of studying the laws of motion of real bodies by studying the motion of an ideal model - a material point - is fundamental in mechanics. Any macroscopic body can be represented as a collection of interacting material points g, with masses equal to the masses of its parts. The study of the movement of these parts comes down to the study of the movement of material points.
The limited application of the concept of a material point is clear from this example: in a rarefied gas at high temperature, the size of each molecule is very small compared to the typical distance between molecules. It would seem that they can be neglected and the molecule can be considered a material point. However, this is not always the case: vibrations and rotations of the molecule are an important reservoir. internal energy"molecule, the "capacity" of which is determined by the size of the molecule, its structure and chemical properties. IN good approximation As a material point, one can sometimes consider a monatomic molecule (inert gases, metal vapors, etc.), but even in such molecules, at a sufficiently high temperature, excitation of electron shells is observed due to collisions of molecules, followed by emission.
Exercise 1
a) a car entering the garage;
b) a car on the Voronezh - Rostov highway?
a) a car entering a garage cannot be taken as a material point, since in these conditions the dimensions of the car are significant;
b) a car on the Voronezh-Rostov highway can be taken as a material point, since the size of the car is much smaller than the distance between cities.
Is it possible to take as a material point:
a) a boy who walks 1 km on his way home from school;
b) a boy doing exercises.
a) When a boy, returning from school, walks a distance of 1 km to home, then the boy in this movement can be considered as a material point, because his size is small compared to the distance he covers.
b) when the same boy performs morning exercises, then he cannot be considered a material point.
Material point
Material point(particle) - the simplest physical model in mechanics - an ideal body whose dimensions are equal to zero; the dimensions of the body can also be considered infinitesimal compared to other sizes or distances within the assumptions of the problem under study. The position of a material point in space is defined as the position of a geometric point.
In practice, a material point is understood as a body with mass, the size and shape of which can be neglected when solving this problem.
At straight motion one body is enough coordinate axis to determine its position.
Peculiarities
The mass, position and speed of a material point at each specific moment in time completely determine its behavior and physical properties.
Consequences
Mechanical energy can be stored by a material point only in the form of the kinetic energy of its movement in space, and (or) the potential energy of interaction with the field. This automatically means that a material point is incapable of deformation (only an absolutely rigid body can be called a material point) and rotation around its own axis and changes in the direction of this axis in space. At the same time, the model of the motion of a body described by a material point, which consists in changing its distance from some instantaneous center of rotation and two Euler angles, which specify the direction of the line connecting this point to the center, is extremely widely used in many branches of mechanics.
Restrictions
The limited application of the concept of a material point is clear from this example: in a rarefied gas at high temperature, the size of each molecule is very small compared to the typical distance between molecules. It would seem that they can be neglected and the molecule can be considered a material point. However, this is not always the case: vibrations and rotations of a molecule are an important reservoir of the “internal energy” of the molecule, the “capacity” of which is determined by the size of the molecule, its structure and chemical properties. To a good approximation, a monatomic molecule (inert gases, metal vapors, etc.) can sometimes be considered as a material point, but even in such molecules, at a sufficiently high temperature, excitation of electron shells is observed due to collisions of molecules, followed by emission.
Notes
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See what a “material point” is in other dictionaries:
A point with mass. In mechanics, the concept of a material point is used in cases where the size and shape of a body do not play a role in the study of its motion, and only mass is important. Almost any body can be considered as a material point if... ... Big Encyclopedic Dictionary
A concept introduced in mechanics to designate an object that is considered as a point with mass. The position of the M. t. in law is defined as the position of the geom. points, which greatly simplifies the solution of mechanics problems. Practically, the body can be considered... ... Physical encyclopedia
material point- A point with mass. [Collection of recommended terms. Issue 102. Theoretical mechanics. Academy of Sciences of the USSR. Committee of Scientific and Technical Terminology. 1984] Topics theoretical mechanics EN particle DE materialle Punkt FR point matériel … Technical Translator's Guide
Modern encyclopedia
In mechanics: infinitesimal body. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910 ... Dictionary of foreign words of the Russian language
Material point- MATERIAL POINT, a concept introduced in mechanics to designate a body whose dimensions and shape can be neglected. The position of a material point in space is defined as the position of a geometric point. The body can be considered material... ... Illustrated Encyclopedic Dictionary
A concept introduced in mechanics for an object of infinitesimal size that has mass. The position of a material point in space is defined as the position of a geometric point, which simplifies the solution of mechanics problems. Almost any body can... ... encyclopedic Dictionary
Material point- a geometric point with mass; material point abstract image a material body that has mass and has no dimensions... The beginnings of modern natural science
material point- materialusis taškas statusas T sritis fizika atitikmenys: engl. mass point; material point vok. Massenpunkt, m; materieller Punkt, m rus. material point, f; point mass, f pranc. point mass, m; point matériel, m … Fizikos terminų žodynas
material point- A point with mass... Polytechnic terminological explanatory dictionary
Books
- Set of tables. Physics. 9th grade (20 tables), . Educational album of 20 sheets. Material point. Coordinates of a moving body. Acceleration. Newton's laws. Law universal gravity. Rectilinear and curvilinear movement. Body movement along...
INTRODUCTION
The didactic material is intended for students of all specialties of the correspondence faculty of GUCMiZ, studying a course in mechanics according to the program for engineering and technical specialties.
The didactic material contains a brief summary of the theory on the topic being studied, adapted to the level of training of part-time students, examples of solving typical problems, questions and assignments similar to those offered to students in exams, and reference material.
The purpose of such material is to help a part-time student independently, in a short time, learn the kinematic description of translational and rotational movements, using the analogy method; learn to solve numerical and qualitative problems, understand issues related to the dimension of physical quantities.
Particular attention is paid to solving qualitative problems, as one of the methods for deeper and more conscious mastery of the fundamentals of physics, necessary when studying special disciplines. They help to understand the meaning of occurring natural phenomena, to understand the essence physical laws and clarify the scope of their application.
Didactic material may be useful to full-time students.
KINEMATICS
The part of physics that studies mechanical motion is called mechanics . Mechanical movement is understood as a change over time in the relative position of bodies or their parts.
Kinematics - the first section of mechanics, it studies the laws of motion of bodies, without being interested in the reasons that cause this movement.
1. Material point. Reference system. Trajectory.
Path. Move vector
The simplest kinematics model is material point . This is a body whose dimensions can be neglected in this problem. Any body can be represented as a collection of material points.
To mathematically describe the movement of a body, it is necessary to decide on a reference system. Reference system (CO) consists of bodies of reference and related coordinate systems And hours. If there are no special instructions in the problem statement, it is considered that the coordinate system is related to the Earth's surface. The most commonly used coordinate system is Cartesian system.
Let it be necessary to describe the movement of a material point in a Cartesian coordinate system XYZ(Fig. 1). At some point in time t 1 point is in position A. The position of a point in space can be characterized by a radius vector r 1 drawn from the origin to the position A, and coordinates x 1 , y 1 , z 1 . Here and below, vector quantities are indicated in bold italics. By the time t 2 = t 1 + Δ t the material point will move to position IN with radius vector r 2 and coordinates x 2 , y 2 , z 2 .
Trajectory of movement called a curve in space along which a body moves. Based on the type of trajectory, rectilinear, curvilinear and circular movements are distinguished.
Path length (or path ) - length of the section AB, measured along the trajectory of movement, is denoted by Δs (or s). Distance in the International System of Units (SI) is measured in meters (m).
Move vector material point Δ r represents the vector difference r 2 And r 1, i.e.
Δ r = r 2 - r 1.
The magnitude of this vector, called displacement, is the shortest distance between positions A And IN(start and end) moving point. Obviously, Δs ≥ Δ r, and equality holds for rectilinear motion.
When a material point moves, the value of the distance traveled, the radius vector and its coordinates change with time. Kinematic equations of motion (further equations of motion) are called their dependences on time, i.e. equations of the form
s=s( t), r=r (t), x=X(t), y=at(t), z=z(t).
If such an equation is known for a moving body, then at any moment of time you can find the speed of its movement, acceleration, etc., which we will verify later.
Any movement of a body can be represented as a set progressive And rotational movements.
2. Kinematics of translational motion
Progressive is a movement in which any straight line rigidly connected to a moving body remains parallel to itself .
Speed characterizes the speed of movement and direction of movement.
Medium speed movements in the time interval Δ t is called the quantity
(1)
where - s is the segment of path traveled by the body in time during time t.
Instant speed movement (speed in this moment time) is a quantity whose modulus is determined by the first derivative of the path with respect to time
(2)
Speed is a vector quantity. The instantaneous velocity vector is always directed along tangent to the trajectory of movement (Fig. 2). The unit of speed is m/s.
The value of the speed depends on the choice of reference system. If a person is sitting in a train carriage, he and the train move relative to the CO connected to the ground, but are at rest relative to the CO connected to the car. If a person walks along a carriage at a speed , then his speed relative to the “ground” CO s depends on the direction of movement. Along the movement of the train z = trains + , against z = trains - .
Projections of the velocity vector on the coordinate axes υ X ,υ y ,υ z are defined as the first derivatives of the corresponding coordinates with respect to time (Fig. 2):
If the projections of velocity on the coordinate axes are known, the velocity modulus can be determined using the Pythagorean theorem:
(3)
Uniform called movement at a constant speed (υ = const). If the direction of the velocity vector does not change v, then the movement will be uniform and rectilinear.
Acceleration - physical quantity characterizing the rate of change in speed in magnitude and direction Average acceleration defined as
(4)
where Δυ is the change in speed over a period of time Δ t.
Vector instantaneous acceleration is defined as the derivative of the velocity vector v by time:
(5)
Since during curvilinear motion the speed can change both in magnitude and direction, it is customary to decompose the acceleration vector into two mutually perpendicular components
A = A τ + A n. (6)
Tangential (or tangential) acceleration A τ characterizes the rate of change in speed in magnitude, its modulus
.(7)
Tangential acceleration is directed tangentially to the trajectory of motion along the speed during accelerated motion and against the speed during slow motion (Fig. 3).
Normal (centripetal) acceleration A n characterizes the change in speed in direction, its module
(8)
Where R- radius of curvature of the trajectory.
The normal acceleration vector is directed to the center of the circle, which can be drawn tangentially to a given point on the trajectory; it is always perpendicular to the tangential acceleration vector (Fig. 3).
The modulus of total acceleration is determined by the Pythagorean theorem
.
(9)
Direction of the total acceleration vector A determined by the vector sum of the normal and tangential acceleration vectors (Fig. 3)
Equally variable called movement with permanent acceleration . If the acceleration is positive, then this is uniformly accelerated motion , if it is negative - equally slow .
When moving in a straight line Aם =0 and A = Aτ. If Aם =0 and Aτ = 0, the body is moving straight and even; at Aם =0 and Aτ = const motion rectilinear uniformly variable.
At uniform motion the distance traveled is calculated using the formula:
d s= d t→ s= ∫d t= ∫d t= t+ s 0 , (10)
Where s 0 - starting path for t = 0. The last formula must be remembered.
Graphic dependencies υ (t) And s(t) are shown in Fig. 4.
For uniformly alternating motion = ∫ A d t = A∫ d t, from here
= At + 0 , (11)
where 0 is the initial speed at t=0.
Distance traveled s= ∫d t = ∫(At + 0)d t. Solving this integral, we get
s = At 2 /2 + 0 t + s 0 , (12)
Where s 0 - initial path (for t= 0). We recommend that you remember formulas (11), (12).
Graphic dependencies A(t), υ (t) And s(t) are shown in Fig. 5.
To uniform motion with acceleration free fall g= 9.81 m/s 2 refers free movement bodies in a vertical plane: bodies fall down from g›0, acceleration when moving up g‹ 0. The speed of movement and the distance traveled in this case changes according to (11):
= 0 + gt; (13)
h = gt 2 /2 + 0 t +h 0 . (14)
Let's consider the movement of a body thrown at an angle to the horizon (ball, stone, cannon shell,...). This complex movement consists of two simple ones: horizontally along the axis OH and verticals along the axis OU(Fig. 6). Along the horizontal axis, in the absence of environmental resistance, the movement is uniform; along the vertical axis - uniformly variable: uniformly slowed down to the maximum lifting point and uniformly accelerated after it. The trajectory of movement has the form of a parabola. Let 0 be the initial speed of a body thrown at an angle α to the horizon from a point A(origin). Its components along the selected axes:
0x = x = 0 cos α = const; (15)
0у = 0 sinα. (16)
According to formula (13) we have for our example at any point of the trajectory to the point WITH
y = 0y - g t= 0 sinα. - g t ;
x = 0х = 0 cos α = const.
At the highest point of the trajectory, point WITH, vertical component of speed y = 0. From here you can find the time of movement to point C:
y = 0y - g t= 0 sinα. - g t = 0 → t = 0 sinα/ g. (17)
Knowing this time, you can determine the maximum height of body lifting using (14):
h max = 0у t- gt 2 /2= 0 sinα 0 sinα/ g– g( 0 sinα /g) 2 /2 = ( 0 sinα) 2 /(2 g) (18)
Since the trajectory of movement is symmetrical, the total time of movement to the end point IN equals
t 1 =2 t= 2 0 sinα / g. (19)
Range of flight AB taking into account (15) and (19) will be determined as follows:
AB= x t 1 = 0 cosα 2 0 sinα/ g= 2 0 2 cosα sinα/ g. (20)
The total acceleration of a moving body at any point on the trajectory is equal to the acceleration of gravity g; it can be decomposed into normal and tangential, as was shown in Fig. 3.
Material point. Reference system.
The mechanical movement of a body is the change in its position relative to other bodies over time.
Almost all physical phenomena are accompanied by the movement of bodies. In physics there is a special section that studies motion - this is Mechanics.
The word “mechanics” comes from the Greek “mechane” - machine, device.
When various machines and mechanisms operate, their parts move: levers, ropes, wheels,... Mechanics also includes finding the conditions under which a body is at rest - the conditions of equilibrium of bodies. These issues play a huge role in the construction business. Not only material bodies can move, but also a sunbeam, a shadow, light signals, and radio signals.
To study movement, you must be able to describe movement. We are not interested in how this movement arose, we are interested in the process itself. The branch of mechanics that studies motion without investigating the cause that causes it is called kinematics.
The movement of each body can be considered in relation to different bodies, and relative to them this body will perform various movements: a suitcase lying in a carriage on the rack of a moving train is at rest relative to the carriage, and moving relative to the Earth. A balloon carried by the wind moves relative to the Earth, but is at rest relative to the air. An aircraft flying in a squadron is at rest relative to other aircraft in the formation, but moves at high speed relative to the Earth.
Therefore, any movement, as well as the rest of the body, is relative.
When answering the question whether a body is moving or at rest, we must indicate in relation to what we are considering the movement.
The body relative to which this movement is considered is called the body of reference.
A coordinate system and a device for measuring time are associated with the reference body. This entire set forms reference system .
What does it mean to describe movement? This means that you need to determine:
1. trajectory, 2. speed, 3. path, 4. body position.
The situation is very simple with a point. From a mathematics course we know that the position of a point can be specified using coordinates. What if we have a body that has size? Each point will have its own coordinates. In many cases, when considering the movement of a body, the body can be taken as a material point, or a point that has the mass of this body. And for a point there is only one way to determine the coordinates.
So, a material point is an abstract concept that is introduced to simplify problem solving.
Condition under which a body can be taken as a material point:
Often a body can be taken as a material point and, provided that its dimensions are comparable to the distance traveled, when at any moment of time all points move the same way. This type of movement is called translational.
A sign of forward motion is the condition that a straight line mentally drawn through any two points of the body remains parallel to itself.
Example: a person moves on an escalator, a needle in a sewing machine, a piston in an internal combustion engine, a car body when driving on a straight road.
Different movements differ in the type of trajectory.
If the trajectory straight line- That linear movement, if the trajectory is a curved line, then the movement is curvilinear.
Moving.
Path and movement: what's the difference?
S = AB + BC + CD
Displacement is a vector (or directed line segment) connecting an initial position to its subsequent position.
Displacement is a vector quantity, which means it is characterized by two quantities: numerical value or module and direction.
It is designated – S, and is measured in meters (km, cm, mm).
If you know the displacement vector, you can unambiguously determine the position of the body.
Vectors and actions with vectors.
VECTOR DEFINITION
Vector called a directed segment, that is, a segment that has a beginning (also called the point of application of the vector) and an end.
VECTOR MODULE
The length of a directed segment representing a vector is called length, or module, vector. The length of the vector is denoted by .
NULL VECTOR
Null vector() - a vector whose beginning and end coincide; its modulus is 0 and its direction is uncertain.
COORDINATE REPRESENTATION
Let a Cartesian coordinate system XOY be specified on the plane.
Then the vector can be specified by two numbers:
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These numbers https://pandia.ru/text/78/050/images/image012_18.gif" width="20" height="25 src="> in geometry are called vector coordinates, and in physics – vector projections to the corresponding coordinate axes.
To find the projection of a vector, you need to: drop perpendiculars from the beginning and end of the vector on the coordinate axes.
Then the projection will be the length of the segment enclosed between the perpendiculars.
Projection can take on both positive and negative meanings.
If the projection turns out with a “-“ sign, then the vector is directed towards the opposite side axis on which it was designed.
With this definition of its vector module, A direction is given by angle a, which is uniquely determined by the relations:
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COLLINEAR VECTORS
D) chess piece
E) chandelier in the room,
G) submarine,
Y) plane on the runway.
8. Do we pay for the journey or transportation when traveling in a taxi?
9. The boat traveled along the lake in a northeast direction for 2 km, and then in a northerly direction for another 1 km. Find geometric construction movement and its modulus.