Directed cut. Trajectory, path length, displacement vector Vector connecting the initial position
a vector connecting the initial position of the body with its subsequent position. and got the best answer
Answer from Winter37[guru]
Mechanical motion is a change in the position of a body in space over time relative to other bodies.
Of all the various forms of motion of matter, this kind of motion is the simplest.
For example: moving the clock hand on the dial, people are walking, tree branches are swaying, butterflies are fluttering, an airplane is flying, etc.
Determining the position of a body at any moment in time is the main task of mechanics.
The movement of the body, in which all points move in the same way, is called translational.
A material point is a physical body, the dimensions of which under the given conditions of motion can be neglected, assuming that all of its mass is concentrated at one point.
A trajectory is a line that a material point describes during its movement.
The path is the length of the trajectory of the material point.
Displacement is a directed straight line segment (vector) connecting the initial position of the body with its subsequent position.
A reference system is: a reference body, a coordinate system associated with it, as well as a device for timing.
An important feature of fur. movement is its relativity.
The relativity of motion is the movement and speed of the body relative to different frames of reference are different (for example, a person and a train). The speed of the body relative to the fixed coordinate system is equal to the geometric sum of the speeds of the body relative to the moving system and the speed of the moving coordinate system relative to the fixed one. (V1 is the speed of a person in the train, V0 is the speed of the train, then V=V1+V0).
The classical law of addition of velocities is formulated as follows: the speed of a material point in relation to the reference system, taken as a fixed one, is equal to the vector sum of the velocities of the point in the moving system and the speed of the moving system relative to the fixed one.
Characteristics of mechanical motion are interconnected by basic kinematic equations.
s = v0t + at2/ 2;
v = v0 + at.
Let us assume that the body moves without acceleration (aircraft on the route), its speed does not change for a long time, a = 0, then the kinematic equations will look like: v = const, s = vt.
Movement in which the speed of the body does not change, i.e., the body moves by the same amount for any equal time intervals, is called uniform rectilinear motion.
During the launch, the speed of the rocket increases rapidly, i.e., the acceleration a > O, a == const.
In this case, the kinematic equations look like this: v = v0 + at, s = V0t + at2/ 2.
In such a motion, the speed and acceleration have the same directions, and the speed changes in the same way for any equal time intervals. This type of motion is called uniformly accelerated.
When the car brakes, the speed decreases equally for any equal time intervals, the acceleration is less than zero; since the speed decreases, the equations take the form: v = v0 + at, s = v0t - at2/ 2. Such a movement is called equally slow.
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).
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.
Definition 1
body trajectory- this is a line that was described by a material point when moving from one point to another over time.
There are several types of motions and trajectories of a rigid body:
- progressive;
- rotational, that is, movement in a circle;
- flat, that is, moving along a plane;
- spherical, characterizing the movement on the surface of the sphere;
- free, in other words, arbitrary.
Picture 1 . Determination of a point using coordinates x = x (t) , y = y (t) , z = z (t) and radius vector r → (t) , r 0 → is the radius vector of the point at the initial time
The position of a material point in space at any time can be set using the law of motion, defined in a coordinate way, through the dependence of coordinates on time x = x(t) , y = y(t) , z = z(t) or from the time of the radius vector r → = r → (t) drawn from the origin to the given point. This is shown in Figure 1.
Definition 2S → = ∆ r 12 → = r 2 → - r 1 → is a directed straight line segment connecting the initial point with the end point of the body trajectory. The value of the path traveled l is equal to the length of the trajectory traveled by the body in a certain period of time t.
Figure 2. Distance traveled l and the displacement vector s → during the curvilinear motion of the body, a and b are the starting and ending points of the path, accepted in physics
Definition 3
Figure 2 shows that when the body moves along a curvilinear trajectory, the module of the displacement vector is always less than the distance traveled.
The path is a scalar value. Considered a number.
The sum of two consecutive movements from point 1 to point 2 and from current 2 to point 3 is the movement from point 1 to point 3, as shown in figure 3.
Picture 3 . The sum of two consecutive movements ∆ r → 13 = ∆ r → 12 + ∆ r → 23 = r → 2 - r → 1 + r → 3 - r → 2 = r → 3 - r → 1
When the radius vector of a material point at a certain time t is r → (t), at the moment t + ∆ t is r → (t + ∆ t) , then its displacement ∆ r → in time ∆ t is equal to ∆ r → = r → (t + ∆t) - r → (t) .
The displacement ∆ r → is considered to be a function of time t: ∆ r → = ∆ r → (t) .
Example 1
By condition, a moving aircraft is given, shown in Figure 4. Determine the type of trajectory of the point M.
Picture 4
Solution
It is necessary to consider the reference system I, called "Aircraft" with the trajectory of the point M in the form of a circle.
Reference system II "Earth" will be set with the trajectory of the existing point M in a spiral.
Example 2
Given a material point that moves from A to B. The value of the radius of the circle R = 1 m. Find S , ∆ r → .
Solution
While moving from A to B, the point travels a path that is equal to half the circle written by the formula:
Substitute the numerical values and get:
S \u003d 3.14 1 m \u003d 3.14 m.
The displacement ∆ r → in physics is considered to be a vector connecting the initial position of the material point with the final one, that is, A with B.
Substituting the numerical values, we calculate:
∆ r → = 2 R = 2 1 = 2 m.
Answer: S = 3, 14 m; ∆r → = 2 m.
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The movement of the body is called a directed segment of a straight line connecting the initial position of the body with its subsequent position. Displacement is a vector quantity.
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Basic concepts of kinematics
kinematics called a branch of mechanics in which the movement of bodies is considered without clarifying the causes of this movement.
Mechanical movement bodies are called a change in ᴇᴦο 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 ᴇᴦο 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 attraction "Giant Wheel", a car on a straight section of the path, etc. move forward. With the translational movement of the body, ᴇᴦο 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).
The displacement of the body is called a directed segment of a straight line connecting the initial position of the body with the subsequent position ᴇᴦο. Displacement is a vector quantity.
The movement of the body is called a directed segment of a straight line connecting the initial position of the body with its subsequent position. Displacement is a vector quantity. - concept and types. Classification and features of the category "Body displacement is a directed line segment connecting the initial position of the body with its subsequent position. Displacement is a vector quantity." 2015, 2017-2018.