That the trajectory of movement is the path of movement. Mechanical movement. Trajectory. Path and movement. Addition of speeds. Special cases of rotational motion

This term has other meanings, see Movement (meanings).

Moving(in kinematics) - change of position physical body in space over time relative to the chosen reference system.

In relation to the movement of a material point moving called the vector characterizing this change. It has the property of additivity. Usually denoted by the symbol S → (\displaystyle (\vec (S))) - from Italian. s postamento (movement).

The vector modulus S → (\displaystyle (\vec (S))) is the displacement modulus, in International system units (SI) are measured in meters; in the GHS system - in centimeters.

You can define movement as a change in the radius vector of a point: Δ r → (\displaystyle \Delta (\vec (r))) .

The displacement module coincides with the distance traveled if and only if the direction of velocity does not change during movement. In this case, the trajectory will be a straight line segment. In any other case, for example, with curvilinear motion, it follows from the triangle inequality that the path is strictly longer.

The instantaneous speed of a point is defined as the limit of the ratio of movement to the small period of time during which it was accomplished. More strictly:

V → = lim Δ t → 0 Δ r → Δ t = d r → d t (\displaystyle (\vec (v))=\lim \limits _(\Delta t\to 0)(\frac (\Delta (\vec (r)))(\Delta t))=(\frac (d(\vec (r)))(dt))) .

III. Trajectory, path and movement

The position of a material point is determined in relation to some other, arbitrarily chosen body, called reference body. Contacts him frame of reference– a set of coordinate systems and clocks associated with a reference body.

In the Cartesian coordinate system, the position of point A in at the moment time in relation to this system is characterized by three coordinates x, y and z or radius vector r– a vector drawn from the origin of the coordinate system to a given point. When a material point moves, its coordinates change over time. r=r(t) or x=x(t), y=y(t), z=z(t) – kinematic equations of a material point.

The main task of mechanics– knowing the state of the system at some initial moment of time t 0 , as well as the laws governing the movement, determine the state of the system at all subsequent moments of time t.

Trajectory movement of a material point - a line described by this point in space. Depending on the shape of the trajectory, there are rectilinear And curvilinear point movement. If the trajectory of a point is a flat curve, i.e. lies entirely in one plane, then the motion of the point is called flat.

The length of the section of the trajectory AB traversed by the material point since the start of time is called path lengthΔs is a scalar function of time: Δs=Δs(t). Unit of measurement – meter(m) – the length of the path traveled by light in a vacuum in 1/299792458 s.

IV. Vector method of specifying movement

Radius vector r– a vector drawn from the origin of the coordinate system to a given point. Vector Δ r=r-r 0 , drawn from the initial position of a moving point to its position at a given time is called moving(increment of the radius vector of a point over the considered period of time).

The average velocity vector v> is the ratio of the increment Δr of the radius vector of a point to the time interval Δt: (1). The direction of the average speed coincides with the direction of Δr. With an unlimited decrease in Δt, the average speed tends to a limiting value, which is called the instantaneous speed v. Instantaneous speed is the speed of a body at a given moment of time and at a given point of the trajectory: (2). Instantaneous speed yes vector quantity, equal to the first derivative of the radius vector of the moving point with respect to time.

To characterize the speed of change of speed v points in mechanics, a vector physical quantity called acceleration.

Medium acceleration uneven movement in the interval from t to t+Δt is called a vector quantity, equal to the ratio speed change Δ v to the time interval Δt:

Instantaneous acceleration a material point at time t will be the limit of average acceleration: (4). Acceleration A is a vector quantity equal to the first derivative of speed with respect to time.

V. Coordinate method of specifying movement

The position of point M can be characterized by the radius vector r or three coordinates x, y and z: M(x,y,z). The radius vector can be represented as the sum of three vectors directed along the coordinate axes: (5).

From the definition of speed (6). Comparing (5) and (6) we have: (7). Taking into account (7) formula (6) we can write (8). The speed module can be found: (9).

Similarly for the acceleration vector:

(10),

(11),

A natural way to define movement (describing movement using trajectory parameters)

The movement is described by the formula s=s(t). Each point of the trajectory is characterized by its value s. The radius vector is a function of s and the trajectory can be given by the equation r=r(s). Then r=r(t) can be represented as complex function r. Let's differentiate (14). Value Δs – distance between two points along the trajectory, |Δ r| - the distance between them in a straight line. As the points get closer, the difference decreases. , Where τ – unit vector tangent to the trajectory. , then (13) has the form v=τ v (15). Therefore, the speed is directed tangentially to the trajectory.

Acceleration can be directed at any angle to the tangent to the trajectory of motion. From the definition of acceleration (16). If τ is tangent to the trajectory, then is a vector perpendicular to this tangent, i.e. directed normally. Unit vector, in the normal direction is denoted n. The value of the vector is 1/R, where R is the radius of curvature of the trajectory.

A point located at a distance from the path and R in the direction of the normal n, is called the center of curvature of the trajectory. Then (17). Taking into account the above, formula (16) can be written: (18).

The total acceleration consists of two mutually perpendicular vectors: directed along the trajectory of motion and called tangential, and acceleration directed perpendicular to the trajectory along the normal, i.e. to the center of curvature of the trajectory and called normal.

We find the absolute value of the total acceleration: (19).

Lecture 2 Movement of a material point in a circle. Angular displacement, angular velocity, angular acceleration. Relationship between linear and angular kinematic quantities. Vectors of angular velocity and acceleration.

Lecture outline

Kinematics rotational movement

In rotational motion, the measure of displacement of the entire body over a short period of time dt is the vector dφ elementary body rotation. Elementary turns (denoted by or) can be considered as pseudovectors (as it were).

Angular movement - a vector quantity whose magnitude is equal to the angle of rotation, and the direction coincides with the direction of translational motion right screw (directed along the axis of rotation so that when viewed from its end, the rotation of the body appears to be occurring counterclockwise). The unit of angular displacement is rad.

The rate of change in angular displacement over time is characterized by angular velocity ω . Angular velocity solid– vector physical quantity that characterizes the rate of change in the angular displacement of a body over time and is equal to the angular displacement performed by the body per unit time:

Directed vector ω along the axis of rotation in the same direction as dφ (according to the right screw rule). The unit of angular velocity is rad/s

The rate of change in angular velocity over time is characterized by angular acceleration ε

(2).

The vector ε is directed along the axis of rotation in the same direction as dω, i.e. with accelerated rotation, with slow rotation.

The unit of angular acceleration is rad/s2.

During the time dt an arbitrary point of a rigid body A move to dr, having walked the path ds. From the figure it is clear that dr equals vector product angular movement dφ to radius – point vector r : dr =[ dφ · r ] (3).

Linear speed of a point is related to the angular velocity and radius of the trajectory by the relation:

In vector form, the formula for linear speed can be written as vector product: (4)

By definition of the vector product its module is equal to , where is the angle between the vectors and , and the direction coincides with the direction of translational motion of the right propeller as it rotates from to .

Let's differentiate (4) with respect to time:

Considering that - linear acceleration, - angular acceleration, and - linear velocity, we obtain:

The first vector on the right side is directed tangent to the trajectory of the point. It characterizes the change in linear velocity modulus. Therefore, this vector is the tangential acceleration of the point: a τ =[ ε · r ] (7). The tangential acceleration module is equal to a τ = ε · r. The second vector in (6) is directed towards the center of the circle and characterizes the change in the direction of linear velocity. This vector is the normal acceleration of the point: a n =[ ω · v ] (8). Its modulus is equal to a n =ω·v or taking into account that v= ω· r, a n = ω 2 · r= v2 / r (9).

Special cases of rotational motion

With uniform rotation: , hence .

Uniform rotation can be characterized rotation period T- the time it takes for a point to complete one full revolution,

Rotational speed - the number of full revolutions made by a body during its uniform motion in a circle, per unit of time: (11)

Speed unit - hertz (Hz).

With uniformly accelerated rotational motion :

(13), (14) (15).

Lecture 3 Newton's first law. Strength. The principle of independence active forces. Resultant force. Weight. Newton's second law. Pulse. Law of conservation of momentum. Newton's third law. Moment of impulse of a material point, moment of force, moment of inertia.

Lecture outline

Newton's first law

Newton's second law

Newton's third law

Moment of impulse of a material point, moment of force, moment of inertia

Newton's first law. Weight. Strength

Newton's first law: There are reference systems relative to which bodies move rectilinearly and uniformly or are at rest if no forces act on them or the action of the forces is compensated.

Newton's first law is true only in inertial system reference and asserts the existence of an inertial reference system.

Inertia- this is the property of bodies to strive to keep their speed constant.

Inertia call the property of bodies to prevent a change in speed under the influence of an applied force.

Body weight– this is a physical quantity that is a quantitative measure of inertia, it is a scalar additive quantity. Additivity of mass is that the mass of a system of bodies is always equal to the sum of the masses of each body separately. Weight– the basic unit of the SI system.

One form of interaction is mechanical interaction. Mechanical interaction causes deformation of bodies, as well as a change in their speed.

Strength– this is a vector quantity that is a measure of the mechanical impact on the body from other bodies, or fields, as a result of which the body acquires acceleration or changes its shape and size (deforms). Force is characterized by its module, direction of action, and point of application to the body.

General methods for determining displacements

 1 =X 1  11 +X 2  12 +X 3  13 +…

 2 =X 1  21 +X 2  22 +X 3  23 +…

 3 =X 1  31 +X 2  32 +X 3  33 +…

Abota standing forces: A=P P, P – generalized force– any load (concentrated force, concentrated moment, distributed load),  P – generalized movement(deflection, rotation angle). The designation  mn means movement in the direction of the generalized force “m”, which is caused by the action of the generalized force “n”. Total displacement caused by several force factors:  P = P P + P Q + P M . Movements caused by a single force or a single moment:  – specific displacement . If a unit force P = 1 caused a displacement  P, then the total displacement caused by the force P will be:  P = P P. If the force factors acting on the system are designated X 1, X 2, X 3, etc. , then movement in the direction of each of them:

where X 1  11 =+ 11; X 2  12 =+ 12 ; Х i  m i =+ m i . Dimension of specific movements:

, J-joules, the dimension of work is 1J = 1Nm.

Job external forces, acting on the elastic system:

.

– the actual work under the static action of a generalized force on an elastic system is equal to half the product of the final value of the force and the final value of the corresponding displacement. Job internal forces(elastic forces) in the case of flat bending:

,

k – coefficient taking into account the uneven distribution of tangential stresses over the area cross section, depends on the shape of the section.

Based on the law of conservation of energy: potential energy U=A.

Work reciprocity theorem (Betley's theorem) . Two states of an elastic system:

 1

1 – movement in direction. force P 1 from the action of force P 1;

 12 – movement in direction. force P 1 from the action of force P 2;

 21 – movement in direction. force P 2 from the action of force P 1;

 22 – movement in direction. force P 2 from the action of force P 2.

A 12 =P 1  12 – work done by the force P 1 of the first state on the movement in its direction caused by the force P 2 of the second state. Similarly: A 21 =P 2  21 – work of the force P 2 of the second state on movement in its direction caused by the force P 1 of the first state. A 12 = A 21. The same result is obtained for any number of forces and moments. Work reciprocity theorem: P 1  12 = P 2  21 .

The work of the forces of the first state on displacements in their directions caused by the forces of the second state is equal to the work of the forces of the second state on displacements in their directions caused by the forces of the first state.

Theorem on the reciprocity of displacements (Maxwell's theorem) If P 1 =1 and P 2 =1, then P 1  12 =P 2  21, i.e.  12 = 21, in the general case  mn = nm.

For two unit states of an elastic system, the displacement in the direction of the first unit force caused by the second unit force is equal to the displacement in the direction of the second unit force caused by the first force.

Universal method for determining displacements (linear and rotation angles) – Mohr's method. A unit generalized force is applied to the system at the point for which the generalized displacement is sought. If the deflection is determined, then the unit force is a dimensionless concentrated force; if the angle of rotation is determined, then it is a dimensionless unit moment. In the case of a spatial system, there are six components of internal forces. The generalized displacement is determined by the formula (Mohr's formula or integral):

The line above M, Q and N indicates that these internal forces are caused by a unit force. To calculate the integrals included in the formula, you need to multiply the diagrams of the corresponding forces. The procedure for determining the movement: 1) for a given (real or cargo) system, find the expressions M n, N n and Q n; 2) in the direction of the desired movement, a corresponding unit force (force or moment) is applied; 3) determine efforts

from the action of a single force; 4) the found expressions are substituted into the Mohr integral and integrated over the given sections. If the resulting mn >0, then the displacement coincides with the selected direction of the unit force, if

For flat design:

Usually, when determining displacements, the influence of longitudinal deformations and shear, which are caused by longitudinal N and transverse Q forces, is neglected; only displacements caused by bending are taken into account. For a flat system it will be:

.

IN

calculation of the Mohr integralVereshchagin's method . Integral

for the case when the diagram for a given load has an arbitrary outline, and for a single load it is rectilinear, it is convenient to determine it using the graph-analytical method proposed by Vereshchagin.

, where is the area of the diagram M r from the external load, y c is the ordinate of the diagram from a unit load under the center of gravity of the diagram M r. The result of multiplying diagrams is equal to the product of the area of one of the diagrams and the ordinate of another diagram, taken under the center of gravity of the area of the first diagram. The ordinate must be taken from a straight-line diagram. If both diagrams are straight, then the ordinate can be taken from any one.

P

moving:

. The calculation using this formula is carried out in sections, in each of which the straight-line diagram should be without fractures. A complex diagram M p is divided into simple ones geometric shapes, for which it is easier to determine the coordinates of the centers of gravity. When multiplying two diagrams that have the form of trapezoids, it is convenient to use the formula:

. The same formula is also suitable for triangular diagrams, if you substitute the corresponding ordinate = 0.

P

Under the action of a uniformly distributed load on a simply supported beam, the diagram is constructed in the form of a convex quadratic parabola, the area of which

(for fig.

, i.e.

, x C =L/2).

D

for “blind” embedding with uniform distributed load we have a concave quadratic parabola for which

;

,

, x C = 3L/4. The same can be obtained if the diagram is represented by the difference between the area of a triangle and the area of a convex quadratic parabola:

. The "missing" area is considered negative.

Castigliano's theorem .

– the displacement of the point of application of the generalized force in the direction of its action is equal to the partial derivative of the potential energy with respect to this force. Neglecting the influence of axial and transverse forces on the movement, we have potential energy:

, where

.

What is the definition of movement in physics?

Sad Roger

In physics there is movement absolute value a vector drawn from the starting point of the body’s trajectory to the final point. In this case, the shape of the path along which the movement took place (that is, the trajectory itself), as well as the size of this path, does not matter at all. Let's say, the movement of Magellan's ships - well, at least the one that eventually returned (one of three) - is equal to zero, although the distance traveled is wow.

Is Tryfon

Displacement can be viewed in two ways. 1. Change in body position in space. Moreover, regardless of the coordinates. 2. The process of movement, i.e. change in position over time. You can argue about point 1, but to do this you need to recognize the existence of absolute (initial) coordinates.

Movement is a change in the location of a certain physical body in space relative to the reference system used.

This definition is given in kinematics - a subsection of mechanics that studies the movement of bodies and the mathematical description of movement.

Displacement is the absolute value of a vector (that is, a straight line) connecting two points on a path (from point A to point B). Displacement differs from path in that it is a vector value. This means that if the object came to the same point from which it started, then the displacement is zero. But there is no way. A path is the distance an object has traveled due to its movement. To better understand, look at the picture:

What is path and movement from a physics point of view? and what is the difference between them....

very necessary) please answer)

User deleted

Alexander kalapats

Path is a scalar physical quantity that determines the length of the trajectory section traveled by the body during a given time. The path is a non-negative and non-decreasing function of time.
Displacement is a directed segment (vector) connecting the position of the body at the initial moment of time with its position at the final moment of time.
Let me explain. If you leave home, go to visit a friend, and return home, then your path will be equal to the distance between your house and your friend’s house, multiplied by two (back and forth), and your movement will be equal to zero, because at the final moment of time you will find yourself in the same place as at the initial moment, i.e. at your home. A path is a distance, a length, i.e. a scalar quantity that has no direction. Displacement is a directional, vector quantity, and the direction is specified by a sign, i.e. displacement can be negative (If we assume that when you reach your friend’s house you have made a movement s, then when you walk from your friend to his house, you will have made a movement -s , where the minus sign means that you walked in the opposite direction to the one in which you walked from the house to your friend).

Forserr33v

Path is a scalar physical quantity that determines the length of the trajectory section traveled by the body during a given time. The path is a non-negative and non-decreasing function of time.
Displacement is a directed segment (vector) connecting the position of the body at the initial moment of time with its position at the final moment of time.
Let me explain. If you leave home, go to visit a friend, and return home, then your path will be equal to the distance between your house and your friend’s house multiplied by two (there and back), and your movement will be equal to zero, because at the final moment of time you will find yourself in the same place as at the initial moment, i.e. at home. A path is a distance, a length, i.e. a scalar quantity that has no direction. Displacement is a directed, vector quantity, and the direction is specified by a sign, i.e., displacement can be negative (If we assume that when you reach your friend’s house you have made a movement s, then when you walk from your friend to his house, you will make a movement -s , where the minus sign means that you walked in the opposite direction to the one in which you walked from the house to your friend).

Displacement, displacement, movement, migration, movement, rearrangement, regrouping, transfer, transportation, transition, relocation, transfer, travel; shifting, moving, telekinesis, epeirophoresis, relocation, rolling, waddle,... ... Dictionary of synonyms

MOVEMENT, movement, cf. (book). 1. Action under Ch. move move. Moving within the service. 2. Action and condition according to Ch. move move. Moving layers earth's crust. Dictionary Ushakova. D.N. Ushakov. 1935 1940 ... Ushakov's Explanatory Dictionary

In mechanics, a vector connecting the positions of a moving point at the beginning and end of a certain period of time; The P vector is directed along the chord of the point's trajectory. Physical encyclopedic dictionary. M.: Soviet encyclopedia. Editor-in-Chief A.M.... ... Physical encyclopedia

MOVE, eat, eat; still (yon, ena); owls, who what. Place, transfer to another place. P. scenery. P. brigade to another site. Displaced persons (persons forcibly displaced from their country). Ozhegov's explanatory dictionary. S.I.... ... Ozhegov's Explanatory Dictionary

- (relocation) Relocation of an office, enterprise, etc. to another place. Often it is caused by a merger or acquisition. Sometimes employees receive a relocation allowance, which is intended to encourage them to stay in their current location... ... Dictionary of business terms

moving- - Telecommunications topics, basic concepts EN redeployment ... Technical Translator's Guide

Moving,- Displacement, mm, the amount of change in the position of any point of an element of a window block (usually a frame impost or vertical bars of sashes) in the direction normal to the plane of the product under the influence of wind load. Source: GOST... ...

moving- Migration of material in the form of a solution or suspension from one soil horizon to another... Dictionary of Geography

moving- 3.14 transfer (in relation to storage location): Changing the storage location of a document. Source: GOST R ISO 15489 1 2007: System of information standards... Dictionary-reference book of terms of normative and technical documentation

moving- ▲ change of position, motionless in space, change of position in space; transformation of a figure that preserves the distances between points of the figure; movement to another place. movement. forward motion... ... Ideographic Dictionary of the Russian Language

Books

GESNm 81-03-40-2001. Part 40. Additional movement of equipment and material resources. State estimate standards. State elemental estimate standards for installation of equipment (hereinafter referred to as GESNm) are intended to determine the need for resources (labor costs of workers,...
Movement of people and cargo in near-Earth space through technical ferrographitization, R. A. Sizov. This publication is the second applied edition to the books by R. A. Sizov “Matter, Antimatter and Energy Environment - Physical Triad real world", in which, based on what was discovered...

In the Cartesian coordinate system, the position of point A at a given time relative to this system is characterized by three coordinates x, y and z or a radius vector r – a vector drawn from the origin of the coordinate system to a given point. When a material point moves, its coordinates change over time. r=r(t) or x=x(t), y=y(t), z=z(t) – kinematic equations of a material point.

IV. Vector method of specifying movement

Radius vector r – a vector drawn from the origin of the coordinate system to a given point. Vector Δ r=r-r 0 , drawn from the initial position of a moving point to its position at a given time is called moving(increment of the radius vector of a point over the considered period of time).

Average speed vector< v> called the increment ratio Δ r radius vector of a point to the time interval Δt: (1). The direction of the average speed coincides with the direction Δ r.With an unlimited decrease in Δt, the average speed tends to a limiting value, which is called instantaneous speedv. Instantaneous speed is the speed of a body at a given moment of time and at a given point of the trajectory: (2). Instantaneous speed v is a vector quantity equal to the first derivative of the radius vector of a moving point with respect to time.

To characterize the speed of change of speed v points in mechanics, a vector physical quantity called acceleration.

Medium acceleration uneven motion in the interval from t to t+Δt is called a vector quantity equal to the ratio of the change in speed Δ v to the time interval Δt:

V. Coordinate method of specifying movement

From the definition of speed (6). Comparing (5) and (6) we have: (7). Taking into account (7), formula (6) can be written (8). The speed module can be found:(9).

Similarly for the acceleration vector:

(10),

(11),

A natural way to define movement (describing movement using trajectory parameters)

The movement is described by the formula s=s(t). Each point of the trajectory is characterized by its value s. The radius vector is a function of s and the trajectory can be given by the equation r=r(s). Then r=r(t) can be represented as a complex function r. Let's differentiate (14). Value Δs – distance between two points along the trajectory, |Δ r| - the distance between them in a straight line. As the points get closer, the difference decreases. , Where τ – unit vector tangent to the trajectory. , then (13) has the form v=τ v (15). Therefore, the speed is directed tangentially to the trajectory.

We find the absolute value of the total acceleration: (19).

Lecture outline

Kinematics of rotational motion

The rate of change in angular displacement over time is characterized by angular velocity ω . The angular velocity of a rigid body is a vector physical quantity that characterizes the rate of change in the angular displacement of a body over time and is equal to the angular displacement performed by the body per unit time:

Directed vector ω along the axis of rotation in the same direction as dφ (according to the right screw rule). Unit of angular velocity - rad/s

The rate of change in angular velocity over time is characterized by angular acceleration ε

(2).

The vector ε is directed along the axis of rotation in the same direction as dω, i.e. with accelerated rotation, with slow rotation.

The unit of angular acceleration is rad/s 2 .

During the time dt an arbitrary point of a rigid body A move to dr, having walked the path ds. From the figure it is clear that dr equal to the vector product of the angular displacement dφ to radius – point vector r : dr =[ dφ · r ] (3).

Linear speed of a point is related to the angular velocity and radius of the trajectory by the relation:

In vector form, the formula for linear speed can be written as vector product: (4)

By definition of the vector product its module is equal to , where is the angle between the vectors and, and the direction coincides with the direction of translational motion of the right propeller as it rotates from to.

Let's differentiate (4) with respect to time:

Considering that - linear acceleration, - angular acceleration, and - linear velocity, we obtain:

Special cases of rotational motion

With uniform rotation: , hence .

Uniform rotation can be characterized rotation period T- the time it takes for a point to complete one full revolution,

Rotational speed - the number of full revolutions made by a body during its uniform motion in a circle, per unit of time: (11)

Speed unit - hertz (Hz).

With uniformly accelerated rotational motion :

Lecture 3 Newton's first law. Strength. The principle of independence of acting forces. Resultant force. Weight. Newton's second law. Pulse. Law of conservation of momentum. Newton's third law. Moment of impulse of a material point, moment of force, moment of inertia.

Lecture outline

Newton's first law

Newton's second law

Newton's third law

Moment of impulse of a material point, moment of force, moment of inertia

Newton's first law. Weight. Strength

Newton's first law: There are reference systems relative to which bodies move rectilinearly and uniformly or are at rest if no forces act on them or the action of the forces is compensated.

Newton's first law is satisfied only in the inertial frame of reference and asserts the existence of the inertial frame of reference.

Inertia- this is the property of bodies to strive to keep their speed constant.

Inertia call the property of bodies to prevent a change in speed under the influence of an applied force.

One form of interaction is mechanical interaction. Mechanical interaction causes deformation of bodies, as well as a change in their speed.

You have already encountered the concept of a path many times. Let us now get acquainted with a new concept for you - moving, which is more informative and useful in physics than the concept of a path.

Let's say you need to transport cargo from point A to point B on the other side of the river. This can be done by car across the bridge, by boat on the river or by helicopter. In each of these cases, the path traveled by the load will be different, but the movement will be the same: from point A to point B.

By moving is a vector drawn from the initial position of a body to its final position. The displacement vector shows the distance the body has moved and the direction of movement. note that direction of movement and direction of movement are two different concepts. Let's explain this.

Consider, for example, the trajectory of a car from point A to the middle of the bridge. Let us designate the intermediate points as B1, B2, B3 (see figure). You see that on segment AB1 the car was traveling northeast (first blue arrow), on segment B1B2 - southeast (second blue arrow), and on segment B2B3 - north (third blue arrow). So, at the moment of passing the bridge (point B3), the direction of movement was characterized by the blue vector B2B3, and the direction of movement was characterized by the red vector AB3.

So, the movement of the body is vector quantity, that is, having a spatial direction and a numerical value (module). Unlike movement, path is scalar quantity, that is, having only a numerical value (and no spatial direction). The path is indicated by the symbol l, movement is indicated by a symbol (important: with an arrow). Symbol s without an arrow indicate the displacement module. Note: the image of any vector in the drawing (in the form of an arrow) or its mention in the text (in the form of a word) makes the presence of an arrow above the designation optional.

Why did physics not limit itself to the concept of path, but introduced a more complex (vector) concept of displacement? Knowing the module and direction of movement, you can always say where the body will be (in relation to its initial position). Knowing the path, the position of the body cannot be determined. For example, knowing only that a tourist has walked 7 km, we cannot say anything about where he is now.

Task. While hiking across the plain, the tourist walked north 3 km, then turned east and walked another 4 km. How far was he from the starting point of the route? Draw its movement.

Solution 1 – using ruler and protractor measurements.

Displacement is a vector connecting the initial and final positions of the body. Let's draw it on checkered paper on a scale: 1 km - 1 cm (drawing on the right). Measuring the module of the constructed vector with a ruler, we get: 5 cm. According to the scale we have chosen, the module of the tourist’s movement is 5 km. But let's remember: to know a vector means to know its magnitude and direction. Therefore, using a protractor, we determine: the direction of movement of the tourist is 53° with the direction to the north (check it yourself).

Solution 2 – without using a ruler or protractor.

Since the angle between the tourist’s movements to the north and east is 90°, we apply the Pythagorean theorem and find the length of the hypotenuse, since it is also the modulus of the tourist’s movement:

As you can see, this value coincides with that obtained in the first solution. Now let’s determine the angle α between the displacement (hypotenuse) and the direction to the north (the adjacent leg of the triangle):

So, the problem was solved in two ways with matching answers.

Class: 9

Lesson objectives:

Educational:
– introduce the concepts of “movement”, “path”, “trajectory”.
Developmental:
- develop logical thinking, correct physical speech, use appropriate terminology.
Educational:
– achieve high class activity, attention, and concentration of students.

Equipment:

plastic bottle with a capacity of 0.33 liters with water and a scale;
medical bottle with a capacity of 10 ml (or small test tube) with a scale.

Demonstrations: Determining displacement and distance traveled.

Lesson progress

1. Updating knowledge.

- Hello, guys! Sit down! Today we will continue to study the topic “Laws of interaction and motion of bodies” and in the lesson we will get acquainted with three new concepts (terms) related to this topic. In the meantime, let's check your homework for this lesson.

2. Checking homework.

Before class, one student writes the solution to the following homework assignment on the board:

Two students are given cards with individual tasks that are completed during the oral test ex. 1 page 9 of the textbook.

1. Which coordinate system (one-dimensional, two-dimensional, three-dimensional) should be chosen to determine the position of bodies:

a) tractor in the field;
b) helicopter in the sky;
c) train
G) chess piece on the board.

2. Given the expression: S = υ 0 t + (a t 2) / 2, express: a, υ 0

1. Which coordinate system (one-dimensional, two-dimensional, three-dimensional) should be chosen to determine the position of such bodies:

a) chandelier in the room;
b) elevator;
c) submarine;
d) plane on the runway.

2. Given the expression: S = (υ 2 – υ 0 2) / 2 · a, express: υ 2, υ 0 2.

3. Study of new theoretical material.

Associated with changes in the coordinates of the body is the quantity introduced to describe the movement - MOVEMENT.

The displacement of a body (material point) is a vector connecting the initial position of the body with its subsequent position.

Movement is usually denoted by the letter . In SI, displacement is measured in meters (m).

– [m] – meter.

Displacement - magnitude vector, those. In addition to the numerical value, it also has a direction. The vector quantity is represented as segment, which begins at a certain point and ends with a point indicating the direction. Such an arrow segment is called vector.

– vector drawn from point M to M 1

Knowing the displacement vector means knowing its direction and magnitude. The modulus of a vector is a scalar, i.e. numerical value. Knowing the initial position and the vector of movement of the body, you can determine where the body is located.

While moving material point occupies different positions in space relative to the chosen reference system. In this case, the moving point “describes” some line in space. Sometimes this line is visible - for example, a high-flying plane can leave a trail in the sky. A more familiar example is the mark of a piece of chalk on a blackboard.

An imaginary line in space along which a body moves is called TRAJECTORY body movements.

The trajectory of a body is a continuous line that is described by a moving body (considered as a material point) in relation to the selected reference system.

The movement in which all points body moving along the same trajectories, called progressive.

Very often the trajectory is an invisible line. Trajectory moving point can be direct or crooked line. According to the shape of the trajectory movement It happens straightforward And curvilinear.

The path length is PATH. The path is a scalar quantity and is denoted by the letter l. The path increases if the body moves. And remains unchanged if the body is at rest. Thus, the path cannot decrease over time.

The displacement module and the path can coincide in value only if the body moves along a straight line in the same direction.

What is the difference between a path and a movement? These two concepts are often confused, although in fact they are very different from each other. Let's look at these differences: ( Appendix 3) (distributed in the form of cards to each student)

The path is a scalar quantity and is characterized only by a numerical value.
Displacement is a vector quantity and is characterized by both a numerical value (module) and direction.
When a body moves, the path can only increase, and the displacement module can both increase and decrease.
If the body returns to the starting point, its displacement is zero, but the path is not zero.

	Path	Moving
Definition	The length of the trajectory described by a body in a certain time	A vector connecting the initial position of the body with its subsequent position
Designation	l [m]	S [m]
Character physical quantities	Scalar, i.e. determined only by numeric value	Vector, i.e. determined by numerical value (modulus) and direction
The need for introduction	Knowing the initial position of the body and the path l traveled during the time interval t, it is impossible to determine the position of the body in given moment time t	Knowing the initial position of the body and S for a period of time t, the position of the body at a given moment of time t is uniquely determined
	l = S in the case of rectilinear motion without returns

4. Demonstration of experience (students perform independently in their places at their desks, the teacher, together with the students, performs a demonstration of this experience)

Fill a plastic bottle with a scale to the neck with water.
Fill the bottle with the scale with water to 1/5 of its volume.
Tilt the bottle so that the water comes up to the neck, but does not flow out of the bottle.
Quickly lower the bottle of water into the bottle (without closing it with the stopper) so that the neck of the bottle enters the water of the bottle. The bottle floats on the surface of the water in the bottle. Some of the water will spill out of the bottle.
Screw the bottle cap on.
Squeeze the sides of the bottle and lower the float to the bottom of the bottle.

By releasing the pressure on the walls of the bottle, make the float float to the surface. Determine the path and movement of the float:__________________________________________________________
Lower the float to the bottom of the bottle. Determine the path and movement of the float:________________________________________________________________________________
Make the float float and sink. What is the path and movement of the float in this case?_______________________________________________________________________________________

5. Exercises and questions for review.

Do we pay for the journey or transportation when traveling in a taxi? (Path)
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 movement of the ball. (Path – 4 m, movement – 2 m.)

6. Lesson summary.

Review of lesson concepts:

– movement;
– trajectory;
- path.

7. Homework.

§ 2 of the textbook, questions after the paragraph, exercise 2 (p. 12) of the textbook, repeat the lesson experience at home.

References

1. Peryshkin A.V., Gutnik E.M.. Physics. 9th grade: textbook for general educational institutions - 9th ed., stereotype. – M.: Bustard, 2005.