Biology 

Trajectory is the way the formula moves. Trajectory

With the help of this video tutorial, you can independently study the topic "Moving", which is included in school course physics for grade 9. From this lecture, students will be able to deepen their knowledge of movement. The teacher will recall the first characteristic of movement - the distance traveled, and then proceed to the definition of movement in physics.

The first characteristic of motion introduced by us earlier was the distance traveled. Recall that it is denoted by the letter S (sometimes the designation L is found) and is measured in SI in meters.

Distance traveled is a scalar quantity, that is, a quantity that is characterized only by a numerical value. This means that we cannot predict where the body will be at the moment we need. We can only talk about the total distance traveled by the body (Fig. 1).

Rice. 1. Knowing only the distance traveled, it is impossible to determine the position of the body at an arbitrary point in time

To characterize the location of the body at an arbitrary moment, a quantity called displacement is introduced. Moving - vector quantity, i.e., this is a quantity that is characterized not only by a numerical value, but also by a direction.

The movement is indicated in the same way as the distance traveled, by the letter S, but, unlike the distance traveled, an arrow is placed above the letter, thereby emphasizing that this is a vector quantity: .

What moving And distance traveled denoted by one letter is somewhat misleading, but we must clearly understand the difference between the path traveled and the movement. Note again that sometimes the path is denoted by L. This avoids confusion.

Definition

Displacement is a vector (directed line segment) that connects the starting point of the body's movement with its end point (Fig. 2).

Rice. 2. Displacement is a vector quantity

Recall that the passed path is the length of the path. This means that the path and displacement are completely different physical quantities, although sometimes there are situations when they coincide numerically.

Rice. 3. Path and displacement module are the same

On fig. 3, the simplest case is considered, when the body moves along a straight line (axis Oh). The body starts its movement from point 0 and gets to point A. In this case, we can say that the displacement modulus is equal to the distance traveled: .

An example of such a movement is an airplane flight (for example, from St. Petersburg to Moscow). If the movement was strictly rectilinear, then the displacement modulus will be equal to the distance traveled.

Rice. 4. The value of the path is greater than the displacement modulus

On fig. 4 the body moves along a curved line, i.e., the movement is curvilinear (from point A to point B). It can be seen from the figure that the displacement module (straight line) will be less than the path traveled, i.e. the length of the traveled path and the length of the displacement vector are not equal.

Rice. 5. Closed trajectory

On fig. 5 the body moves along a closed curve. It leaves point A and returns to the same point. The displacement modulus is , and distance traveled is the length of the entire curve, .

This case can be characterized by the following example. The student left the house in the morning, went to school, spent the whole day studying, besides this, he visited several other places (shop, gym, library) and returned home. Please note: as a result, the student ended up at home, which means that his displacement is 0 (Fig. 6).

Rice. 6. Student displacement is zero

When we are talking about moving, it is important to remember that moving depends on the frame of reference in which the motion is considered.


Rice. 7. Determination of the modulus of displacement of the body

The body moves in a plane XOY. Point A is the initial position of the body. Her coordinates. The body moves to a point. The vector is the displacement of the body: .

You can calculate the modulus of displacement as the hypotenuse right triangle, using the Pythagorean theorem: . To find the displacement vector, it is necessary to find the angle between the axis Oh and the displacement vector.

We can choose the system arbitrarily, that is, direct the coordinate axes in the way that is convenient for us, the main thing is to consider the projections of all vectors in the future in the same chosen coordinate system.

Conclusion

In conclusion, it can be noted that we have become acquainted with an important quantity - displacement. Once again, note that the displacement and the path can only coincide in the case of rectilinear movement, without changing the direction of such movement.

Bibliography

  1. Kikoin I.K., Kikoin A.K. Physics: textbook for grade 9 high school. - M.: Enlightenment.
  2. Peryshkin A.V., Gutnik E.M., Physics. Grade 9: textbook for general education. institutions/A. V. Peryshkin, E. M. Gutnik. - 14th ed., stereotype. - M.: Bustard, 2009. - 300.
  3. Sokolovich Yu.A., Bogdanova G.S.. Physics: Handbook with examples of problem solving. - 2nd edition redistribution. - X .: Vesta: Publishing house "Ranok", 2005. - 464 p.
  1. Internet portal "vip8082p.vip8081p.beget.tech" ()
  2. Internet portal "foxford.ru" ()

Homework

  1. What is path and movement? What is the difference?
  2. The motorcyclist pulled out of the garage and headed north. Drove 5 km, then turned west and drove also 5 km. How far from the garage will it be?
  3. The minute hand has gone full circle. Determine the displacement and distance traveled for the point that is at the end of the arrow (the radius of the clock is 10 cm).

Trajectory- this is the line that the body describes when moving.

Bee trajectory

Path is the length of the path. That is, the length of that possibly curved line along which the body moved. Path scalar ! moving- vector quantity ! This is a vector that is drawn from the starting point of the body to the end point. Has a numerical value equal to the length of the vector. Distance and displacement are essentially different physical quantities.

You can find different path and movement designations:

Amount of movements

Let the body move s 1 during the time interval t 1 , and move s 2 during the next time interval t 2 . Then for the entire time of movement, the displacement s 3 is the vector sum

Uniform movement

Movement with a constant modulo and direction speed. What does it mean? Consider the movement of the car. If she is driving in a straight line, the speedometer shows the same speed value (module of speed), then this movement is uniform. Should the car change direction (turn), this will mean that the velocity vector has changed its direction. The velocity vector is directed towards the direction the car is going. Such movement cannot be considered uniform, despite the fact that the speedometer shows the same number.

The direction of the velocity vector always coincides with the direction of motion of the body

Can the movement on the carousel be considered uniform (if there is no acceleration or deceleration)? It is impossible, the direction of movement is constantly changing, and hence the velocity vector. From the reasoning, we can conclude that uniform motion - it is always moving in a straight line! So, with uniform motion, the path and displacement are the same (explain why).

It is easy to imagine that with uniform motion for any equal intervals of time, the body will move the same distance.

Position material point is defined 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 the body of reference.

In the Cartesian coordinate system, the position of point A in this moment time with respect to this system is characterized by three coordinates x, y and z or the radius vector r – a vector drawn from the origin of the coordinate system to 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 time t 0 , as well as the laws governing the movement, determine the state of the system at all subsequent times t.

Trajectory motion 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 the point is a plane curve, i.e. lies entirely in one plane, then the movement of the point is called flat.

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

IV. Vector way to define motion

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

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

To characterize the rate of change of speed v point in mechanics, a vector physical quantity is introduced, called acceleration.

Average acceleration non-uniform movement 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:

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

V. Coordinate method of motion assignment

The position of the point M can be characterized by the radius - the 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). Given (7), formula (6) can be written (8). The speed modulus can be found:(9).

Similarly for the acceleration vector:

(10),

(11),

    Natural way of specifying motion (description of motion using trajectory parameters)

The movement is described by the formula s=s(t). Each point of the trajectory is characterized by its value s. Radius - the 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 us differentiate (14). The value Δs is the distance between two points along the trajectory, |Δ r| is the distance between them in a straight line. As the points get closer, the difference decreases. , Where τ is the 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 motion path. From the definition of acceleration (16). If τ - tangent to the trajectory, then - vector perpendicular to this tangent, i.e. directed along the normal. The unit vector, in the direction of the normal is denoted n. The value of the vector is 1/R, where R is the radius of curvature of the trajectory.

Point away from the path at a distance and R in the direction of the normal n, is called the center of curvature of the trajectory. Then (17). Given 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 is called normal.

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

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

Lecture plan

    Kinematics of rotary motion

During rotational motion, the vector dφ elementary rotation of the body. Elementary turns (denoted or) can be seen as pseudovectors (as it were).

Angular movement - vector quantity, the module of which 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 seems to be 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 body– vector physical quantity, which characterizes the rate of change in the angular displacement of the 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 rule of the right screw). Unit of angular velocity - rad/s

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

(2).

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

The unit of angular acceleration is rad/s 2 .

During dt arbitrary point of the rigid body A move to dr, passing the way ds. It can be seen from the figure that dr equals vector product angular displacement dφ by radius – point vector r : dr =[ dφ · r ] (3).

Point Linear Speed is related to the angular velocity and radius of the trajectory by the relation:

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

By definition of a vector product its modulus is , where is the angle between the vectors and, and the direction coincides with the direction of the translational motion of the right screw when it rotates from to .

Differentiate (4) with respect to time:

Considering that - linear acceleration, - angular acceleration, and - linear speed, we get:

The first vector on the right side is directed tangentially to the point trajectory. It characterizes the change in the linear velocity modulus. Therefore, this vector is the tangential acceleration of the point: a τ =[ ε · r ] (7). The tangential acceleration modulus is 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 the 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 given that v = ω· r, a n = ω 2 · r = v 2 / r (9).

    Particular cases of rotational motion

With uniform rotation: , hence .

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

Rotation frequency - the number of complete revolutions made by the body during its uniform motion in a circle, per unit time: (11)

Speed ​​unit - hertz (Hz).

With uniformly accelerated rotational motion :

Lecture 3 Newton's first law. Force. 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 momentum of a material point, moment of force, moment of inertia.

Lecture plan

    Newton's first law

    Newton's second law

    Newton's third law

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

    Newton's first law. Weight. Force

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

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

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

inertia called the property of bodies to prevent a change in speed under the action of an applied force.

Body mass is a physical quantity that is a quantitative measure of inertia, it is a scalar additive quantity. Mass additivity consists in the fact that the mass of a system of bodies is always equal to the sum of the masses of each body separately. Weight is 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.

Force- 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 module, direction of action, point of application to the body.

You have already encountered the concept of a path more than once. Now let's get acquainted with a new concept for you - moving, which is more informative and useful in physics than the notion of a path.

Let's say that from point A to point B on the other side of the river you need to transport cargo. 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.

moving called the vector drawn from the initial position of the 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's denote the intermediate points - B1, B2, B3 (see figure). You can see that on the segment AB1 the car was moving to the northeast (first blue arrow), on the segment B1B2 - to the southeast (second blue arrow), and on the segment B2B3 - to the 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 vector quantity, that is, having a spatial direction and a numerical value (modulus). Unlike movement, the path - scalar, that is, having only a numerical value (and not having a spatial direction). The path is marked with 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 it optional to have an arrow above the designation.

Why in physics they did not limit themselves to the concept of a path, but introduced a more complex (vector) concept of displacement? Knowing the modulus and the direction of movement, you can always tell 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 traveled 7 km, we cannot say anything about where he is now.

Task. In a hike along the plain, the tourist walked north for 3 km, then turned east and walked another 4 km. How far from the starting point of the route was it? Draw his movement.

Solution 1 - with 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). Having measured the module of the constructed vector with a ruler, we get: 5 cm. According to the scale we have chosen, the tourist movement module is 5 km. But let's remind: to know a vector means to know its modulus and direction. Therefore, using a protractor, we determine: the direction of movement of the tourist is 53 ° with the direction to the north (check for yourself).

Solution 2 - without using a ruler and protractor.

Since the angle between the movements of the tourist 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 movement of the tourist:

As you can see, this value is the same as 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 is solved in two ways with coinciding answers.

Displacement, shift, movement, migration, movement, permutation, regrouping, transfer, transportation, transition, relocation, transfer, travel; shifting, moving, telekinesis, epeirophoresis, rebasing, rolling, waddling, ... ... Synonym dictionary

MOVEMENT, displacement, cf. (book). 1. Action according to Ch. move move. Service movement. 2. Action and status according to Ch. move move. Seam movement earth's crust. Dictionary Ushakov. D.N. Ushakov. 1935 1940 ... Explanatory Dictionary of Ushakov

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

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

- (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 should encourage them to stay in the service in this ... ... Glossary of business terms

moving- - Telecommunication topics, basic concepts of EN redeployment ... Technical Translator's Handbook

moving,- Displacement, mm, the amount of change in the position of any point of the element of the window block (usually, the impost of the frame or vertical bars of the sashes) in the direction of the 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 ... Geography Dictionary

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

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

Books

  • GESNm 81-03-40-2001. Part 40. Additional movement of equipment and material resources,. State budget standards. The state elemental estimated norms for the installation of equipment (hereinafter referred to as GESNm) are designed to determine the need for resources (labor costs of workers, ...
  • Movement of people and goods in near-Earth space by means of technical ferrographitization, R. A. Sizov. Present 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 the discovered ...