Lesson Objectives:

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

Equipment:

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

Demos: Determination of displacement and distance travelled.

During the classes

1. Actualization of 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, 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 assignments that are performed during the oral check ex. 1 page 9 of the textbook.

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

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

2. An expression is given: S \u003d υ 0 t + (a t 2) / 2, express: a, υ 0

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

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

2. An expression is given: S \u003d (υ 2 - υ 0 2) / 2 a, express: υ 2, υ 0 2.

3. The study of new theoretical material.

The value introduced to describe the motion is associated with changes in body coordinates, – MOVEMENT.

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

The 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 starts at some point and ends with a point that indicates 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 module. The modulus of a vector is a scalar, i.e. numerical value. Knowing the initial position and the displacement vector of the body, it is possible to determine where the body is located.

In the process of motion, the 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 aircraft 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 describes a moving body (considered as a material point) with respect to the selected reference system.

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

Very often the trajectory is an invisible line. Trajectory moving point can be straight or crooked line. According to the shape of the trajectory motion it happens straightforward and curvilinear.

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

The modulus of displacement and the path can have the same value only if the body moves along a straight line in the same direction.

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

1. Way - scalar and is characterized only numerical value.
2. Displacement is a vector quantity and is characterized by both a numerical value (modulus) and a direction.
3. When the body moves, the path can only increase, and the displacement modulus can both increase and decrease.
4. If the body has returned to the starting point, its displacement is zero, and the path is not equal to zero.

	Way	moving
Definition	The length of the trajectory described by the body for 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. defined only by numeric value	Vector, i.e. defined by numerical value (modulus) and direction
The need for an introduction	Knowing the initial position of the body and the distance l traveled over a period of time t, it is impossible to determine the position of the body at a given time t	Knowing the initial position of the body and S for the time interval t, the position of the body at a given 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)

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

1. By releasing the pressure on the walls of the bottle, achieve the ascent of the float. Determine the path and movement of the float: ______________________________________________________________
2. Lower the float to the bottom of the bottle. Determine the path and movement of the float:______________________________________________________________________________
3. Make the float float and sink. What is the path and movement of the float in this case?

5. Exercises and questions for repetition.

1. Do we pay for the journey or transportation when traveling in a taxi? (Way)
2. The ball fell from a height of 3 m, bounced off the floor and was caught at a height of 1 m. Find the path and move the ball. (Path - 4 m, movement - 2 m.)

6. The result of the lesson.

Repetition of the concepts of the lesson:

– movement;
– trajectory;
- way.

7. Homework.

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

Bibliography

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

What is a trajectory?

Trajectory definition

Trajectory definition:

A trajectory is a line along which a body moves.

In the picture, the body moves from point A to point B along a curved line.

This curved line is the trajectory.

The displacement vector links the start and end points.

And the trajectory is a sequence of points along which the body moves.

Relationship between trajectory and frame of reference

The trajectory depends on the frame of reference. This should be understood as follows: if a body in one frame of reference moves in a straight line, then in another frame of reference it can have a curvilinear trajectory.

To understand how the trajectory depends on the frame of reference, let's give an example.

Consider the trajectory of a point on the surface of a car wheel.

Regarding the driver, i.e. in the reference frame associated with this driver, a point on the surface of the wheel performs a rotational movement in a circle when the car moves.

Relative to the observer outside the car, the point makes two movements: it rotates around the circumference of the wheel and moves forward.

A trajectory is a line along which a body moves. In our example, it turns out that one and the same point moves along different trajectories at the same time. And it is right.

Trajectory

Trajectory of a material point- a line in three-dimensional space, which is a set of points where a material point was, is or will be when it moves in space. . It is significant that the concept of a trajectory has physical meaning even in the absence of any movement along it.

In addition, even in the presence of an object moving along it, the trajectory depicted in a predetermined system of spatial coordinates cannot by itself say anything definite about the reasons for its movement until an analysis of the configuration of the field of forces acting on it in the same coordinate system.

It is no less important that the shape of the trajectory is inextricably linked and depends on the specific frame of reference in which the motion is described.

It is possible to observe the trajectory when the object is stationary, but when the frame of reference is moving. So, starry sky is considered a good model of an inertial and fixed frame of reference. However, with a long exposure, these stars appear to move along circular paths (Fig. 2)

The case is also possible when the body is clearly moving, but the trajectory in the projection onto the observation plane is one fixed point. This is, for example, the case of a bullet flying directly into the observer's eye or a train leaving him.

Trajectory of a free material point

According to Newton's First Law, sometimes called the law of inertia, there must be a system in which free body retains (as a vector) its velocity. Such a frame of reference is called inertial. The trajectory of such a movement is a straight line, and the movement itself is called uniform and rectilinear.

Description of the trajectory

Fig.2 Rectilinear uniformly accelerating motion in one inertial system in general case will be parabolic in another uniformly moving inertial frame of reference. The decomposition of the acting force into components is formally correct and is discussed in the text

It is customary to describe the trajectory of a material point in a predetermined coordinate system using a radius vector , the direction, length and starting point of which depend on time . In this case, the curve described by the end of the radius vector in space can be represented as conjugate arcs of different curvature , located in the general case in intersecting planes . In this case, the curvature of each arc is determined by its radius of curvature directed to the arc from the instantaneous center of rotation, which is in the same plane as the arc itself. Moreover, a straight line is considered as a limiting case of a curve, the radius of curvature of which can be considered equal to infinity. And therefore, the trajectory in the general case can be represented as a set of conjugate arcs.

It is essential that the shape of the trajectory depends on the reference system chosen to describe the motion of a material point. So rectilinear uniformly accelerating motion in an inertial frame will generally be parabolic (as long as the accelerating speed of the body is comparable in magnitude to the relative speed of a uniformly moving inertial reference frame. See Figure 2).

Relationship with speed and normal acceleration

Fig.3 The daily movement of the luminaries in the reference system associated with the camera in the projection onto the drawing plane

The velocity of a material point is always directed tangentially to the arc used to describe the trajectory of the point. In this case, there is a relationship between the magnitude of the speed , normal acceleration and the radius of curvature of the trajectory at a given point:

However, not every movement famous curved speed famous radius and found by the above formula normal(centripetal) acceleration is associated with the manifestation of a force directed along the normal to the trajectory (centripetal force). So, found according to the photo diurnal movement the acceleration of any of the stars does not at all indicate the existence of a force that causes this acceleration, attracting it to polar star as the center of rotation.

Connection with the equations of dynamics

Representing the trajectory as a trace left by movement material points, connects the purely kinematic concept of the trajectory, as a geometric problem, with the dynamics of the motion of a material point, that is, the problem of determining the causes of its motion. In fact, the solution of Newton's equations (in the presence of complete set of initial data) gives the trajectory of a material point.

In the general case, the body is not free in its movement, and restrictions are imposed on its position, and in some cases on speed, - connections. If the links impose restrictions only on the coordinates of the body, then such links are called geometric. If they also propagate at speeds, then they are called kinematic. If the constraint equation can be integrated over time, then such a constraint is called holonomic.

The action of bonds on a system of moving bodies is described by forces called reactions of bonds. In this case, the force included in the left side of equation (1) is the vector sum of the active (external) forces and the reaction of the bonds.

It is essential that in the case of holonomic constraints it becomes possible to describe the motion mechanical systems in generalized coordinates included in the Lagrange equations. The number of these equations depends only on the number of degrees of freedom of the system and does not depend on the number of bodies included in the system, the position of which must be determined for complete description movement.

If the bonds acting in the system are ideal, that is, they do not transfer the energy of motion into other types of energy, then when solving the Lagrange equations, all unknown reactions of the bonds are automatically excluded.

Finally, if active forces belong to the class of potential, then with an appropriate generalization of concepts it becomes possible to use the Lagrange equations not only in mechanics, but also in other areas of physics.

Operating on material point forces in this understanding uniquely determine the shape of the trajectory of its movement (under known initial conditions). The converse statement is not true in the general case, since the same trajectory can take place with different combinations active forces and communication reactions.

Motion under the action of external forces in a non-inertial frame of reference

If the frame of reference is non-inertial (that is, it moves with some acceleration relative to the inertial frame of reference), then expression (1) can also be used in it, however, on the left side it is necessary to take into account the so-called inertial forces (including centrifugal force and Coriolis force, associated with the rotation of a non-inertial frame of reference) .

Illustration

Trajectories of the same motion in stationary and rotating frames of reference. At the top of the inertial frame, you can see that the body is moving in a straight line. Below in the non-inertial it is seen that the body turned away from the observer along the curve.

As an example, consider a theater worker moving in the grate space above the stage in relation to the theater building evenly and straightforward and carrying over rotating scene of a leaky bucket of paint. It will leave a mark on it from falling paint in the form unwinding spiral(if moving from scene rotation center) and swirling- in the opposite case. At this time, his colleague, who is responsible for the cleanliness of the rotating stage and is on it, will therefore be forced to carry a non-leaky bucket under the first, constantly being under the first. And its movement in relation to the building will also be uniform and straightforward, although with respect to the scene, which is non-inertial system, its movement will be twisted and uneven. Moreover, in order to counteract drift in the direction of rotation, he must overcome the action of the Coriolis force with muscular effort, which his upper colleague does not experience above the stage, although the trajectories of both in inertial system theater buildings will represent straight lines.

But one can imagine that the task of the colleagues considered here is precisely the application straight lines on rotating stage. In this case, the bottom must require the top to move along a curve that is mirror image a trace of previously spilled paint, while remaining above any point of a straight line passing in a chosen radial direction. Hence, rectilinear motion in non-inertial system reference will not be for the observer in inertial system.

Furthermore, uniform body movement in one system, can be uneven in another. So, two drops of paint that fell into different moments of time from a leaky bucket, both in its own frame of reference and in the frame of the lower colleague immobile in relation to the building (on the stage that has already stopped rotating), will move in a straight line (towards the center of the Earth). The difference will be that for the observer below this motion will be accelerated, and for his upper colleague, if he, having stumbled, will fall, moving along with any of the drops, the distance between the drops will increase proportionally first degree time, that is, the mutual motion of drops and their observer in his accelerated coordinate system will be uniform with a speed determined by the delay between the moments of falling drops:

.

Where is the free fall acceleration.

Therefore, the shape of the trajectory and the speed of the body along it, considered in a certain frame of reference, about which nothing is known in advance, does not give an unambiguous idea of ​​the forces acting on the body. It is possible to decide whether this system is sufficiently inertial only on the basis of an analysis of the causes of the occurrence of acting forces.

Thus, in a non-inertial system:

• The curvature of the trajectory and / or the inconsistency of the speed are insufficient arguments in favor of the assertion that a body moving along it is affected by external forces, which in the final case can be explained by gravitational or electromagnetic fields.
• The straightness of the trajectory is an insufficient argument in favor of the assertion that no forces act on a body moving along it.

Notes

In physics, there is another formula for measuring the trajectory (path): s=4Atv, where A is the amplitude, t is the time, v is the oscillation frequency

Literature

• Newton I. Mathematical principles of natural philosophy. Per. and approx. A. N. Krylova. Moscow: Nauka, 1989
• Frish S. A. and Timoreva A. V. Well general physics, Textbook for physics and mathematics and physics and technology faculties public universities, Volume I. M .: GITTL, 1957

Links

• Trajectory and displacement vector, section of the textbook on physics [ non-authoritative source?]

Wikimedia Foundation. 2010 .

Synonyms:

• It doesn't hurt me (film)
• American History X (film)

See what "Trajectory" is in other dictionaries:

TRAJECTORY- (from Latin trajicere to throw, cross), in geometry: a straight or curved line that describes a moving or falling body, for example, a core, after exiting a cannon. 2) a curve intersecting a system of homogeneous curves at the same angle. ... ... Vocabulary foreign words Russian language

It is a set of points through which a certain object has passed, passes or passes. By itself, this line points the way this object. It cannot be used to find out whether the object began to move or why its path was curved. But the relationship between the forces and parameters of the object allows you to calculate the trajectory. In this case, the object itself must be significantly less than the path it has traveled. Only in this case it can be considered a material point and speak of a trajectory.

The line of motion of an object is necessarily continuous. In mathematics, it is customary to talk about the movement of a free or non-free material point. Only forces act on the first. A non-free point is under the influence of connections with other points, which also affect its movement and, ultimately, its track.

To describe the trajectory of one or another material point, it is necessary to determine the frame of reference. Systems can be inertial and non-inertial, and the trail from the motion of the same object will look different.

The way to describe the trajectory is the radius vector. Its parameters depend on time. To the data, to describe the trajectory, the starting point of the radius vector, its length and direction. The end of the radius vector describes in space a curve that consists of one or more arcs. The radius of each arc is extremely important because it allows you to determine the acceleration of an object at a particular point. This acceleration is calculated as the quotient of the square of the normal speed divided by the radius. That is, a=v2/R, where a is the acceleration, v is the normal speed, and R is the radius of the arc.

A real object is almost always under the action of certain forces that can initiate its movement, stop it, or change direction and speed. Forces can be both external and internal. For example, when moving, it is affected by the force of gravity of the Earth and other space objects, engine power and many more factors. They determine the trajectory.

The ballistic trajectory is free movement object under the influence of gravity alone. Such an object can be a projectile, apparatus, bomb, and others. In this case, there is neither thrust nor other forces capable of changing the trajectory. This type of movement is ballistics.

You can conduct a simple experiment that allows you to see how the ballistic trajectory changes depending on the initial acceleration. Imagine that you are dropping a rock from a high . If you don't tell the stone initial speed, but just release it, the movement of this material point will be rectilinear vertically. If you throw it in a horizontal direction, then under the influence various forces(in this case force of your throw and gravity) the trajectory of movement will be a parabola. In this case, the rotation of the Earth can be ignored.

Trajectory of a material point- a line in space, along which the body moves, which is a set of points at which a material point was, is or will be when it moves in space relative to the selected reference system. It is essential that the concept of a trajectory has a physical meaning even in the absence of any movement along it.

In addition, even in the presence of an object moving along it, the trajectory depicted in a predetermined system of spatial coordinates cannot by itself say anything definite about the reasons for its movement until an analysis of the configuration of the field of forces acting on it in the same coordinate system is carried out. .

It is no less important that the shape of the trajectory is inextricably linked and depends on the specific frame of reference in which the motion is described.

It is possible to observe the trajectory when the object is stationary, but when the frame of reference is moving. Thus, the starry sky can serve as a good model for an inertial and fixed frame of reference. However, with a long exposure, these stars appear to move in circular paths (Fig. 3)

The case is also possible when the body is obviously moving, but the trajectory in the projection onto the observation plane is one fixed point. This is, for example, the case of a bullet flying directly into the observer's eye or a train leaving him.

Trajectory of a free material point

According to Newton's First Law, sometimes called the law of inertia, there must be such a system in which a free body maintains (as a vector) its velocity. Such a frame of reference is called inertial. The trajectory of such a movement is a straight line, and the movement itself is called uniform and rectilinear.

Description of the trajectory

It is customary to describe the trajectory of a material point in a predetermined coordinate system using a radius vector , the direction, length and starting point of which depend on time . In this case, the curve described by the end of the radius vector in space can be represented as conjugate arcs of different curvature , located in the general case in intersecting planes . In this case, the curvature of each arc is determined by its radius of curvature directed to the arc from the instantaneous center of rotation, which is in the same plane as the arc itself. Moreover, a straight line is considered as a limiting case of a curve, the radius of curvature of which can be considered equal to infinity. And therefore, the trajectory in the general case can be represented as a set of conjugate arcs.

It is essential that the shape of the trajectory depends on the reference system chosen to describe the motion of a material point. Thus, rectilinear uniformly accelerating motion in one inertial frame will generally be parabolic in another uniformly moving inertial reference frame.

A section of the trajectory of a material point in physics is usually called a path and is usually denoted by the symbol S- from ital. s postamento(movement).

Relationship with speed and normal acceleration

The velocity of a material point is always directed tangentially to the arc used to describe the trajectory of the point. There is a relationship between the speed v (\displaystyle v), normal acceleration a n (\displaystyle a_(n)) and the radius of curvature of the trajectory R (\displaystyle R) at this point:

a n = v 2 R (\displaystyle a_(n)=(\frac (v^(2))(R)))

However, not every movement famous curved speed famous radius and found by the above formula normal(centripetal) acceleration is associated with the manifestation of a force directed along the normal to the trajectory (centripetal force). Thus, the acceleration of any of the stars found from photographs of the daily motion of the luminaries does not at all indicate the existence of a force that causes this acceleration, attracting it to the Polar Star, as the center of rotation.

Connection with the equations of dynamics

Representing the trajectory as a trace left by movement material points, connects the purely kinematic concept of the trajectory, as a geometric problem, with the dynamics of the motion of a material point, that is, the problem of determining the causes of its motion. In fact, the solution of Newton's equations (in the presence of complete set of initial data) gives the trajectory of a material point.

In the general case, the body is not free in its movement, and restrictions are imposed on its position, and in some cases on speed, - connections. If the links impose restrictions only on the coordinates of the body, then such links are called geometric. If they also propagate at speeds, then they are called kinematic. If the constraint equation can be integrated over time, then such a constraint is called holonomic.

The action of bonds on a system of moving bodies is described by forces called reactions of bonds. In this case, the force included in the left side of equation (1) is the vector sum of the active (external) forces and the reaction of the bonds.

It is essential that in the case of holonomic constraints it becomes possible to describe the motion of mechanical systems in generalized coordinates , included in the Lagrange equations . The number of these equations depends only on the number of degrees of freedom of the system and does not depend on the number of bodies included in the system, the position of which must be determined for a complete description of the motion.

If the bonds acting in the system are ideal, that is, they do not transfer the energy of motion into other types of energy, then when solving the Lagrange equations, all unknown reactions of the bonds are automatically excluded.

Finally, if the acting forces belong to the class of potential , then with an appropriate generalization of concepts, it becomes possible to use the Lagrange equations not only in mechanics, but also in other areas of physics.

The forces acting on a material point in this understanding uniquely determine the shape of the trajectory of its movement (under known initial conditions). The converse statement is generally not true, since the same trajectory can take place with different combinations of active forces and coupling reactions.

Motion under the action of external forces in a non-inertial frame of reference

If the frame of reference is non-inertial (that is, it moves with some acceleration relative to the inertial frame of reference), then expression (1) can also be used in it, however, on the left side it is necessary to take into account the so-called inertial forces (including centrifugal force and Coriolis force, associated with the rotation of a non-inertial frame of reference) .

Illustration

As an example, consider a theater worker moving in the grate space above the stage in relation to the theater building evenly and straightforward and carrying over rotating scene of a leaky bucket of paint. It will leave a mark on it from falling paint in the form unwinding spiral(if moving from scene rotation center) and swirling- in the opposite case. At this time, his colleague, who is responsible for the cleanliness of the rotating stage and is on it, will therefore be forced to carry a non-leaky bucket under the first, constantly being under the first. And its movement in relation to the building will also be uniform and straightforward, although with respect to the scene, which is non-inertial system, its movement will be twisted and uneven. Moreover, in order to counteract drift in the direction of rotation, he must overcome the action of the Coriolis force with muscular effort, which his upper colleague does not experience above the stage, although the trajectories of both in inertial system theater buildings will represent straight lines.

But one can imagine that the task of the colleagues considered here is precisely the application straight lines on rotating stage. In this case, the bottom should require the top to move along a curve that is a mirror image of the trace from the previously spilled paint, while remaining above any point of a straight line passing in the chosen radial direction. Hence, rectilinear motion in non-inertial system reference will not be for the observer in inertial system.

Furthermore, uniform body movement in one system, can be uneven in another. So, two drops of paint that fell into different moments of time from a leaky bucket, both in its own frame of reference and in the frame of the lower colleague immobile in relation to the building (on the stage that has already stopped rotating), will move in a straight line (towards the center of the Earth). The difference will be that for the observer below this motion will be accelerated, and for his upper colleague, if he, having stumbled, will fall, moving along with any of the drops, the distance between the drops will increase proportionally first degree time, that is, the mutual motion of drops and their observer in his accelerated coordinate system will be uniform with speed v (\displaystyle v), determined by the delay Δt (\displaystyle \Delta t) between the moments of falling drops:

v = g Δ t (\displaystyle v=g\Delta t).

Where g (\displaystyle g)- acceleration of gravity .

Therefore, the shape of the trajectory and the speed of the body along it, considered in a certain frame of reference, about which nothing is known in advance, does not give an unambiguous idea of ​​the forces acting on the body. It is possible to decide whether this system is sufficiently inertial only on the basis of an analysis of the causes of the occurrence of acting forces.

Thus, in a non-inertial system:

• The curvature of the trajectory and/or the inconsistency of the speed are insufficient arguments in favor of the assertion that external forces act on a body moving along it, which in the final case can be explained by gravitational or electromagnetic fields.
• The straightness of the trajectory is an insufficient argument in favor of the assertion that no forces act on a body moving along it.