What is a trajectory briefly. Particular cases of rotational motion

In many problems, I will be interested not only in the movement of material points in space, but also in the trajectories of their movement.

Definition

The line that a particle describes as it moves is called trajectory.

Depending on the shape of the trajectory mechanical movement can be divided into:

rectilinear motion, the trajectory of the point in this case is a straight line;
and curvilinear movement (trajectory - curved line).

The shape of the trajectory depends on the choice of reference system. AT different systems reference trajectories can be represented by different lines, can be straight and curved.

When moving a point with constant acceleration, which describes the equation:

\[\overline(r)\left(t\right)=(\overline(r))_0+(\overline(v))_0t+\frac(\overline(a)t^2)(2)\left(1 \right),\]

(where $\overline(r)\left(t\right)$ is the radius vector of the point at time $t$; $(\overline(v))_0$ is the initial speed of the point; $\overline(a) $ - point acceleration,) the motion trajectory is a flat curve, which means that all points of this curve are in the same plane. The position of this plane in space is given by the acceleration and initial velocity vectors. The orientation of the coordinate axes is most often chosen so that the plane of motion coincides with one of coordinate planes. In this case, the vector equation (1) can be reduced to two scalar equations.

Motion trajectory equation

Consider free movement bodies near the earth's surface. The origin of coordinates will be placed at the point of throwing the body (Fig. 1). Let's direct the coordinate axes as shown in Fig.1.

Then the equation of motion of the body (1) in projections onto coordinate axes Cartesian coordinate system takes the form of a system of two equations:

\[\left\( \begin(array)(c) x=v_0t(\cos \alpha \left(2\right),\ ) \\ y=v_0t(\sin \alpha \ )-\frac(gt^ 2)(2)\left(3\right).\end(array)\right.\]

In order to obtain the equation for the trajectory of the body's motion ($y=y(x)$), the time of the body's motion should be excluded from equations (2) and (3). We express $t$ from equation (2) and substitute it into expression (3), we get:

Expression (4) is the equation of a parabola passing through the origin. Its ropes are directed downwards, since the coefficient of $x^2$ is less than zero.

The vertex of this parabola is at the point with coordinates:

\[\left\( \begin(array)(c) x=\frac(v^2_0(\sin \alpha (\cos \alpha \ )\ ))(g) \\ y=\frac(v^2_0 (sin)^2\alpha )(2g) \end(array) \right.\left(5\right).\]

You can find the coordinates of the vertex of the trajectory using known rules studies of functions to an extremum. Thus, the position of the maximum of the function $y(x)$ is determined by equating to zero the first derivative ($\frac(dy)(dx)$) of it with respect to $x$.

Movement reversibility

From the notion of a trajectory, one can concretize the meaning of the reversibility of mechanical motion.

Let a particle move in a force field such that its acceleration at any point has a certain value, independent of the speed. How will this particle move if, at some point of its trajectory, the direction of the velocity is replaced by the opposite one? Mathematically, this is equivalent to replacing $t\ $ with $-t$ for all equations. The trajectory equation does not contain time, it turns out that the particle will move "backward" along the same trajectory. In this case, the time intervals between any points of the trajectory will be the same for forward and reverse motion. Each point of the trajectory is assigned certain value velocity values regardless of the direction of movement along a given trajectory. These properties are visible in oscillatory movements pendulum.

All of the above is true when any resistance to movement can be neglected. The reversibility of motion exists when the law of conservation of mechanical energy is fulfilled.

Path Options

The position of the points of the reference system can be determined using different ways. In accordance with these methods, the motion of a point or body is also described:

Coordinate form of motion description. A coordinate system is chosen, in which the position of a point is characterized by three coordinates (in three-dimensional space). These can be coordinates $x_1=x,x_2=y,x_3=z$, in Cartesian system coordinates. $x_1=\rho ,x_2=\varphi ,x_3=\ z$ in a cylindrical system, etc. When moving a point, the coordinates are functions of time. To describe the movement of a point means to indicate these functions:
When describing motion in vector form, the position of a material point specifies the radius vector ($\overline(r)$) with respect to the point, which is taken as the initial one. In this case, a reference point (body) is entered. As the point moves, the vector $\overline(r)$ is constantly changing. The end of this vector describes the trajectory. Movement defines the expression:
The third way to describe the movement is the description using the parameters of the trajectory.

The path is scalar, equal to the length trajectories.

If the trajectory is given, then the problem of describing the motion is reduced to determining the law of motion along it. In this case, the starting point of the trajectory is selected. Any other point is characterized by the distance $s$ along the trajectory from the starting point. In this case, the movement is described by the expression:

Let a point move uniformly along a circle of radius R. The law of movement of a point along a circle in the method under consideration can be written as:

where $s$ is the path of the point along the trajectory; $t$ - movement time; $A$ - coefficient of proportionality. Known are the circle and the starting point of the movement. The count of positive values $s$ coincides with the direction of moving the point along the trajectory.

Knowing the trajectory of the body in many cases greatly simplifies the process of describing the movement of the body.

Examples of problems with a solution

Example 1

Exercise: The point moves in the XOY plane from the origin with the speed $\overline(v)=A\overline(i)+Bx\overline(j)\ ,\ $where $\overline(i)$, $\overline(j)$ - orts of the X and Y axes; $A$,B- constants. Write the equation for the trajectory of the point ($y(x)$). Draw a trajectory. \textit()

Decision: Consider the equation for changing the particle velocity:

\[\overline(v)=A\overline(i)+Bx\overline(j)\ \left(1.1\right).\]

From this equation it follows that:

\[\left\( \begin(array)(c) v_x=A, \\ v_y=Bx \end(array) \right.\left(1.2\right).\]

From (1.2) we have:

To obtain the trajectory equation, one should solve the differential equation (1.3):

We have obtained the equation of a parabola, the branches of which are directed upwards. This parabola passes through the origin. The minimum of this function is at the point with coordinates:

\[\left\( \begin(array)(c) x=0 \\ y=0. \end(array) \right.\]

Example 2

Exercise: The motion of a material point in the plane is described by the system of equations: $\left\( \begin(array)(c) x=At. \\ y=At(1+Bt) \end(array) \right.$, where $A$ and $B$ are positive constants Write the equation for the trajectory of the point.

Decision: Consider the system of equations that is specified in the condition of the problem:

\[\left\( \begin(array)(c) x=At. \\ y=At\left(1+Bt\right) \end(array) \right.\left(2.1\right).\]

Let us exclude time from the equations of the system. To do this, we express the time from the first equation of the system, we get:

Let us substitute the right (2.2) part instead of $t$ into the second equation of system (2.1), we have:

Answer:$y=x+\frac(B)(A)x^2$

Since ancient times, humanity has tried to achieve victory in a collision with the enemy at the maximum possible distance, so as not to destroy its own warriors. Slings, bows, crossbows, then guns, now bombs - they all need an accurate calculation of the ballistic trajectory. And if the point of impact could be tracked visually with the old military “equipment”, which made it possible to learn and shoot more accurately next time, then modern world the destination is usually so far away that it is simply impossible to see it without additional instruments.

What is ballistic trajectory

This is the path that some object overcomes. It must have a certain initial speed. It is affected by air resistance and gravity, which excludes the possibility of movement in a straight line. Even in space, such a trajectory will be distorted under the influence of the gravity of various objects, although not as significantly as on our planet. If we ignore the resistance air masses, then most of all such a process of displacement will resemble an ellipse.

Another option is hyperbole. And only in some cases it will be a parabola or a circle (upon reaching the second and first space velocity respectively). In most cases, such calculations are carried out for missiles. They tend to fly in the upper atmosphere, where the influence of air is minimal. As a result, most often the ballistic trajectory still resembles an ellipse. Depending on many factors, such as speed, mass, type of atmosphere, temperature, rotation of the planet, and so on, individual parts of the path can take on a wide variety of forms.

Ballistic trajectory calculation

In order to understand exactly where the released body will fall, apply differential equations and method numerical integration. The ballistic trajectory equation depends on many variables, but there is also a certain universal version that does not give the required accuracy, but is quite sufficient for an example.

y=x-tgѲ 0 -gx 2 /2V 0 2 -Cos 2 Ѳ 0, where:

y is maximum height above the surface of the earth.
X is the distance from the starting point to the moment when the body reaches highest point.
Ѳ 0 - throwing angle.
V 0 - initial speed.

Thanks to formula it becomes possible to describe the ballistic flight path in airless space. It will turn out in the form of a parabola, which is typical for most options for free movement in such conditions and in the presence of gravity. The following can be distinguished characteristics this trajectory:

The most optimal elevation angle for maximum distance is 45 degrees.
The object has the same speed of movement both during the start and at the moment of landing.
The angle of the throw is identical to the angle at which the fall will occur.
The object flies to the top of the trajectory in exactly the same time, during which it then falls down.

In the vast majority of calculations of this kind, it is customary to neglect the resistance of air masses and some other factors. If they are taken into account, then the formula will turn out to be too complicated, and the error is not so large as to significantly affect the effectiveness of the hit.

Differences from flat

This name means another variant of the path of the object. Flat and ballistic trajectory are several different concepts, although general principle they have the same. In fact, this type of movement implies the maximum possible displacement in horizontal plane. And throughout the path, the object maintains sufficient acceleration. The ballistic version of the movement is necessary for moving over long distances. For example, flat trajectory is most important for a bullet. She must fly straight enough for as long as possible and punch through everything that gets in her way. On the other hand, a rocket or a projectile from a cannon inflicts maximum damage precisely at the end of the movement, as it gains the maximum possible speed. In the interval of their movement, they are not so crushing.

Usage in modern times

Ballistic trajectory is most often used in military sphere. bullets and so on - they all fly far, and for an accurate shot you need to take into account many variables. In addition, the space program is also based on ballistics. Without it, it is impossible to accurately launch a rocket so that it ultimately does not fall to the ground, but makes several turns around the planet (or even breaks away from it and goes further into space). In general, almost everything that can fly (regardless of how it does it) is somehow connected with a ballistic trajectory.

Conclusion

The ability to calculate all the elements and launch any object in the right place is extremely important in modern times. Even if you do not take the military, which traditionally needs such capabilities more than anyone else, there will still be many quite civilian applications.

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 in certain 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, the force of the engine, and many other factors. They determine the trajectory.

A ballistic trajectory is the free movement of an 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.

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 the body of reference.

In the Cartesian coordinate system, the position of point A at a given moment of time with respect 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 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 scalar function time: Δs=Δs(t). Unit of measurement - meter(m) – path length, traversed by light in vacuum for 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 (sixteen). If a 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 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 is a vector physical quantity that 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 equal to the vector product of the angular displacement dφ by radius – point vector r : dr =[ dφ · r ] (3).

Point Linear Speed connected with angular velocity and the radius of the trajectory by the ratio:

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 direction linear speed. This vector is normal acceleration points: a n =[ ω · v ] (eight). 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 - number full revolutions performed 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.

Lesson Objectives:

Educational:
– introduce the concepts of “displacement”, “path”, “trajectory”.
Developing:
- develop logical thinking, correct physical speech, use appropriate terminology.
Educational:
- achieve high activity class, 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
d) a chess piece on the board.

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 the initial position of the 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.

On the move material point occupies different positions in space relative to the chosen frame of reference. 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)

The path is a scalar quantity and is characterized only by numerical value.
Displacement is a vector quantity and is characterized by both a numerical value (modulus) and a direction.
When the body moves, the path can only increase, and the displacement modulus can both increase and decrease.
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]
The nature of 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)

Fill a plastic bottle with a scale up to the neck with water.
Fill the bottle with a 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 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.
Screw on the bottle cap.
While squeezing the sides of the bottle, lower the float to the bottom of the bottle.

By releasing the pressure on the walls of the bottle, achieve the ascent of the float. 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 repetition.

Do we pay for the journey or transportation when traveling in a taxi? (Way)
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.