Define displacement and distance travelled. Path and movement of the body
Mechanics.
weight(kg)
Electric charge(C)
Trajectory
Distance traveled or just path l) -
moving- it's a vectorS
Define and indicate the unit of speed.
Speed- a vector physical quantity characterizing the speed of moving a point and the direction of this movement. [V]=m s
Define and indicate the unit of measurement for acceleration.
Acceleration- vector physical quantity characterizing the speed of change in the module and direction of speed and equal to the increment of the speed vector per unit of time:
Define and indicate the unit of measure for the radius of curvature.
Radius of curvature- a scalar physical quantity, inverse to the curvature C at a given point of the curve and equal to the radius of the circle tangent to the trajectory at this point. The center of such a circle is called the center of curvature for the given point on the curve. The radius of curvature is determined: R \u003d C -1 \u003d, [R]=1m/rad.
Define and indicate the unit of curvature
Trajectories.
Curvature of the trajectory is a physical quantity equal to , where is the angle between the tangents drawn at 2 points of the trajectory; - the length of the trajectory between these points. How< , тем кривизна меньше. В окружности 2 пи радиант = .
Define and indicate the unit of measure for angular velocity.
Angular velocity- vector physical quantity characterizing the rate of change of the angular position and equal to the angle rotation per unit time: . [w]= 1 rad/s=1s -1
Define and indicate the unit of measure for a period.
Period(T) - a scalar physical quantity equal to the time of one complete rotation of the body around its axis or the time of a complete rotation of a point along the circumference. where N is the number of revolutions for a time equal to t. [T]=1s.
Define and indicate the unit of frequency.
Frequency of circulation- scalar physical quantity equal to the number revolutions per unit of time: . =1/s.
Give a definition and indicate the unit of measurement of the momentum of the body (momentum).
Pulse is a vector physical quantity equal to the product of the mass and the velocity vector. . [p]=kg m/s.
Give a definition and indicate the unit of measurement of the momentum of force.
Impulse of force- a vector physical quantity equal to the product of the force and the time of its action. [N]=N.s.
Define and indicate the unit of measure for work.
Force work- a scalar physical quantity characterizing the action of a force and equal to the scalar product of the force vector and the displacement vector: where is the projection of the force on the direction of displacement, is the angle between the directions of force and displacement (velocity). [A] \u003d \u003d 1N m.
Define and indicate the unit of measure for power.
Power- a scalar physical quantity characterizing the speed of work and equal to the work produced per unit of time: . [N]=1W=1J/1s.
Define potential forces.
Potential or conservative forces - forces, the work of which, when moving the body, does not depend on the trajectory of the body and is determined only by the initial and final positions of the body.
Define dissipative (nonpotential) forces.
Non-potential forces are forces under the action of which on a mechanical system its total mechanical energy decreases, passing into other non-mechanical forms of energy.
Define leverage.
Shoulder of strength called distance between the axis and the straight line along which the force acts(distance x counted along the O axis x perpendicular to the given axis and force).
Define the moment of force about a point.
Moment of force about some point O- vector physical quantity equal to the vector product of the radius vector drawn from a given point O to the point of application of the force and the force vector. M= r * F= . [M] SI \u003d 1N m \u003d 1 kg m 2 / s 2
Define a perfectly rigid body.
Absolutely rigid body is a body whose deformations can be neglected.
Conservation of momentum.
Law of conservation of momentum:momentum of a closed system of bodies is a constant value.
Mechanics.
1. Specify the unit of measurement for concepts: force (1 N \u003d 1 kg m / s 2)
weight(kg)
Electric charge(C)
Define the concepts: displacement, path, trajectory.
Trajectory- an imaginary line along which the body moves
Distance traveled or just path l) -the length of the path along which the body moved
moving- it's a vectorS, directed from the start point to the end point
Class: 9
Lesson Objectives:
- Educational:
– introduce the concepts of “displacement”, “path”, “trajectory”. - Developing:
- develop logical thinking, correct physical speech, use the 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 tasks that are performed during the oral test of exercise. 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, – MOVING.
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 1Knowing 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's motion is a continuous line, which is described by a moving body (considered as a material point) with respect to the chosen frame of reference.
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 traffic happens straightforward and curvilinear.
The path length is PATH. 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. In this way, 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 value and is characterized only by a numeric 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.
Path | moving | |
Definition | The length of the trajectory described by the body in a 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 path l traveled in a time interval 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 with water up to the neck plastic bottle with a scale.
- 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? (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 move the ball. (Path - 4 m, movement - 2 m.)
6. The result of the lesson.
Repetition of the concepts of the lesson:
– movement;
– trajectory;
- path.
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.
A trajectory is a continuous line along which a material point moves in a given reference system. Depending on the shape of the trajectory, rectilinear and curvilinear motion of a material point are distinguished.
lat.Trajectorius - related to movement
Path - the length of the section of the trajectory of a material point, passed by it in a certain time.
Distance traveled - the length of the trajectory section from the start to the end point of the movement.
Displacement (in kinematics) is a change in the location of a physical body in space relative to the selected frame of reference. Also, displacement is a vector that characterizes this change. It has the additivity property. The length of the segment is the displacement modulus, measured in meters (SI).
You can define displacement as a change in the radius vector of a point: .
The displacement modulus coincides with the distance traveled if and only if the direction of the velocity does not change during the 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 displacement to a small period of time for which it is completed. More strictly:
Average ground speed. Average speed vector. Instant speed.
Average ground speed
The average (ground) speed is the ratio of the length of the path traveled by the body to the time during which this path was traveled:
Average ground speed, unlike instantaneous speed, is not a vector quantity.
The average speed is equal to the arithmetic mean of the speeds of the body during the movement only if the body moved with these speeds for equal periods of time.
At the same time, if, for example, the car moved half the way at a speed of 180 km/h, and the second half at a speed of 20 km/h, then the average speed would be 36 km/h. In examples like this, the average speed is equal to the harmonic mean of all speeds on separate, equal sections of the path.
Average speed is the ratio of the length of a section of the path to the period of time during which this path has been traveled.
Average body speed
With uniformly accelerated motion
With uniform motion
Here we used:
Average body speed
Initial body speed
body acceleration
body movement time
The speed of a body after a certain period of time
The instantaneous speed is the first derivative of the path with respect to time =
v=(ds/dt)=s"
where the symbols d/dt or the stroke at the top right of a function denote the derivative of this function.
Otherwise, it is the speed v =s/t as t tends to zero... :)
In the absence of acceleration at the time of measurement, the instantaneous is equal to the average during the period of movement without acceleration Vmgn. = Vav. =S/t for this period.
If we take into account the physical processes in the domestic sphere, then many of them seem very fine. Therefore, the concepts of path and movement are perceived as one and the same, the only difference is that the first is a description of the action, and the second is the result of the action. But if you turn to information sources for clarification, you can immediately find a significant difference between these operations.
What is a path?
A path is a movement that results in a change in the location of an object or person. This value is scalar, so it has no direction, but it can be used to determine the distance traveled.
The path can be executed in the following ways:
- In a straight line.
- curvilinear.
- Round.
- Other methods are possible (for example, a zigzag path).
The path can never be negative and decrease over time. The distance is measured in meters. Most often, in physics, the letter is used to indicate the path S, in rare cases they use the letter L. With the help of the path, it is impossible to foresee where the object we need will be at a certain point in time.
Movement Features
Displacement is the difference between the start and end points of the location of a person or object in space after some path has been overcome.
The displacement value is always positive and also has a clear direction.
Coincidence between movement and path is possible only if the path was carried out in a straight line, and the direction did not change.
With the help of displacement, you can calculate where a person or object was at a certain point in time.
The letter S is used to denote displacement, but since displacement is a vector quantity, an arrow → is placed above this letter, which indicates that displacement is a vector. Unfortunately, adding to the confusion between path and displacement is the fact that both concepts can also be denoted by the letter L.
What is common between the concepts of path and movement?
Despite the fact that the path and movement are completely different concepts, there are certain elements that contribute to the fact that the concepts are confused:
- Distance and displacement can always be only positive values.
- The same letter L can be used to indicate the path and movement.
Even considering the fact that these concepts have only two common element their meaning is so great that it causes confusion for many people. Especially there are problems for schoolchildren during the study of physics.
What are the main differences between the concepts of path and movement?
These concepts have a number of differences that will always help determine what value is in front of you, the path or movement:
- The path is the primary concept, and the movement is secondary. For example, displacement determines the difference between the start and end points of a person's location in space after overcoming a path. Accordingly, it is impossible to obtain the amount of displacement without using the path initially.
- For the path, the beginning of the movement plays a huge role, and for determining the movement, the beginning of the movement is absolutely not necessary.
- The main difference between these values is that the path has no direction, while the movement has one. For example, the path is carried out only directly - forward, and the movement allows movement backward.
- In addition, the concepts differ in appearance. The path refers to a scalar value, and the displacement refers to a vector.
- calculus method. For example, the path is calculated using the total distance traveled, and the movement, in turn, is calculated using the change in the location of the object in space.
- The path can never be equal to zero, and the value equal to zero is allowed in the displacement.
Having studied these differences, you can immediately understand what the difference between the concepts of path and movement is, and never confuse them again.
Difference Between Path and Move by Example
In order to quickly understand the difference between path and movement, you can use certain examples:
- The car moved 2 meters forward and 2 meters back. The path is the sum of the entire distance traveled, respectively, it is 4 meters. And displacement is a start and end point, so in this case it is zero.
- In addition, the difference between the path and the movement can be seen on their own experience. It is necessary to stand at the start of the 400-meter treadmill and run two laps (the second lap will end at the starting point). The result is that the path is 800 meters (400+400) and the displacement is 0 because the start and end points are the same.
- The ball thrown up reached a height of 15 meters, and then fell to the ground. In this case, the path will be equal to 30 meters, since 15 meters up and 15 meters down are added. And the displacement will be 0, due to the fact that the ball returned to its original position.
Path is a physical quantity equal to the length
trajectory between the initial position of the body and
its final position. Denoted l.
Track units are units of length (m, cm, km,…)
but the basic unit of length is in SI meter. Written like this
The distance between points A and C is not equal to the length of the path. This is another physical quantity. It's called movement. The movement has not only a numerical value, but also a certain direction, which depends on the location of the start and end points of the body's movement. Quantities that have not only a module (numerical value), but also a direction are called vector quantities or simply vectors.
moving – this is a vector physical quantity characterizing the change in the position of the body in space, equal to the length of the segment connecting the point of the initial position of the body with the point of its final position. Move from start position to end position is directed.
Designated . Unit .
Quantities that have no direction, such as distance, mass, temperature, are called scalars or scalars.
Can path and displacement be equal?
If a body or material point (MT) moves along a straight line, and always in the same direction, then the path and movement coincide, i.e. they are numerically equal. So if a stone falls vertically in a gorge 100 m deep, then its movement will be directed downward and s = 100 m. Path l \u003d 100 m.
If the body makes several movements, then they are added, but not in the same way as numerical values are added, but according to other rules, according to the rules of vector addition. You will soon pass them in the course of mathematics. For now, let's look at an example.
To get to the bus stop, Petr Sergeevich walks first through the courtyard 300 m to the west, and then along the avenue 400 m to the north. Find the displacement of Peter Sergeevich and compare it with the distance traveled.
Given: s 1 = 300 m; s 2 \u003d 400 m.
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Solution:
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l \u003d s 1 + s 2 \u003d 300 m + 400 m \u003d 700 m.
To find the displacement, you need to know the length of the segment connecting the initial position of the body and the final position. This is the length of the vector s.
Before us is a right triangle with known legs (300 and
400 m). Let's use the Pythagorean theorem to find the length of the hypotenuse s:
Thus, the path traveled by a person is 200 m more than the displacement.
If, suppose, Petr Sergeevich, having reached a stop, suddenly decided to go back and moved in the opposite direction, then the length of his path would be 1400 m, and his displacement would be 0 m.
Reference system.
To solve the basic problem of mechanics means to indicate where the body will be at any given moment in time. In other words, calculate the coordinates of the body. Yes, here's the catch: where will we count the coordinates from?
You can, of course, take geographic coordinates - longitude and latitude, but! Firstly, the body (MT) can also move outside the planet Earth. Secondly, the system of geographical coordinates does not take into account the three-dimensionality of our space.
First you need to choose reference body. This is so important that otherwise we will find ourselves in a situation similar to that presented in R. Stevenson's novel Treasure Island. Having buried the main part of the treasure, Captain Flint left a map and a description of the place.
The tall tree of the Lookout Mountain. Direction - from the tree in the shade at noon. Walk a hundred feet. Turn towards the west. Walk ten fathoms. Dig to a depth of ten inches.
The shortcoming of the description of the place where the treasure lies lies in the fact that the tree, which in this problem is the reference body, cannot be found by the indicated signs.
This example shows the importance of choice reference body – any body from which the coordinates of the position of a moving material point are counted.
Consider the drawing. As a moving object, take: 1) a yacht; 2) a seagull. Take for the reference body: a) a rock on the shore; b) the captain of the yacht; c) a flying seagull. How does the nature of the movement of a moving object, its coordinates, depend on the choice of the reference body?
When describing the features of the movement of a particular body, it is important to indicate in relation to which reference body the characteristics are given.
Let's try to enter the coordinates of the body or MT. Let's use a rectangular Cartesian XYZ coordinate system with the origin at point O. We place the origin of the reference system where the reference body is located. From this point we draw three mutually perpendicular coordinate axes OX,OY,OZ. Now the coordinates of the material point (x;y;z) can be specified relative to the reference body.
To study the movement of the body (MT), you also need a watch or a device for measuring time. We will associate the beginning of the countdown with a certain event. Most often, this is the beginning of body movement (MT).
The set of the reference body, the coordinate system associated with the reference body and the instrument for measuring time intervals is called frame of reference (CO) .
If a motionless body is chosen as the reference body, then the frame of reference will also be motionless (FRS). Most often, the surface of the Earth is chosen as a fixed reference body. One can choose a moving body as the reference body and obtain moving frame of reference(PSO).
Look at Figure 1. The 3D coordinate system allows you to set the position in space of any point. For example, the coordinates of the point F located on the column are (6; 3; 1).
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Think! Which coordinate system will you choose when solving problems related to movement:
1) a cyclist participates in competitions on a cycle track;
2) a fly crawls on glass;
3) a fly flies around the kitchen;
4) the truck is moving along a straight section of the highway;
5) a person goes up in an elevator;
6) the projectile takes off and flies from the muzzle of the gun.
Exercise 1.
1. Select in Fig. 3 the cases in which a mechanical movement is performed.
3. There are two operators in the mission control center. One controls the parameters of the orbit of the Mir station, and the other carries out the docking of the Progress spacecraft with this station. Which of the operators can consider the station "Mir material point?
4. To study the movement of a fighter aircraft and hot air balloon(Fig. 4) the XOYZ rectangular coordinate system is chosen. Describe the reference system used here. Could you use more simple systems coordinates?
5. The athlete ran a 400-meter distance (Fig. 5). Find the movement of the athlete and the path traveled by him.
6. Figure 6 shows a leaf of a plant on which a snail is crawling. Calculate, using the scale grid, the path traveled by the snail from point A to point B and from point B to point C.
7. The car, having driven along a straight section of the highway from a gas station to the nearest settlement, returned back. Calculate the displacement modulus of the car and the distance traveled by it. What can be said about the relationship between the displacement module and the distance traveled by it, if the car traveled only from a gas station to a settlement?
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