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.
« Physics - Grade 10 "
How do vector quantities differ from scalar quantities?
The line along which a point moves in space is called trajectory.
Depending on the shape of the trajectory, all movements of the point are divided into rectilinear and curvilinear.
If the path is a straight line, the movement of the point is called straightforward, and if the curve is curvilinear.
Let at some point in time the moving point occupies the position M 1 (Fig. 1.7, a). How to find its position after a certain period of time after this moment?
Suppose we know that the point is at a distance l relative to its initial position. Will we be able to uniquely determine the new position of the point in this case? Obviously not, since there are an infinite number of points that are at a distance l from the point M 1. To unambiguously determine the new position of the point, one must also know in which direction from the point M 1 a segment of length l should be laid.
Thus, if the position of a point at some point in time is known, then its new position can be found using a certain vector (Fig. 1.7, b).
The vector drawn from the initial position of a point to its final position is called displacement vector or simply moving a point
Since displacement is a vector quantity, the displacement shown in Figure (1.7, b) can be denoted
Let us show that with the vector method of specifying the motion, the displacement can be considered as a change in the radius vector of the moving point.
Let the radius vector 1 set the position of the point at time t 1 , and the radius vector 2 at time t 2 (Fig. 1.8). To find the change in the radius vector over a period of time Δt = t 2 - t 1, it is necessary to subtract the initial vector 1 from the final vector 2 . Figure 1.8 shows that the movement made by a point during the time interval Δt is a change in its radius vector during this time. Therefore, denoting the change in the radius vector through Δ , we can write: Δ = 1 - 2 .
Path s- the length of the trajectory when moving the point from position M 1 to position M 2.
The displacement modulus may not be equal to the path traveled by the point.
For example, in Figure 1.8, the length of the line connecting the points M 1 and M 2 is greater than the displacement modulus: s > |Δ|. The path is equal to the displacement only in the case of rectilinear unidirectional motion.
Body displacement Δ - vector, path s - scalar, |Δ| ≤ s.
Source: "Physics - Grade 10", 2014, textbook Myakishev, Bukhovtsev, Sotsky
Kinematics - Physics, textbook for grade 10 - Classroom physics
Physics and knowledge of the world --- What is mechanics ---
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! And that means at uniform motion 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.
At first glance, movement and path are concepts that are close in meaning. However, in physics there are key differences between movement and path, although both concepts are associated with a change in the position of the body in space and are often (usually in rectilinear motion) numerically equal to each other.
To understand the differences between movement and path, let us first give them the definitions that physics gives them.
body movement- This directed line segment (vector), whose beginning coincides with the initial position of the body, and whose end coincides with the final position of the body.
body path- This distance passed by the body in a certain period of time.
Let's imagine that you stood at your entrance to a certain point. We walked around the house and returned to the starting point. So: your displacement will be equal to zero, but the path will not. The path will be equal to the length of the curve (for example, 150 m) that you walked around the house.
But back to the coordinate system. Let a point body move rectilinearly from point A with coordinate x 0 \u003d 0 m to point B with coordinate x 1 \u003d 10 m. The displacement of the body in this case will be 10 m. body way.
If the body moved in a straight line from the initial (A) point with coordinate x 0 = 5 m, to the final (B) point with coordinate x 1 = 0, then its displacement will be -5 m, and the path will be 5 m.
The displacement is found as a difference, where the initial coordinate is subtracted from the final one. If the end coordinate is less than the start coordinate, i.e. the body moved in the opposite direction with respect to the positive direction of the X axis, then the displacement will be a negative value.
Since displacement can be both positive and negative, displacement is a vector quantity. In contrast, the path is always positive or zero magnitude (path is a scalar magnitude), since distance cannot be negative in principle.
Let's consider one more example. The body moved in a straight line from point A (x 0 \u003d 2 m) to point B (x 1 \u003d 8 m), then it also moved straight from B to point C with coordinate x 2 \u003d 5 m. What are the common path (A →B→C) done by this body and its total displacement?
Initially, the body was at a point with a coordinate of 2 m, at the end of its movement it ended up at a point with a coordinate of 5 m. Thus, the movement of the body was 5 - 2 = 3 (m). It is also possible to calculate the total displacement as the sum of two displacements (vectors). The displacement from A to B was 8 - 2 = 6 (m). The displacement from point B to C was 5 - 8 = -3 (m). Adding both displacements we get 6 + (-3) = 3 (m).
The total path is calculated by adding the two distances traveled by the body. The distance from point A to B is 6 m, and from B to C the body has traveled 3 m. In total, we get 9 m.
Thus, in this problem, the path and displacement of the body differ from each other.
The considered problem is not entirely correct, since it is necessary to indicate the moments of time at which the body is at certain points. If x 0 corresponds to the time t 0 = 0 (the start of observations), then let x 1 correspond to t 1 = 3 s, and x 2 corresponds to t 2 = 5 s. That is, the time interval between t 0 and t 1 is 3 s, and between t 0 and t 2 is 5 s. In this case, it turns out that the path of the body for a period of time of 3 seconds was 6 meters, and for a period of 5 seconds - 9 meters.
Time is involved in determining the path. In contrast, for movement, time does not really matter.
Section 1 MECHANICS
Chapter 1: Fundamentals of kinematics
mechanical movement. Trajectory. Path and movement. Addition of speeds
mechanical movement of the body called the change in its position in space relative to other bodies over time.
The mechanical movement of bodies studies Mechanics. Branch of mechanics that describes geometric properties motion without taking into account the masses of bodies and acting forces is called kinematics .
Mechanical movement is relative. To determine the position of a body in space, you need to know its coordinates. To determine the coordinates material point it is necessary, first of all, to choose a reference body and associate a coordinate system with it.
Reference bodya body is called, relative to which the position of other bodies is determined. The reference body is chosen arbitrarily. It can be anything: land, building, car, ship, etc.
The coordinate system, the body of reference with which it is associated, and the indication of the time reference form reference system , relative to which the motion of the body is considered (Fig. 1.1).
A body whose dimensions, shape and structure can be neglected when studying a given mechanical movement, is called material point . A material point can be considered a body whose dimensions are much smaller than the distances characteristic of the motion considered in the problem.
Trajectoryis the line along which the body moves.
Depending on the type of trajectory of movement, they are divided into rectilinear and curvilinear.
Pathis the length of the trajectory ℓ(m) ( fig.1.2)
The vector drawn from the initial position of the particle to its final position is called moving this particle for a given time.
Unlike the path, the displacement is not a scalar, but a vector quantity, since it shows not only how far, but also in what direction the body has moved in a given time.
Displacement vector modulus(that is, the length of the segment that connects the start and end points of the movement) can be equal to the distance traveled or less than the distance traveled. But the displacement module can never be greater than the distance traveled. For example, if a car moves from point A to point B along a curved path, then the absolute value of the displacement vector is less than the distance traveled ℓ. The path and the displacement modulus are equal only in one single case, when the body moves in a straight line.
Speedis a vector quantitative characteristic of the movement of the body
average speed- This physical quantity, equal to the ratio of the point displacement vector to the time interval
The direction of the average velocity vector coincides with the direction of the displacement vector.
instant speed, that is, the speed at a given moment of time is a vector physical quantity equal to the limit to which average speed with an infinite decrease in the time interval Δt.
Vector instantaneous speed directed tangentially to the trajectory of motion (Fig. 1.3).
In the SI system, speed is measured in meters per second (m / s), that is, the unit of speed is considered to be the speed of such a uniform rectilinear motion, at which in one second the body travels a distance of one meter. Speed is often measured in kilometers per hour.
or 1 
Addition of speeds
Any mechanical phenomena are considered in some frame of reference: movement makes sense only relative to other bodies. When analyzing the motion of the same body in different systems reference, all kinematic characteristics of motion (path, trajectory, movement, speed, acceleration) are different.
For example, a passenger train is moving along a railroad at a speed of 60 km/h. A person is walking along the carriage of this train at a speed of 5 km/h. If we consider the railway to be stationary and take it as a frame of reference, then the speed of a person is relatively railway, will be equal to the addition of the speeds of the train and the person, that is
60km/h + 5km/h = 65km/h if the person is walking in the same direction as the train and
60km/h - 5km/h = 55km/h if the person is walking against the direction of the train.
However, this is only true in this case, if the person and the train are moving along the same line. If a person moves at an angle, then this angle must be taken into account, and the fact that speed is a vector quantity.
Let's consider the example described above in more detail - with details and pictures.
So, in our case, the railway is a fixed frame of reference. The train that moves along this road is a moving frame of reference. The car on which the person is walking is part of the train. The speed of a person relative to the car (relative to the moving frame of reference) is 5 km/h. Let's denote it with a letter. The speed of the train (and hence the wagon) relative to a fixed frame of reference (that is, relative to the railway) is 60 km/h. Let's denote it with a letter. In other words, the speed of the train is the speed of the moving frame relative to the fixed frame.
The speed of a person relative to the railway (relative to a fixed frame of reference) is still unknown to us. Let's denote it with a letter.
Let's associate with the fixed frame of reference (Fig. 1.4) the XY coordinate system, and with the moving frame of reference - X p O p Y p. Let us now determine the speed of a person relative to the fixed frame of reference, that is, relative to the railway.
For a short period of time Δt, the following events occur:
The person moves relative to the car at a distance
The wagon moves relative to the railway for a distance
Then for this period of time the movement of a person relative to the railway:
This displacement addition law . In our example, the movement of a person relative to the railway is equal to the sum of the movements of a person relative to the wagon and the wagon relative to the railway.
Dividing both parts of the equality by a small period of time Dt, during which the movement occurred:
We get:
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