With the help of this video lesson, you can independently study the topic “Displacement”, which is included in the school physics course for grade 9. From this lecture, students will be able to deepen their knowledge of movement. The teacher will remind you of the first characteristic of movement - the distance traveled, and then move on to the definition of movement in physics.
The first movement characteristic we introduced earlier was the distance traveled. Let us recall that it is denoted by the letter S (sometimes the designation L is found) and is measured in SI meters.
Distance traveled is a scalar quantity, i.e. a quantity that is characterized only by a numerical value. This means that we will not be able to predict where the body will be at the moment of time we need. We can only talk about the total distance traveled by the body (Fig. 1).
Rice. 1. Knowing only the distance traveled, it is impossible to determine the position of the body at an arbitrary moment in time
To characterize the location of a body at an arbitrary moment, a quantity called displacement is introduced. Displacement is a vector quantity, i.e. it is a quantity that is characterized not only by a numerical value, but also by direction.
The movement is indicated in the same way as the distance traveled, by the letter S, but, unlike the distance traveled, an arrow is placed above the letter, thereby emphasizing that this is a vector quantity: .
What moving And distance traveled denoted by one letter is somewhat misleading, but we must clearly understand the difference between the path traveled and movement. Once again, we note that sometimes the path is designated L. This avoids confusion.
Definition
Displacement is a vector (directed line segment) that connects the starting point of a body’s movement with its end point (Fig. 2).
Rice. 2. Displacement is a vector quantity
Let us remind you that the passed path is the length of the trajectory. This means that path and movement are completely different physical quantities, although sometimes there are situations when they coincide numerically.
Rice. 3. The path and the moving module are the same
In Fig. 3, the simplest case is considered when the body moves along a straight line (axis Oh). The body begins its movement from point 0 and ends up at point A. In this case, we can say that the module of displacement is equal to the distance traveled: .
An example of such a movement is an airplane flight (for example, from St. Petersburg to Moscow). If the movement was strictly linear, then the displacement module will be equal to the distance traveled.
Rice. 4. The distance is greater than the displacement module
In Fig. 4 the body moves along a curved line, i.e. the movement is curvilinear (from point A to point B). The figure shows that the displacement module (straight line) will be less than the distance traveled, i.e. the length of the distance traveled and the length of the displacement vector are not equal.
Rice. 5. Closed trajectory
In Fig. 5 the body moves along a closed curve. It leaves point A and returns to the same point. The displacement module is equal to , and distance traveled is the length of the entire curve, .
This case can be characterized by the following example. The student left home in the morning, went to school, studied all day, besides this, visited several other places (shop, gym, library) and returned home. Please note: in the end the student ended up at home, which means his displacement is 0 (Fig. 6).
Rice. 6. The student's displacement is zero.
When we're talking about about moving, it is important to remember that moving depends on the frame of reference in which the motion is considered.
Rice. 7. Determination of the body displacement modulus
The body moves in a plane XOY. Point A is the initial position of the body. Its coordinates. The body moves to point . A vector is the movement of a body: .
The displacement modulus can be calculated as the hypotenuse right triangle, using the Pythagorean theorem: . To find the displacement vector, it is necessary to find the angle between the axis Oh and the displacement vector.
We can choose the system arbitrarily, that is, direct coordinate axes in the way that is convenient for us, the main thing is to consider the projections of all vectors in the future in the same chosen coordinate system.
Conclusion
In conclusion, it can be noted that we have become acquainted with an important quantity - displacement. Please note again that displacement and path can only coincide if rectilinear movement, without changing the direction of such movement.
Bibliography
- Kikoin I.K., Kikoin A.K. Physics: textbook for 9th grade high school. - M.: Enlightenment.
- Peryshkin A.V., Gutnik E.M., Physics. 9th grade: textbook for general education. institutions/A. V. Peryshkin, E. M. Gutnik. - 14th ed., stereotype. - M.: Bustard, 2009. - 300.
- Sokolovich Yu.A., Bogdanova G.S.. Physics: A reference book with examples of problem solving. - 2nd edition repartition. - X .: Vesta: Ranok Publishing House, 2005. - 464 p.
- Internet portal “vip8082p.vip8081p.beget.tech” ()
- Internet portal “foxford.ru” ()
Homework
- What is path and movement? What is the difference?
- The motorcyclist left the garage and headed north. I drove 5 km, then turned west and drove another 5 km. How far from the garage will it be?
- The minute hand has gone full circle. Determine the displacement and distance traveled for the point that is located at the end of the hand (the radius of the clock is 10 cm).
At first glance, movement and path are similar concepts. However, in physics, there are key differences between displacement and path, although both concepts are associated with a change in the position of a body in space and are often (usually with linear 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.
Moving the body- This directed straight line segment (vector), the beginning of which coincides with the initial position of the body, and the end coincides with the final position of the body.
Body path- This distance, which the body has passed over a certain period of time.
Let's imagine that you stand at your entrance at a certain point. We walked around the house and returned to the starting point. So: your displacement will be zero, but your path will not be. There will be a way equal to length curve (for example, 150 m) along which you walked around the house.
However, let's return to the coordinate system. Let a point body move rectilinearly from point A with coordinate x 0 = 0 m to point B with coordinate x 1 = 10 m. The displacement of the body in this case will be 10 m. Since the movement was rectilinear, then the movement done will be equal to 10 meters body way.
If the body moved rectilinearly 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 the difference, where the initial coordinate is subtracted from the final coordinate. If the final coordinate is less than the initial one, that is, 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 have both positive and negative values, displacement is a vector quantity. In contrast, path is always a positive or zero quantity (path is a scalar quantity), since distance cannot be negative in principle.
Let's look at another example. The body moved rectilinearly from point A (x 0 = 2 m) to point B (x 1 = 8 m), then it also moved rectilinearly from B to point C with coordinate x 2 = 5 m. What are the equal and different common paths (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). You can also calculate total movement 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 movements 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 are different.
The problem considered 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 moment of time t 0 = 0 (the moment of the beginning of observations), then let, for example, x 1 correspond to t 1 = 3 s, and x 2 correspond 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 in a period of time of 3 seconds was 6 meters, and in a period of 5 seconds - 9 meters.
Time is involved in determining the path. In contrast, time is not particularly important for movement.
You have already encountered the concept of a path many times. Let us now get acquainted with a new concept for you - moving, which is more informative and useful in physics than the concept of a path.
Let's say you need to transport cargo from point A to point B on the other side of the river. 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.
By moving is a vector drawn from the initial position of a 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 us designate the intermediate points as B1, B2, B3 (see figure). You see that on segment AB1 the car was traveling northeast (first blue arrow), on segment B1B2 - southeast (second blue arrow), and on segment B2B3 - 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 is vector quantity, that is, having a spatial direction and numeric value(module). Unlike movement, path is scalar quantity, that is, having only a numerical value (and no spatial direction). The path is indicated by 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 the presence of an arrow above the designation optional.
Why did physics not limit itself to the concept of path, but introduced a more complex (vector) concept of displacement? Knowing the module and direction of movement, you can always say 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 has walked 7 km, we cannot say anything about where he is now.
Task. While hiking on the plain, the tourist walked north 3 km, then turned east and walked another 4 km. How far was he from the starting point of the route? Draw its movement.
Solution 1 – using 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). Measuring the module of the constructed vector with a ruler, we get: 5 cm. According to the scale we have chosen, the module of the tourist’s movement is 5 km. But let's remember: to know a vector means to know its magnitude and direction. Therefore, using a protractor, we determine: the direction of movement of the tourist is 53° with the direction to the north (check it yourself).
Solution 2 – without using a ruler or protractor.
Since the angle between the tourist’s movements 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 tourist’s movement:
As you can see, this value coincides with that 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 was solved in two ways with matching answers.
« Physics - 10th grade"
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 a point are divided into rectilinear and curvilinear.
If the trajectory 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 occupy position M 1 (Fig. 1.7, a). How to find its position after a certain period of time after this moment?
Suppose it is known that the point is at a distance l relative to its initial position. In this case, will we be able to unambiguously determine the new position of the point? Obviously not, since there are countless points that are distant from point M 1 at a distance l. In order to unambiguously determine the new position of the point, you also need to know in which direction from the point M 1 you should lay a segment of length l.
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 the 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 movement, movement can be considered as a change in the radius vector of a moving point.
Let radius vector 1 specify the position of the point at time t 1, and 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. From Figure 1.8 it is clear that the movement made by a point during the time period Δt is the change in its radius vector during this time. Therefore, denoting the change in radius vector through Δ, we can write: Δ = 1 - 2.
Way s- the length of the trajectory when moving a point from position M 1 to position M 2.
The displacement module may not be equal to the path traveled by the point.
For example, in Figure 1.8, the length of the line connecting points M 1 and M 2 is greater than the displacement module: s > |Δ|. The path is equal to the displacement only in the case of rectilinear unidirectional movement.
The displacement of the body Δ is a vector, the path s is a scalar, |Δ| ≤ s.
Source: “Physics - 10th grade”, 2014, textbook Myakishev, Bukhovtsev, Sotsky
Kinematics - Physics, textbook for grade 10 - Cool physics
Physics and knowledge of the world --- What is mechanics ---
Section 1 MECHANICS
Chapter 1: BASIC KINEMATICS
Mechanical movement. Trajectory. Path and movement. Speed addition
Mechanical body movement is called the change in its position in space relative to other bodies over time.
Mechanical movement of bodies studies Mechanics. The section of mechanics that describes the geometric properties of motion without taking into account the masses of bodies and acting forces is called kinematics .
Mechanical motion is relative. To determine the position of a body in space, you need to know its coordinates. To determine the coordinates of a material point, you must first select a reference body and associate a coordinate system with it.
Body of referencecalled a body 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 reference body with which it is associated, and the indication of the time reference form frame of reference , relative to which the movement 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.
Trajectoryit is the line along which the body moves.
Depending on the type of trajectory, movements 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 of this particle for a given time.
Unlike a path, 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 during a given time.
Motion vector module(that is, the length of the segment that connects the starting and ending 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 magnitude of the displacement vector is less than the distance traveled ℓ. The path and the modulus of displacement are equal only in one single case, when the body moves in a straight line.
Speedis a vector quantitative characteristic of body movement
average speed– this is a physical quantity equal to the ratio of the vector of movement of a point to the period of time
The direction of the average speed vector coincides with the direction of the displacement vector.
Instant speed, that is, the speed in this moment time is a vector physical quantity equal to the limit to which it tends average speed with an infinite decrease in the time interval Δt.
The instantaneous velocity vector is directed tangentially to the motion trajectory (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 uniform rectilinear motion in which a body travels a distance of one meter in one second. Speed is often measured in kilometers per hour.
or 1 
Speed addition
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 reference systems, all kinematic characteristics of motion (path, trajectory, displacement, speed, acceleration) turn out to be different.
For example, a passenger train moves along the railway 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 stationary and take it as a reference system, then the speed of a person is relative railway, will be equal to the addition of the speeds of the train and the person, that is
60 km/h + 5 km/h = 65 km/h if a person is walking in the same direction as the train and
60 km/h - 5 km/h = 55 km/h if a 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 it is necessary to take into account this angle, and the fact that speed is a vector quantity.
Let's look at the example described above in more detail - with details and pictures.
So, in our case, the railway is a stationary frame of reference. The train that moves along this road is a moving frame of reference. The carriage on which the person is walking is part of the train. The speed of a person relative to the carriage (relative to the moving frame of reference) is 5 km/h. Let's denote it with the letter . The speed of the train (and therefore the carriage) relative to a fixed frame of reference (that is, relative to the railway) is 60 km/h. Let's denote it with the letter . In other words, the speed of the train is the speed of the moving frame of reference relative to the stationary frame of reference.
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 the letter .
Let us associate the coordinate system XOY with the fixed reference system (Fig. 1.4), and with the moving reference system – X p O p Y p. Let us now determine the speed of a person relative to the fixed reference system, that is, relative to the railway.
Over a short period of time Δt the following events occur:
A person moves relative to the carriage at a distance
· The car moves relative to the railway at a distance
Then, during this period of time, the movement of a person relative to the railway is:
This law of addition of displacements . In our example, the movement of a person relative to the railway is equal to the sum of the movements of the person relative to the carriage and the carriage relative to the railway.
Dividing both sides of the equality by a small period of time Dt during which the movement occurred:
We get:
|