Trajectory(from Late Latin trajectories - related to movement) is the line along which a body (material point) moves. The trajectory of movement can be straight (the body moves in one direction) and curved, that is, mechanical movement can be rectilinear and curvilinear.
Trajectory rectilinear motion in this coordinate system it is a straight line. For example, we can assume that the trajectory of a car on a flat road without turns is straight.
Curvilinear movement is the movement of bodies in a circle, ellipse, parabola or hyperbola. Example curvilinear movement– the movement of a point on the wheel of a moving car or the movement of a car in a turn.
The movement can be difficult. For example, the trajectory of a body at the beginning of its journey can be rectilinear, then curved. For example, at the beginning of the journey a car moves along a straight road, and then the road begins to “wind” and the car begins to move in a curved direction.
Path
Path is the length of the trajectory. The path is a scalar quantity and in international system SI units are measured in meters (m). Path calculation is performed in many physics problems. Some examples will be discussed later in this tutorial.
Move vector
Move vector(or just moving) is a directed straight line segment connecting the initial position of the body with its subsequent position (Fig. 1.1). Displacement is a vector quantity. The displacement vector is directed from the starting point of movement to the ending point.
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 magnitude of the displacement vector can never be greater than the distance traveled.
The magnitude of the displacement vector is equal to the distance traveled when the path coincides with the trajectory (see sections Trajectory and Path), for example, if a car moves from point A to point B along a straight road. The magnitude of the displacement vector is less than the distance traveled when a material point moves along a curved path (Fig. 1.1).
Rice. 1.1. Displacement vector and distance traveled.
In Fig. 1.1:
Another example. If the car drives in a circle once, it turns out that the point at which the movement begins will coincide with the point at which the movement ends, and then the displacement vector will be equal to zero, and the distance traveled will be equal to the length of the circle. Thus, path and movement are two different concepts.
Vector addition rule
The displacement vectors are added geometrically according to the vector addition rule (triangle rule or parallelogram rule, see Fig. 1.2).
Rice. 1.2. Addition of displacement vectors.
Figure 1.2 shows the rules for adding vectors S1 and S2:
a) Addition according to the triangle rule
b) Addition according to the parallelogram rule
Motion vector projections
When solving problems in physics, projections of the displacement vector onto coordinate axes are often used. Projections of the displacement vector onto the coordinate axes can be expressed through the differences in the coordinates of its end and beginning. For example, if a material point moves from point A to point B, then the displacement vector (Fig. 1.3).
Let us choose the OX axis so that the vector lies in the same plane with this axis. Let's lower the perpendiculars from points A and B (from the starting and ending points of the displacement vector) until they intersect with the OX axis. Thus, we obtain the projections of points A and B onto the X axis. Let us denote the projections of points A and B, respectively, as A x and B x. The length of the segment A x B x on the OX axis is displacement vector projection on the OX axis, that is
S x = A x B x
IMPORTANT!
I remind you for those who do not know mathematics very well: do not confuse a vector with the projection of a vector onto any axis (for example, S x). A vector is always indicated by a letter or several letters, above which there is an arrow. In some electronic documents, the arrow is not placed, as this may cause difficulties when creating an electronic document. In such cases, be guided by the content of the article, where the word “vector” may be written next to the letter or in some other way they indicate to you that this is a vector, and not just a segment.
Rice. 1.3. Projection of the displacement vector.
The projection of the displacement vector onto the OX axis is equal to the difference between the coordinates of the end and beginning of the vector, that is
S x = x – x 0 Similarly, the projections of the displacement vector on the OY and OZ axes are determined and written: S y = y – y 0 S z = z – z 0
Here x 0 , y 0 , z 0 are the initial coordinates, or the coordinates of the initial position of the body (material point); x, y, z - final coordinates, or coordinates of the subsequent position of the body (material point).
The projection of the displacement vector is considered positive if the direction of the vector and the direction of the coordinate axis coincide (as in Fig. 1.3). If the direction of the vector and the direction of the coordinate axis do not coincide (opposite), then the projection of the vector is negative (Fig. 1.4).
If the displacement vector is parallel to the axis, then the modulus of its projection is equal to the modulus of the Vector itself. If the displacement vector is perpendicular to the axis, then the modulus of its projection is equal to zero (Fig. 1.4).
Rice. 1.4. Motion vector projection modules.
The difference between the subsequent and initial values of some quantity is called the change in this quantity. That is, the projection of the displacement vector onto the coordinate axis is equal to the change in the corresponding coordinate. For example, for the case when the body moves perpendicular to the X axis (Fig. 1.4), it turns out that the body DOES NOT MOVE relative to the X axis. That is, the movement of the body along the X axis is zero.
Let's consider an example of body motion on a plane. The initial position of the body is point A with coordinates x 0 and y 0, that is, A(x 0, y 0). The final position of the body is point B with coordinates x and y, that is, B(x, y). Let's find the modulus of body displacement.
From points A and B we lower perpendiculars to the coordinate axes OX and OY (Fig. 1.5).
Rice. 1.5. Movement of a body on a plane.
Let us determine the projections of the displacement vector on the OX and OY axes:
S x = x – x 0 S y = y – y 0
In Fig. 1.5 it is clear that triangle ABC is a right triangle. It follows from this that when solving the problem one can use Pythagorean theorem, with which you can find the module of the displacement vector, since
AC = s x CB = s y
According to the Pythagorean theorem
S 2 = S x 2 + S y 2
Where can you find the module of the displacement vector, that is, the length of the body’s path from point A to point B:
And finally, I suggest you consolidate your knowledge and calculate a few examples at your discretion. To do this, enter some numbers in the coordinate fields and click the CALCULATE button. Your browser must support the execution of JavaScript scripts and script execution must be enabled in your browser settings, otherwise the calculation will not be performed. In real numbers, the integer and fractional parts must be separated by a dot, for example, 10.5.
How to determine the displacement module? (mechanics) and got the best answer
Answer from Ivan Vyazigin[newbie]
according to the Pythagorean theorem = root (16+9) = 5
Reply from Marinas[guru]
Three main ways to describe body movement
Vector method
t. O - reference body; t. A - material point (particle); - radius vector (this is a vector connecting the origin with the position of a point at an arbitrary moment in time)
Trajectory (1-2) - a line describing the movement of a body (material point A) over a period of time
Displacement () is a vector connecting the positions of a moving point at the beginning and end of a certain period of time.
Path () – length of the trajectory section.
Let's write the equation of motion of a point in vector form:
The speed of a point is the limit of the ratio of movement to the period of time during which this movement occurred, when this period of time tends to zero.
That is, instantaneous speed
Acceleration (or instantaneous acceleration) is a vector physical quantity equal to the limit of the ratio of the change in speed to the period of time during which this change occurred.
Acceleration, like the change in speed, is directed towards the concavity of the trajectory and can be decomposed into two components - tangential - tangent to the trajectory of movement - and normal - perpendicular to the trajectory.
- full acceleration;
- normal acceleration (characterizes the change in speed in direction);
- tangential acceleration (characterizes the change in speed in magnitude);
, where is the unit normal vector ()
R1 - radius of curvature.
,
Where;
Coordinate method of describing movement
At coordinate method descriptions of motion, the change in the coordinates of a point over time is written in the form of functions of all three of its coordinates versus time:
kinematic levels of motion of a point)
Projections on the axis:
A natural way to describe movement
Reply from Av paap[newbie]
thank you
Reply from Olga Gavrilova[active]
Why is this so?
Reply from 3 answers[guru]
Hello! Here is a selection of topics with answers to your question: How to determine the displacement module? (mechanics)
Class: 9
Lesson objectives:
- Educational:
– introduce the concepts of “movement”, “path”, “trajectory”. - Developmental:
- develop logical thinking, correct physical speech, use appropriate terminology. - Educational:
– achieve high class activity, attention, and concentration of students.
Equipment:
- plastic bottle with a capacity of 0.33 liters with water and a scale;
- medical bottle with a capacity of 10 ml (or small test tube) with a scale.
Demonstrations: Determining displacement and distance traveled.
Lesson progress
1. Updating 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, let's 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 completed during the oral test ex. 1 page 9 of the textbook.
1. Which coordinate system (one-dimensional, two-dimensional, three-dimensional) should be chosen to determine the position of bodies:
a) tractor in the field;
b) helicopter in the sky;
c) train
d) chess piece on the board.
2. Given the expression: S = υ 0 t + (a t 2) / 2, express: a, υ 0
1. Which coordinate system (one-dimensional, two-dimensional, three-dimensional) should be chosen to determine the position of such bodies:
a) chandelier in the room;
b) elevator;
c) submarine;
d) plane on the runway.
2. Given the expression: S = (υ 2 – υ 0 2) / 2 · a, express: υ 2, υ 0 2.
3. Study of new theoretical material.
Associated with changes in the coordinates of the body is the quantity introduced to describe the movement - MOVEMENT.
The displacement of a body (material point) is a vector connecting the initial position of the body with its subsequent position.
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 begins at a certain point and ends with a point indicating 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 magnitude. The modulus of a vector is a scalar, i.e. numerical value. Knowing the initial position and the vector of movement of the body, you can determine where the body is located.
In the process of movement, a 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 plane 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 is described by a moving body (considered as a material point) in relation to the selected reference system.
The movement in which all points body moving along the same trajectories, called progressive.
Very often the trajectory is an invisible line. Trajectory moving point can be direct or crooked line. According to the shape of the trajectory movement It happens straightforward And curvilinear.
The path length is PATH. The path is a scalar quantity and is denoted by the letter l. The path increases if the body moves. And remains unchanged if the body is at rest. Thus, the path cannot decrease over time.
The displacement module and the path can coincide in value only if the body moves along a straight line in the same direction.
What is the difference between a path and a movement? These two concepts are often confused, although in fact they are very different from each other. Let's 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 a numerical value.
- Displacement is a vector quantity and is characterized by both a numerical value (module) and direction.
- When a body moves, the path can only increase, and the displacement module can both increase and decrease.
- If the body returns to the starting point, its displacement is zero, but the path is not zero.
| Path | Moving | |
| Definition | The length of the trajectory described by a body in a certain time | A vector connecting the initial position of the body with its subsequent position |
| Designation | l [m] | S [m] |
| Nature of physical quantities | Scalar, i.e. determined only by numeric value | Vector, i.e. determined by numerical value (modulus) and direction |
| The need for introduction | Knowing the initial position of the body and the path l traveled over a period of time t, it is impossible to determine the position of the body at a given moment in time t | Knowing the initial position of the body and S for a period of time t, the position of the body at a given moment of 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 to the neck with water.
- Fill the bottle with the 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 closing it with the stopper) so that the neck of the bottle enters the water of the bottle. The bottle floats on the surface of the water in the bottle. Some of the water will spill out of the bottle.
- Screw the bottle cap on.
- Squeeze the sides of the bottle and lower the float to the bottom of the bottle.
- By releasing the pressure on the walls of the bottle, make the float float to the surface. 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 review.
- 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 movement of the ball. (Path – 4 m, movement – 2 m.)
6. Lesson summary.
Review of lesson concepts:
– movement;
– trajectory;
- path.
7. Homework.
§ 2 of the textbook, questions after the paragraph, exercise 2 (p. 12) of the textbook, repeat the lesson experience at home.
References
1. Peryshkin A.V., Gutnik E.M.. Physics. 9th grade: textbook for general educational institutions - 9th ed., stereotype. – M.: Bustard, 2005.
In kinematics, to find various quantities, we use mathematical methods. In particular, to find the magnitude of the displacement vector, you need to apply a formula from vector algebra. It contains the coordinates of the beginning and end points of the vector, i.e. initial and final body position.
Instructions
During movement, a material body changes its position in space. Its trajectory can be a straight line or arbitrary; its length is the path of the body, but not the distance over which it has moved. These two quantities coincide only in the case of rectilinear motion.
So, let the body make some movement from point A (x0, y0) to point B (x, y). To find the magnitude of the displacement vector, you need to calculate the length of the vector AB. Draw coordinate axes and mark on them the known points of the initial and final positions of the body A and B.
Draw a line from point A to point B, indicate the direction. Lower the projections of its ends onto the axis and plot on the graph parallel and equal segments passing through the points under consideration. You will see that the figure shows a right triangle with projection sides and hypotenuse displacement.
Using the Pythagorean theorem, find the length of the hypotenuse. This method is widely used in vector algebra and is called the triangle rule. First, write down the lengths of the legs; they are equal to the differences between the corresponding abscissas and ordinates of points A and B:
ABx = x – x0 – projection of the vector onto the Ox axis;
ABy = y – y0 – its projection onto the Oy axis.
Define the displacement |AB|:
|AB| = ?(ABx? + ABy?) = ((x – x0)? + (y – y0)?).
For three-dimensional space, add a third coordinate to the formula - applicate z:
|AB| = ?(ABx? + ABy? + ABz?) = ((x – x0)? + (y – y0)? + (z – z0)?).
The resulting formula can be applied to any trajectory and type of movement. In this case, the magnitude of the displacement has an important property. It is always less than or equal to the length of the path; in the general case, its line does not coincide with the trajectory curve. Projections are mathematical quantities that can be either greater or less than zero. However, this does not matter, since they participate in the calculation to an even degree.
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 displacement module 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). It can be seen from the figure 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 it comes to moving, it's 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: .
You can calculate the displacement modulus as the hypotenuse of a 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 a system arbitrarily, that is, direct the 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 selected coordinate system.
Conclusion
In conclusion, it can be noted that we have become acquainted with an important quantity - displacement. Please note again that movement and path can coincide only in the case of rectilinear movement, without changing the direction of such movement.
References
- 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? How are they different?
- 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).
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