Physics
Displacement Vector
A displacement vector in physics represents the change in position of an object from its initial to final location. It has both magnitude and direction, and is typically represented by an arrow. The length of the arrow corresponds to the magnitude of the displacement, while the direction of the arrow indicates the direction of the displacement.
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10 Key excerpts on "Displacement Vector"
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
Examples of vectors: position (e.g., 1.5 km north or 3.5 m in the positive x direction), velocity (2.0 m/s in the negative y direction), and force (5.5 N in the negative z direction). Going Deeper Scalar Quantities and Vector Quantities I N T E R A C T I V E F E A T U R E Position and Displacement: Vectors in One Dimension | 45 Displacement is the difference between two vectors, so it is also a vector. In mathematics, the symbol ∆ usually means change in or difference in and is equal to the final value minus the initial value of the changing quantity. Figure 2.1.2 When the initial position is +9.00 m and the final position is +2.00 m, the displace- ment is −7.00 m. +x 0 x = +2.00 m x 0 = +9.00 m Definition of Displacement When an object moves from place to place, the magnitude of its displacement is equal to the straight-line distance between the initial and final positions. The direction of the displacement is measured from the initital position toward the final position. The displacement depends only on the initial and final positions, and not on the details of the path taken. In Figure 2.1.2, the bear is initially facing right, suggesting that the bear first rode some unspecified distance in the positive direction before turning around and riding back to the final position. We distinguish between the distance (the total length of travel) and the magnitude of the displacement. Unless the path from the initial to the final position was a straight line with no backtracking, the distance traveled will be greater than the magnitude of the displacement. Adding and Subtracting Vectors in One Dimension Sometimes it is necessary or useful to depict vectors graphically, in which case they are represented as arrows. The direction of the arrow should be consistent with the coordinate system in the diagram, and the length of the arrow should be proportional to the vector’s magnitude.
eBook - PDFCalculus
Multivariable
- William G. McCallum, Deborah Hughes-Hallett, Andrew M. Gleason, David O. Lomen, David Lovelock, Jeff Tecosky-Feldman, Thomas W. Tucker, Daniel E. Flath, Joseph Thrash, Karen R. Rhea, Andrew Pasquale, Sheldon P. Gordon, Douglas Quinney, Patti Frazer Lock(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
Dallas Pittsburgh Oshkosh Buffalo, SD Los Angeles Albuquerque Figure 13.1: Displacement Vectors between cities Notation and Terminology The Displacement Vector is our first example of a vector. Vectors have both magnitude and direction; in comparison, a quantity specified only by a number, but no direction, is called a scalar. 1 For instance, the time taken by the flight from Dallas to Pittsburgh is a scalar quantity. Displacement is a vector since it requires both distance and direction to specify it. In this book, vectors are written with an arrow over them, v , to distinguish them from scalars. Other books use a bold v to denote a vector. We use the notation −−→ PQ to denote the Displacement Vector from a point P to a point Q. The magnitude, or length, of a vector v is written ‖v ‖. Addition and Subtraction of Displacement Vectors Suppose NASA commands a robot on Mars to move 75 metres in one direction and then 50 metres in another direction. (See Figure 13.2.) Where does the robot end up? Suppose the displacements are represented by the vectors v and w , respectively. Then the sum v + w gives the final position. 1 So named by W. R. Hamilton because they are merely numbers on the scale from -∞ to ∞. 13.1 Displacement VectorS 737 The sum, v + w , of two vectors v and w is the combined displacement resulting from first applying v and then w . (See Figure 13.3.) The sum w + v gives the same displacement. Combined displacement 75 m 50 m Start Finish Figure 13.2: Sum of displacements of robots on Mars v w v + w v w Start Finish Figure 13.3: The sum v + w = w + v Suppose two different robots start from the same location. One moves along a Displacement Vector v and the second along a Displacement Vector w . What is the Displacement Vector, x , from the first robot to the second? (See Figure 13.4.) Since v + x = w , we define x to be the difference x = w − v .
eBook - PDF- Deborah Hughes-Hallett, William G. McCallum, Andrew M. Gleason, Eric Connally, Daniel E. Flath, Selin Kalaycioglu, Brigitte Lahme, Patti Frazer Lock, David O. Lomen, David Lovelock, Guadalupe I. Lozano, Jerry Morris, David Mumford, Brad G. Osgood, Cody L. Patterson, Douglas Quinney, Karen R. Rhea, Ayse Arzu Sahin, Ad(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
62. The Displacement Vector from (1, 1, 1) to (1, 2, 3) is − − 2 . 63. The Displacement Vector from (, ) to (, ) is the same as the Displacement Vector from (, ) to (, ). 13.2 VECTORS IN GENERAL Besides displacement, there are many quantities that have both magnitude and direction and are added and multiplied by scalars in the same way as displacements. Any such quantity is called a vector and is represented by an arrow in the same manner we represent displacements. The length of the arrow is the magnitude of the vector, and the direction of the arrow is the direction of the vector. 13.2 VECTORS IN GENERAL 711 Velocity Versus Speed The speed of a moving body tells us how fast it is moving, say 80 km/hr. The speed is just a number; it is therefore a scalar. The velocity, on the other hand, tells us both how fast the body is moving and the direction of motion; it is a vector. For instance, if a car is heading northeast at 80 km/hr, then its velocity is a vector of length 80 pointing northeast. The velocity vector of a moving object is a vector whose magnitude is the speed of the object and whose direction is the direction of its motion. The velocity vector is the Displacement Vector if the object moves at constant velocity for one unit of time. Example 1 A car is traveling north at a speed of 100 km/hr, while a plane above is flying horizontally southwest at a speed of 500 km/hr. Draw the velocity vectors of the car and the plane. Solution Figure 13.19 shows the velocity vectors. The plane’s velocity vector is five times as long as the car’s, because its speed is five times as great. ✲ Velocity vector of car ✲ Velocity vector of plane North ✻ Figure 13.19: Velocity vector of the car is 100 km/hr north and of the plane is 500 km/hr southwest The next example illustrates that the velocity vectors for two motions add to give the velocity vector for the combined motion, just as displacements do.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
67 4.1 POSITION AND DISPLACEMENT Learning Objectives After reading this module, you should be able to . . . 4.1.1 Draw two-dimensional and three-dimensional position vectors for a particle, indicating the components along the axes of a coordinate system. 4.1.2 On a coordinate system, determine the direction and magnitude of a particle’s position vector from its components, and vice versa. 4.1.3 Apply the relationship between a particle’s dis- placement vector and its initial and final position vectors. Key Ideas ● The location of a particle relative to the origin of a coordinate system is given by a position vector r → , which in unit-vector notation is r → = x ˆ i + y ˆ j + z k ̂ . Here x ˆ i, y ˆ j, and z k ̂ are the vector components of posi- tion vector r → , and x, y, and z are its scalar components (as well as the coordinates of the particle). ● A position vector is described either by a magnitude and one or two angles for orientation, or by its vector or scalar components. ● If a particle moves so that its position vector changes from r → 1 to r → 2 , the particle’s displacement Δ r → is Δ r → = r → 2 − r → 1 . The displacement can also be written as Δ r → = ( x 2 − x 1 ) ˆ i + ( y 2 − y 1 ) ˆ j + ( z 2 − z 1 )k ̂ = Δx ˆ i + Δy ˆ j + Δz k ̂ . What Is Physics? In this chapter we continue looking at the aspect of physics that analyzes motion, but now the motion can be in two or three dimensions. For example, medical researchers and aeronautical engineers might concentrate on the physics of the two- and three-dimensional turns taken by fighter pilots in dogfights because a modern high-performance jet can take a tight turn so quickly that the pilot immediately loses consciousness. A sports engineer might focus on the physics of basketball. For example, in a free throw (where a player gets an uncontested shot at the basket from about 4.3 m), a player might employ the overhand push shot, in which the ball is pushed away from about shoulder height and then released.
eBook - PDF- Daniel A. Fleisch(Author)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
1 Vectors 1.1 Definitions (basic) There are many ways to define a vector. For starters, here’s the most basic: A vector is the mathematical representation of a physical entity that may be characterized by size (or “magnitude”) and direction. In keeping with this definition, speed (how fast an object is going) is not rep-resented by a vector, but velocity (how fast and in which direction an object is going) does qualify as a vector quantity. Another example of a vector quantity is force, which describes how strongly and in what direction something is being pushed or pulled. But temperature, which has magnitude but no direction, is not a vector quantity. The word “vector” comes from the Latin vehere meaning “to carry;” it was first used by eighteenth-century astronomers investigating the mechanism by which a planet is “carried” around the Sun. 1 In text, the vector nature of an object is often indicated by placing a small arrow over the variable representing the object (such as F ), or by using a bold font (such as F ), or by underlining (such as F or F ∼ ). When you begin hand-writing equations involving vectors, it’s very important that you get into the habit of denoting vectors using one of these techniques (or another one of your choosing). The important thing is not how you denote vectors, it’s that you don’t simply write them the same way you write non-vector quantities. A vector is most commonly depicted graphically as a directed line seg-ment or an arrow, as shown in Figure 1.1 (a). And as you’ll see later in this section, a vector may also be represented by an ordered set of N numbers, 1 The Oxford English Dictionary . 2nd ed. 1989. 1 2 Vectors (b) (a) Figure 1.1 Graphical depiction of a vector (a) and a vector field (b). where N is the number of dimensions in the space in which the vector resides. Of course, the true value of a vector comes from knowing what it represents.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
We can apply the same basic equations for displacement, velocity, and acceleration we derived in Motion Along a Straight Line to describe the motion of the jets in two and three dimensions, but with some modifications—in particular, the inclusion of vectors. In this chapter we also explore two special types of motion in two dimensions: projectile motion and circular motion. Last, we conclude with a discussion of relative motion. In the chapter-opening picture, each jet has a relative motion with respect Chapter 4 | Motion in Two and Three Dimensions 157 to any other jet in the group or to the people observing the air show on the ground. 4.1 | Displacement and Velocity Vectors Learning Objectives By the end of this section, you will be able to: • Calculate position vectors in a multidimensional displacement problem. • Solve for the displacement in two or three dimensions. • Calculate the velocity vector given the position vector as a function of time. • Calculate the average velocity in multiple dimensions. Displacement and velocity in two or three dimensions are straightforward extensions of the one-dimensional definitions. However, now they are vector quantities, so calculations with them have to follow the rules of vector algebra, not scalar algebra. Displacement Vector To describe motion in two and three dimensions, we must first establish a coordinate system and a convention for the axes. We generally use the coordinates x, y, and z to locate a particle at point P(x, y, z) in three dimensions. If the particle is moving, the variables x, y, and z are functions of time (t): (4.1) x = x(t) y = y(t) z = z(t). The position vector from the origin of the coordinate system to point P is r → (t). In unit vector notation, introduced in Coordinate Systems and Components of a Vector, r → (t) is (4.2) r → (t) = x(t) i ^ + y(t) j ^ + z(t) k ^ . Figure 4.2 shows the coordinate system and the vector to point P, where a particle could be located at a particular time t.
eBook - PDFMathematical Methods for Physicists
A Concise Introduction
- Tai L. Chow(Author)
- 2000(Publication Date)
- Cambridge University Press(Publisher)
1 Vector and tensor analysis Vectors and scalars Vector methods have become standard tools for the physicists. In this chapter we discuss the properties of the vectors and vector fields that occur in classical physics. We will do so in a way, and in a notation, that leads to the formation of abstract linear vector spaces in Chapter 5. A physical quantity that is completely specified, in appropriate units, by a single number (called its magnitude) such as volume, mass, and temperature is called a scalar. Scalar quantities are treated as ordinary real numbers. They obey all the regular rules of algebraic addition, subtraction, multiplication, division, and so on. There are also physical quantities which require a magnitude and a direction for their complete specification. These are called vectors if their combination with each other is commutative (that is the order of addition may be changed without aecting the result). Thus not all quantities possessing magnitude and direction are vectors. Angular displacement, for example, may be characterised by magni-tude and direction but is not a vector, for the addition of two or more angular displacements is not, in general, commutative (Fig. 1.1). In print, we shall denote vectors by boldface letters (such as A ) and use ordin-ary italic letters (such as A ) for their magnitudes; in writing, vectors are usually represented by a letter with an arrow above it such as ~ A . A given vector A (or ~ A ) can be written as A A ^ A ; 1 : 1 where A is the magnitude of vector A and so it has unit and dimension, and ^ A is a dimensionless unit vector with a unity magnitude having the direction of A . Thus ^ A A = A . 1 A vector quantity may be represented graphically by an arrow-tipped line seg-ment. The length of the arrow represents the magnitude of the vector, and the direction of the arrow is that of the vector, as shown in Fig.
eBook - PDF- Stephen Lee(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
Vectors But the principal failing occurred in the sailing And the bellman, perplexed and distressed, Said he had hoped, at least when the wind blew due East That the ship would not travel due West. Lewis Carroll 3.1 Adding vectors All these questions involve vectors – displacement, velocity and force. When you are concerned only with the magnitude and direction of these vectors, the three problems can be reduced to one. They can all be solved using the same vector techniques. Displacement Vectors The instruction ‘walk 12 m east and then 5 m north’ can be modelled mathematically using a scale diagram, as in figure 3.1. The arrowed lines AB and BC are examples of vectors. We write the vectors as AB ⎯→ and BC ⎯→ . The arrow above the letters is very important as it indicates the direction of the vector. AB ⎯→ means from A to B. AB ⎯→ and BC ⎯→ are examples of Displacement Vectors . Their lengths represent the magnitude of the displacements. Figure 3.1 12 m 5 m q p C B A N 3 Q UESTION 3.1 If you walk 12 m east and then 5 m north, how far and in what direction will you be from your starting point? A bird is caught in a wind blowing east at 12 ms 1 and flies so that its speed would be 5 ms 1 north in still air. What is its actual velocity? A sledge is being pulled with forces of 12 N east and 5 N north. What single force would have the same effect? It is often more convenient to use a single letter to denote a vector. For example in textbooks and exam papers you might see the Displacement Vectors AB ⎯→ and BC ⎯→ written as p and q (i.e. in bold print). When writing these vectors yourself, you should underline your letters, e.g. p and q . The magnitudes of p and q are then shown as p and q or p and q (in italics). These are scalar quantities. Figure 3.2 The combined effect of the two displacements AB ⎯→ ( p ) and BC ⎯→ ( q ) is AC ⎯→ and this is called the resultant vector . It is marked with two arrows to distinguish it from p and q .
eBook - PDF- Daniel Kleppner, Robert Kolenkow(Authors)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
Our first application of vectors will be to the description of position and motion in familiar three-dimensional space. 5 4 3 2 1 4 3 2 1 0 1 2 3 4 y x z To locate the position of a point in space, we start by setting up a co-ordinate system. For convenience we choose a three-dimensional Carte-sian system with axes x , y , and z , as shown. In order to measure position, the axes must be marked in some convenient unit of length—meters, for instance. The position of the point of interest is given by listing the val-ues of its three coordinates, x 1 , y 1 , z 1 , which we can write compactly as a position vector r ( x 1 , y 1 , z 1 ) or more generally as r ( x , y , z ). This nota-tion can be confusing because we normally label the axes of a Cartesian coordinate system by x , y , z . However, r ( x , y , z ) is really shorthand for r ( x -axis , y -axis , z -axis). The components of r are the coordinates of the point referred to the particular coordinate axes. The three numbers ( x , y , z ) do not represent the components of a vec-tor according to our previous discussion because they specify only the position of a single point, not a magnitude and direction. Unlike other physical vectors such as force and velocity, r is tied to a particular coor-dinate system. 1.7 THE POSITION VECTOR R AND DISPLACEMENT 13 The position of an arbitrary point P at ( x , y , z ) is written as r = ( x , y , z ) = x ˆ i + y ˆ j + z ˆ k . If we move from the point x 1 , y 1 , z 1 to some new position, x 2 , y 2 , z 2 , then the displacement defines a true vector S with coordinates S x = x 2 − x 1 , S y = y 2 − y 1 , S z = z 2 − z 1 . S x y z ( x 2, y 2, z 2 ) ( x 1, y 1, z 1 ) S is a vector from the initial position to the final position—it defines the displacement of a point of interest. Note, however, that S contains no information about the initial and final positions separately—only about the relative position of each.
eBook - PDFFrom Atoms to Galaxies
A Conceptual Physics Approach to Scientific Awareness
- Sadri Hassani(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
Do not confuse average speed with average velocity. Average Velocity A vector quantity defined as displacement of an object in some time interval divided by that time interval. Do not confuse average velocity with average speed. Centripetal Acceleration The acceleration of an object moving with constant speed on a circle. Its direction is toward the center (thus the name centr ipetal). Displacement A directed line segment (arrow) drawn from the initial position of an object in motion to its final position. The initial and final positions are determined by the beginning and end of a time interval. Distance The length of the path taken by an object in motion in some time interval. Instantaneous Speed Distance traveled in some time interval divided by that time inter-val when the time interval is taken to be as short as possible. Instantaneous Velocity A vector quantity defined as displacement of an object in some time interval divided by that time interval when the time interval is taken to be as short as possible. Observer A point with respect to which the motion of an object is considered. Parallax The change in the angle of the line of sight of an object in motion relative to an observer. Position Vector A directed line segment (arrow) drawn from the observer to the object in motion. Reference Frame The collection of all objects (including people) which do not move relative to one another; i.e., the position vector of each object relative to any other object does not change. 6.5.3 Review Questions 6.1. What is motion? When we say that a car is moving, what (who) is the observer? 6.2. What is a reference frame? Do all objects (people) in a lecture hall constitute a reference frame? What about the lecturer who is pacing the width of the hall? 6.3. Is it possible for one observer to detect a simple motion for an object while a second observer detects a complicated motion for the same object? Give an example.
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