Motion Along a Line

9 Key excerpts on "Motion Along a Line"

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  eBook - PDF

  Fundamentals of Physics, Extended

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  There are countless other Motion Along a Straight Line 14 CHAPTER 2 MOTION ALONG A STRAIGHT LINE examples. In this chapter, we study the basic physics of motion where the object (race car, tectonic plate, blood cell, or any other object) moves along a single axis. Such motion is called one-dimensional motion. Motion The world, and everything in it, moves. Even seemingly stationary things, such as a roadway, move with Earth’s rotation, Earth’s orbit around the Sun, the Sun’s orbit around the center of the Milky Way galaxy, and that galaxy’s migration relative to other galaxies. The classification and comparison of motions (called kinematics) is often challenging. What exactly do you measure, and how do you compare? Before we attempt an answer, we shall examine some general properties of motion that is restricted in three ways. 1. The motion is along a straight line only. The line may be vertical, horizontal, or slanted, but it must be straight. 2. Forces (pushes and pulls) cause motion but will not be discussed until Chapter 5. In this chapter we discuss only the motion itself and changes in the motion. Does the moving object speed up, slow down, stop, or reverse direction? If the motion does change, how is time involved in the change? 3. The moving object is either a particle (by which we mean a point-like object such as an electron) or an object that moves like a particle (such that every portion moves in the same direction and at the same rate). A stiff pig slipping down a straight playground slide might be considered to be moving like a par- ticle; however, a tumbling tumbleweed would not. Position and Displacement To locate an object means to find its position relative to some reference point, often the origin (or zero point) of an axis such as the x axis in Fig. 2.1.1. The positive direction of the axis is in the direction of increasing numbers (coordi- nates), which is to the right in Fig.

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  Halliday and Resnick's Principles of Physics

  • David Halliday, Robert Resnick, Jearl Walker(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  12 CHAPTER 2 MOTION ALONG A STRAIGHT LINE Motion The world, and everything in it, moves. Even seemingly stationary things, such as a roadway, move with Earth’s rotation, Earth’s orbit around the Sun, the Sun’s orbit around the center of the Milky Way galaxy, and that galaxy’s migration relative to other galaxies. The classification and comparison of motions (called kinematics) is often challenging. What exactly do you measure, and how do you compare? Before we attempt an answer, we shall examine some general properties of motion that is restricted in three ways. 1. The motion is along a straight line only. The line may be vertical, horizontal, or slanted, but it must be straight. 2. Forces (pushes and pulls) cause motion but will not be discussed until Chapter 5. In this chapter we discuss only the motion itself and changes in the motion. Does the moving object speed up, slow down, stop, or reverse direction? If the motion does change, how is time involved in the change? 3. The moving object is either a particle (by which we mean a point-like object such as an electron) or an object that moves like a particle (such that every portion moves in the same direction and at the same rate). A stiff pig slipping down a straight playground slide might be considered to be moving like a par- ticle; however, a tumbling tumbleweed would not. Position and Displacement To locate an object means to find its position relative to some reference point, often the origin (or zero point) of an axis such as the x axis in Fig. 2-1. The positive direction of the axis is in the direction of increasing numbers (coordinates), which is to the right in Fig. 2-1. The opposite is the negative direction. For example, a particle might be located at x = 5 m, which means it is 5 m in the positive direction from the origin. If it were at x = −5 m, it would be just as far from the origin but in the opposite direction.

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  Essential Physics

  • John Matolyak, Ajawad Haija(Authors)
  • 2013(Publication Date)
  • CRC Press

    (Publisher)

  25 © 2010 Taylor & Francis Group, LLC Motion in One Dimension This chapter addresses deriving and using the equations of motion that describe the time depen-dence of an object’s displacement, velocity, and acceleration. Relationships between displacement, velocity, and acceleration are also of importance and will be derived. This chapter starts with the basic definitions of displacement and average velocity of an object moving in one dimension. Such a simplified start will help to lead a complete set of equations of motion for an object moving along one dimension, east–west, north–south, or up and down. In the context of coordinate systems that were treated in the previous chapter, the one-dimensional motion will reduce the time and effort needed on the study of motion in two dimensions, which is the subject of the next chapter. 2.1 DISPLACEMENT Motion can be defined as a continuous change in position and that change could occur in one, two, or three dimensions. As the treatment here will be limited to motions only in one dimension, one axis of the coordinate system described in Chapter 1 will suffice. Choosing this axis as the x-axis, we depict on it a point, O, which will be considered as an origin of zero coordinate. Any point on the right of the origin O will have a positive coordinate and any point on the left of O will have a negative coordinate (Figure 2.1). The displacement, denoted by Δ x, of an object as it moves from an initial position x i to a final position x f along the x-axis can be defined as the change in the object’s position along the x-axis. That is Δ x = x f – x i . (2.1) 2.1.1 S PECIAL R EMARKS The quantities in Equation 2.1 are treated as vectors. x i is the initial position vector, x f the final posi-tion vector, and Δ x the displacement vector. Accordingly, all these quantities have a direction and magnitude.

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  Particle Mechanics

  • Chris Collinson, Tom Roper(Authors)
  • 1995(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  2

  Straight Line Motion

  Some of the intuitive ideas met in the previous chapter are developed in the context of the motion of a particle moving on a straight line. Newton’s second law of motion is expressed as an equation, the general solution of which describes all possible motions of a particle consistent with the given forces acting on that particle. The importance of initial conditions to the determination of the actual motion of a given particle is stressed. The concepts of kinetic and potential energies are introduced and the conservation of the total energy used as a powerful tool for a qualitative discussion of motion. The theme of linear approximation again plays a role, particularly in the discussion of the gravitational acceleration experienced by a particle moving close to the earth’s surface. The motion of a particle moving in a resisting medium is studied and the existence of a limiting speed is predicted.

  2.1 Kinematics of a Particle Moving on a Straight Line

  Consider a particle P moving on a straight line l as illustrated. The first step in modelling the observed motion of such a particle is to decide on a method for

  Fig 2.1 Straight line motion.

  specifying the position of the particle at any given instant of time. If the particle models a car in which you are travelling then you might well specify your position by observing features of the surrounding landscape – a distant spire, a roadside telephone box, intersections with other roads, etc. However, we do not usually follow this procedure when discussing straight line motion. Instead we consider the straight line in isolation rather than in relation to the rest of the space; the motion is thought of as being one dimensional. In these circumstances the position of P can only be specified in relation to some other point lying on the straight line itself. Such a point O will be referred to as an origin.

  As the particle P moves on the straight line the distance of P from O will vary. This distance is not sufficient to specify the position of P relative to O ; knowing the distance alone does not specify whether P is to the left or right of O. The use of the phrase to the left or right of O is itself ambiguous and depends very much on how you look at the straight line. Turning the page upside down interchanges “left of O ” with “right of O ”. What we really require is an unambiguous method for specifying the orientation of P relative to O. The simplest procedure is to first orientate the line itself by placing an arrowhead on it, as has been done in Fig 2.2 . The orientation of P relative to O can then be specified according as to whether a translation from O to P is in the direction of the arrowhead or not. In what follows the direction of the arrowhead specifying the orientation of the straight line will be called the positive direction , the opposite direction will be called the negative direction

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  From Atoms to Galaxies

  A Conceptual Physics Approach to Scientific Awareness

  • Sadri Hassani(Author)
  • 2010(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 6 Kinematics: Describing Motion Chapter 4 taught us some basic knowledge of motion along a straight line. This chapter generalizes the concept of motion and introduces some fundamental quantities necessary for its description. 6.1 Position, Displacement, and Distance What is motion? Vaguely speaking, it is the change in the state of an object. More precisely, it is the change in the position of an object with time. Still more precisely, we must speak of the motion of an object relative to an observer , although the “object” could be a person, and the “observer” a thing. To analyze the motion, draw an arrow from the “observer” O to the “object” A , and call the arrow the position vector . 1 The very definition of the position vector assumes Position vector. that both O and A are points. The position vector, denoted commonly by r , determines the instantaneous position of the object A [Figure 6.1(a)] relative to an observer O . The word “instantaneous” is important because the position of the point object A is, in general, constantly changing. If we were to take snapshots of A at various times, t 1 , t 2 , t 3 , etc., and label the corresponding points at which A is located by A 1 , A 2 , A 3 , etc, we would have a situation depicted in Figure 6.1(b), with position vectors r 1 , r 2 , r 3 , etc. In this figure, only three out of an infinitude of possible snapshots are shown. Bear in mind that every directed line segment from O to a point on the curve in Figure 6.1 is a possible position vector. What do you know? 6.1. Can you say that the observer—as defined in the descrip-tion of motion—does not move? If the point A does not change, i.e., if the position vector r does not vary with time, we say that the object is stationary relative to O . You have to make a clear distinction between the distance between O and A , which is the length of r , and the position vector, which is the directed line segment r .

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  Engineering Mechanics

  Problems and Solutions

  • Arshad Noor Siddiquee, Zahid A. Khan, Pankul Goel(Authors)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  1 2 3 4 B A Fig. 10.1 500 Engineering Mechanics The average acceleration is defined as the ratio of average velocity to the time internal Δ t and expressed as a v t avg avg = ∆ The instantaneous acceleration is defined as the ratio of instantaneous velocity to the time interval Δ t and expressed as lim . t inst velocity v t dv dt → 0 ∆ or 10.3 Rectilinear Motion During motion, a particle can have three types of motion i.e., translational, rotational or general plane motion. In this chapter, we will study translational motion. Rotational motion and general plane motion are discussed in chapter 15. In translational motion, the particle travels along a straight line path without rotation; it is also known as rectilinear motion. In this type of motion, all particles of a body travel along a straight line only i.e., bears only one dimensional motion. The rectilinear motion may lie along X-axis or Y-axis. The examples of rectilinear motion along X-axis are motion of vehicles on a straight track (road without speed breaker), motion of piston in a horizontal cylinder. However, the motion of lift in a building and falling of a stone vertically are examples of rectilinear motion along Y-axis. 10.4 Rectilinear Motion in Horizontal Direction (X-axis) A particle moving in a straight line can have either uniform or variable acceleration. Generally the particles consist of variable acceleration in actual practice. The motion of a body under variable acceleration can be analyzed by using differential and integral equations of motion. The motion under uniform acceleration is analyzed by using equations of motion as discussed in next section (10.4.2). 10.4.1 Motion with variable acceleration Differential equations of motion: These equations are used to analyze the motion of a body having variable acceleration. The equations for four parameters of motion i.e., displacement, time, velocity and acceleration are given by v dx dt a dv dt d x dt v dv dx = = or or 2 2

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  Applied Mechanics

  Made Simple

  • George E. Drabble(Author)
  • 2013(Publication Date)
  • Made Simple

    (Publisher)

  CHAPTER TWO THE NATURE OF MOTION Without concerning ourselves about the reasons why motion takes place (these we shall deal with in Chapter Four), let us consider a familiar case of a moving object: an aircraft crossing the Atlantic. The first consideration is where it is going to and where it has come from. In terms of applied mechanics, we call this the displacement of the aircraft. The second consideration (a very important one) is how long it takes to perform the journey. This is deter-mined by the average rate of travel along the route chosen, and we call this the average velocity. Thirdly, and finally, we know that, although it is con-venient for our schedules to speak of an average speed of, say 550 miles an hour, under practical conditions the actual speed is bound to vary from this. To take the two most obvious divergencies, the plane cannot start at 550 miles an hour, nor can it finish at this speed. The velocity, then, has to change from time to time, and the rate at which it does this is termed the acceleration. Let us look at each of these aspects of motion in turn. (1) Displacement Displacement seems to be a very simple concept. But if we are going to deal mathematically with it, which is exactly what the study of applied mechanics purports to do with physical situations, we have to be very careful to make sure that the rules of mathematics work when applied to each and every situation. Let us try to apply a simple mathematical rule of addition to displacement. We know that, mathematically, 2 added to 3 makes 5. This is easy; but implicit in this simple statement are quite a few important conditions. First, the two things added must be of the same kind. Two years added to three months does not add up to five of anything. Secondly, applying the rule to displacement, it seems obvious at first that two feet added to three feet makes five feet; and indeed this is true if, for instance, we are measuring out material or merely measuring distance travelled.

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  Classical Mechanics

  • Hiqmet Kamberaj(Author)
  • 2021(Publication Date)
  • De Gruyter

    (Publisher)

  4  Two- and three-dimensional motion

  In this chapter, we will discuss the kinematics of a particle moving in two and three dimensions. Utilizing two- and three-dimensional motion, we will be able to examine a variety of movements, starting with the motion of satellites in orbit to the flow of electrons in a uniform electric field. We will begin studying in more detail the vector nature of displacement, velocity, and acceleration. Similar to one-dimensional motion, we will also derive the kinematic equations for three-dimensional motion from these three quantities’ fundamental definitions. Then the projectile motion and uniform circular motion will be described in detail as particular cases of the movements in two dimensions.

  4.1  The displacement, velocity, and acceleration vectors

  When we discussed the one-dimensional motion (see Chapter 3 ), we mentioned that the movement of an object along a straight line is thoroughly described in terms of its position as a function of time,

  x ( t )

  . For the two-dimensional motion, we will extend this idea to the movement in the

  x y

  plane.

  As a start, we describe a particle’s position by the position vector r pointing from the origin of some coordinate system to the particle located in the

  x y

  plane, as shown in Fig. 4.1 . At time

  t i

  the particle is at point P, and at some later time

  t f

  it is at the position Q. The path from P to Q generally is not a straight line. As the particle moves from P to Q in the time interval

  Δ t =

  t f

  −

  t i

  , its position vector changes from

  r i

  to

  r f

  .

  Definition 4.1 (Displacement vector).

  The displacement is a vector, and the displacement of the particle is the difference between its final position and its initial position. We now formally define the displacement vector for the particle as the difference between its final position vector and its initial position vector:

  (4.1)

  Δ r =

  r f

  −

  r i

  .

  The direction of

  Δ r

  is indicated in Fig. 4.1 from P to Q. Note that the magnitude of

  Δ r

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  Reeds Vol 2: Applied Mechanics for Marine Engineers

  • Paul Anthony Russell(Author)
  • 2021(Publication Date)
  • Reeds

    (Publisher)

  The study of kinematics concentrates on describing motion in words, numbers, diagrams, graphs, and equations. These help the engineer develop cognitive understanding about the way objects behave in the material world. The abstract realism will not be divorced from the object and forces involved; although these are Kinematics • 51 not part of the discipline, some reference to force and objects does help in shaping the engineer’s thought processes. Case A represents a body that was moving at 5 m/s due east, having its velocity changed to 12 m/s due east; the vector of each velocity is drawn from a common point; the difference between the free ends of the vectors is the change of velocity – in this case it is 7 m/s. Case B is a body with an initial velocity of 9 m/s due east, being changed to 2 m/s due west; the vector diagram shows the vector of each velocity drawn from a common point; the difference between their free ends is the change of velocity, which is 11 m/s. Case C is that of a body with an initial velocity of 6 m/s due east changed to 8 m/s due south. The vector diagram is constructed on the same principle of the two vectors drawn from a common point. The change of velocity is, as always, the difference between the free ends of the two vectors, this is, 8 6 10 2 2 + = m/s. The direction for change of velocity is S 36° 52’ W due to change in velocity taking place in the direction of the applied force, which in this case is east to south-west. In all cases, the vector diagrams are constructed by drawing the velocity vectors from a common point. This technique is called vector subtraction. Space diagrams Vector diagrams A 5 m/s 9 m/s 2 m/s 6 m/s 8 m/s 12 m/s B C N S 5 7 12 W E 9 6 8 Change of velocity 11 2 ▲ Figure 2.10 Space and vector diagrams for a change in velocity 52 • Applied Mechanics Acceleration is the rate of change of velocity; therefore, in all of these cases the value of acceleration can be obtained by dividing change of velocity by time.

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