Motion in Space

4 Key excerpts on "Motion in Space"

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  eBook - ePub

  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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  eBook - PDF

  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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  eBook - ePub

  Particle Mechanics

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

    (Publisher)

  4

  Motion in Space

  Many of the ideas discussed in Chapter 2 are generalized to the motion of a particle moving in space. A vectorial approach is used throughout, coordinates, when used, being adapted to the problem being discussed. The importance of the choice of frame of reference used to describe the motion of a given particle is stressed and the role played by Newton’s first law of motion in the choice of the frame of reference is discussed. Three different types of conservation laws are derived. These are important in that they provide first integrals of the equation of motion which can be written down without the need for carrying out actual integrations. Newton’s third law of motion is stated in both a weak and strong form. Much of this Chapter is concerned with development of the theory and the opportunity for tackling exercises is therefore somewhat restricted.

  4.1 Kinematics of a Particle Moving in Space

  Consider a particle moving in space. The definitions and results which we are about to discuss are generalizations of those discussed in Section 2.1 for a particle P moving on a straight line. In Section 2.1 the position of P was specified, relative to a given origin O lying on the line of motion, by a directed number x , the displacement of P relative to O. You will have noticed the fact that if the line of motion is chosen as the x-axis of a cartesian coordinate system with origin O , then the displacement of P relative to O is just the × coordinate of P. An obvious generalization for a particle P moving in space would be to specify the position of P relative to a given origin by the cartesian coordinates (x, y, z) of P relative to a given set of axes passing through O. In order to simplify later calculations we confine attention here to right handed rectangular cartesian coordinates for which the axes are mutually perpendicular and orientated with O z in the direction of the thumb of the right hand when the curled fingers of that hand are scooped from O x to O y, as in Fig 4.1 . The axes are said to form a frame of reference or frame

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  General Physics

  • Nelson Bolívar(Author)
  • 2020(Publication Date)
  • Arcler Press

    (Publisher)

  General Physics 72 3.10. VELOCITY Let’s understand the motion of simplest body, a particle, or material . Let us consider the circumstance when its dimensions are quite small compared to the distances from the other objects. This is obviously an idealization but still it works frequently in practice (Alder and Wainwright, 1970; Bussi et al., 2007). For instance, the planets are indeed not point-like, but in a mathematical description the motions of planets around the sun can be well-thought-out as such in good estimate, as long as one doesn’t take into account rotations about their axes, or changes of the directions of axes, or tides on the surfaces (Da Rold and Pomarol, 2006; Spurr, 2008). The ship can be thought as a point when it is very far from the shore, but when ship enters the harbor its dimension should be precisely known (Tanner and Whitehouse, 1976; Loarie et al., 2009). As already defined, the motion is to be examined in the given reference frame. The particle defines in its motion the curve, which is known as the trajectory , as displayed in Figure 3.14a. The position vector is the function of time r ( t ), in other words, co-ordinates are the 3 functions of time x ( t ), y ( t ), z ( t ). If these functions are known, the motion of particle is completely known. It can be said that the system possesses 3° of freedom. Figure 3.14. (a) The trajectory of the particle, (b) velocity. Source: http://www.softouch.on.ca/kb/data/Course%20in%20Classical%20 Physics%201--Mechanics%20(A). Let’s consider position vector at instant of the time t , r ( t ) as signified in Figure 3.14a and an instantaneously following instant t + Δt , r ( t + Δt ), Δt is the short time interval. In this short time interval the particle has progressed by Δ s , which is the step in space having the magnitude and direction, it is the vector (Alder and Wainwright, 1967; Perrine and Edgerton, 1978).

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