Motion in One Dimension

9 Key excerpts on "Motion in One Dimension"

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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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  College Physics

  • Michael Tammaro(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  Here in Chapter 2, we begin the study of mechanics—the study of how objects move in response to forces. Before we consider forces, though, we will learn to simply describe motion, which is the subject of kinematics. Chapter 2 deals exclusively with one- dimensional kinematics, which is most simply defined as motion along a straight line. Pictured here is the Drop Tower thrill-ride found at Kings Dominion amusement park in Richmond, Virginia. It is the largest of its kind, with an overall height of 93 m, a drop distance of 83 m, and a maximum speed of 116 km/h. It dramatically illustrates motion along a straight line. Joel Bullock (TheCoasterCritic.com) 2 Kinematics in One Dimension 42 Position and Displacement: Vectors in One Dimension | 43 2.1 Calculate the distance and displacement when an object moves from place to place. 2.1.1 Match a vector sum to a diagram representing the graphical sum of those vectors. The world around us moves. Some motion is slow and steady, like the motion of the pas- sengers on a slow-moving escalator. Some motion is fast and unsteady, like the motion of a zigzagging gazelle being chased by a cheetah. Describing motion is the subject of kinematics, which brings together the ideas of distance and time to form the various quan- tities that are required for a complete description of motion. Here in Chapter 2 we focus on one-dimensional motion—the simplest case. Understanding motion is a prerequisite for many topics in this text, such as the conservation of energy, thermodynamics, and special relativity. Position When specifying the position of an object, there is some uncertainty based on the object’s finite size. An automobile, for example, is about 5 m long, 2 m wide, and 2 m high, and it can assume a variety of orientations—facing north, south, and so on. Specifying a single point in space to be the “location” of the car introduces some uncertainties.

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  Available until 16 Feb |Learn more

  Classical Mechanics

  A Computational Approach with Examples Using Mathematica and Python

  • Christopher W. Kulp, Vasilis Pagonis(Authors)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  CHAPTER 2

  Single-Particle Motion in One Dimension

  In this chapter, we will examine one-dimensional motion, i.e., motion along a line. It is sometimes the case that a particle’s motion need only to be described along one direction. Furthermore, a careful study of one-dimensional motion will be a useful foundation for understanding more general motion in higher dimensions. In this chapter, we will give several examples of solving Newton’s second law, F = ma in one dimension. We will consider several types of forces: both constant and those which depend on time F(t), velocity F(v), and position F(x). In addition, we will discuss and demonstrate two different uses of computers to solve physics problems: how to use computer algebra systems (CAS) to obtain the analytical solutions of Newton’s second law, and how to obtain numerical solutions of ordinary differential equations (ODE) using software packages and by using the Euler method.

  2.1Equations of motion

  To begin our study of one-dimesional motion, we first need to make some assumptions about the object whose motion we are examining. One fundamental assumption in this chapter is that the object being studied is a point particle. In order to mathematically describe the motion of a particle under the influence of a force, we need to find the particle’s equations of motion. The equations of motion of a particle are the equations which describe its position, velocity, and acceleration as a functions of time. Equations of motion can be in the form of algebraic equations, or in the form of differential equations.

  As we will see, the equations of motion of a particle can be found by solving Newton’s second law as a differential equation. In this chapter, we will focus on one-dimensional motion, where the force vector and the particle’s displacement are along the same line (but not necessarily in the same direction—the direction could be horizontal or vertical). Because all vectors in a given problem lay along the same line, we drop the vector notation in all the equations. A negative sign between two quantities will denote vectors that lay in opposite directions along the same line.

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  eBook - ePub

  A Modern Approach to Classical Mechanics

  • Harald Iro(Author)
  • 2015(Publication Date)
  • WSPC

    (Publisher)

  3

  One-dimensional motion of a particle

  The simplest system consists of a single particle whose motion only has one degree of freedom. In the following, we investigate the constants of the motion and the conserved quantities for Newton’s equation in one dimension. Since the representation in phase space here is two-dimensional, the particle’s motion in phase space can be visualized graphically.

  3.1  Examples of one-dimensional motion

  Figure 3.1: The inclined track.

  i) The inclined track

  : A particle of mass m slides without friction on a track inclined at angle α to the direction of the gravitational force F = mg. Since forces obey vector addition (see Page 17), the force of gravity can be split into components parallel and perpendicular to the track1 :

  where (see Fig. 3.1 )

  Motion along the track is influenced only by F|| ; the perpendicular component F is exactly counterbalanced by the track2 (see Fig. 3.1 ). Let s be the distance along the track (with respect to some initial point s0 = 0). The momentum of the particle is p = m . Hence, the equation of motion is

  Figure 3.2: The plane pendulum.

  ii) The plane mathematical pendulum

  : A particle of mass m is attached to the end of a massless bar of length l that can swing freely about a fixed point. The particle is pushed such that its motion always remains in the plane (i.e. the initial velocity vector lies in the plane containing the bar and gravitational force vector). Since lϕ is the arclength, where ϕ is the angle between the bar and the vertical, the equation of motion reads (see also Section 3.3.2 below):

  Figure 3.3: An oscillating mass.

  iii) The harmonic oscillator

  : A particle of mass m is confined to move along the x-axis. It is attached to a spring, with equilibrium position x

  equ

  . Assume that the spring obeys Hooke’s Law, |F| ∝ |x − x

  equ

  |, meaning that the force is harmonic, i.e. it is given by F = −k(x − x

  equ

  ). If x

  equ

  is chosen as the origin, i.e. x

  equ

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  eBook - PDF

  Workshop Physics Activity Guide Module 1

  Mechanics I

  • Priscilla W. Laws, David P. Jackson, Brett J. Pearson(Authors)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  In addition, we will focus on motion confined to a straight line, what we refer to as one-dimensional motion. For the activities in this unit, we will use an ultrasonic motion sensor interfaced with a computer that measures the distance to an object. The program displays how an object’s position changes over time in the form of a graph, and calculates other quantities such as velocity and acceleration. Our goal in this unit is to develop an understanding of the concepts of position, velocity, and acceleration, as well as their formal mathematical definitions. UNIT 3: INTRODUCTION TO ONE-DIMENSIONAL MOTION 59 DESCRIBING MOTION WITH WORDS AND GRAPHS 3.2 POSITION CHANGES The activities in this first section on kinematics will help us learn to describe changes in position using both words and graphs. Activities in the next section will involve descriptions of changes in the velocity of an object. For the activities in this section, you will need: • 1 ruler • 1 ultrasonic motion sensor with attached computer interface Motion Along a Line In Section 1.8, we introduced the position vector, which is a vector whose tail is located at the origin and whose head points to some location in space. In the case of the solar system, the Sun is the natural location of the origin, and each planet can be specified by a three-dimensional (or two-dimensional) position vector relative to this origin. Of course, the planets move through space as time changes, so each position vector is actually a function of time as well. In fact, position and time are the two most fundamental measurements in the study of motion, and we will spend much of the next few weeks discussing them. While a position vector is inherently three-dimensional, we can often work in a reduced number of dimensions. For example, in our solar system model, we assumed the planets all lied in a plane, in which case each planet could be speci- fied by a two-dimensional position vector.

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  eBook - ePub

  Doing Physics with Scientific Notebook

  A Problem Solving Approach

  • Joseph Gallant(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Chapter 2

  One-Dimensional Kinematics

  We start Doing Physics with SNB in the branch of physics known as Classical Mechanics. Our study of Classical Mechanics begins with one-dimensional kinematics, the description of motion in a straight line. This includes horizontal motion to the right or left and vertical motion straight up or down. This description will tell us where an object is, where it is going, and how much time it took to get there.

  Constant Acceleration

  Studying motion with a constant acceleration is a good place to start. We can describe this motion completely without using calculus. Let’s start with some definitions and important distinctions, and then we’ll solve some one-dimensional problems.

  Displacement and Position

  To describe an object’s motion, we need to establish a coordinate system so we can specify location. For 1-dimensional motion the coordinate system is just the x-axis with the origin at x = 0 and positive values to the right. The object’s position, specified by its x-coordinate, tells us how far from the origin it is and in which direction.

  The displacement is the object’s change in position.

  (2.1)

  This difference between the object’s final and initial position tells us how far from its original position it ends. A positive displacement means the object ends to the right of x0 and a negative displacement means the object ends to the left of x0 . For a round trip, the initial and final positions are equal and the displacement is zero. The SI unit for both position and displacement is the meter (m).

  Note The use of the upper case Greek letter delta (“Δ”) to mean “change in” is standard mathematical notation, but it has no special significance in SNB. In a calculation, SNB interprets the expression Δf as Δ x f, the product of two variables.

  Even though they have the same unit, displacement and distance are different. Suppose you start 2 meters from the origin, walk 9 meters to the right, turn around and walk 5 meters back toward the origin. Your initial position was x0 = 2 m and your final position is x = 6m, so your displacement is Δx

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  The pilots in the United States Navy’s Blue Angels can perform high-speed maneuvers in perfect unison. They do so by controlling the displace- ment, velocity, and acceleration of their jet aircraft. These three concepts and the relationships among them are the focus of this chapter. 2 | Kinematics in One Dimension Chapter | 2 LEARNING OBJECTIVES After reading this module, you should be able to... 2.1 | Define one-dimensional displacement. 2.2 | Discriminate between speed and velocity. 2.3 | Define one-dimensional acceleration. 2.4 | Use one-dimensional kinematic equations to predict future or past values of variables. 2.5 | Solve one-dimensional kinematic problems. 2.6 | Solve one-dimensional free-fall problems. 2.7 | Predict kinematic quantities using graphical analysis. 2.1 | Displacement There are two aspects to any motion. In a purely descriptive sense, there is the movement itself. Is it rapid or slow, for instance? Then, there is the issue of what causes the motion or what changes it, which requires that forces be considered. Kinematics deals with the concepts that are needed to describe motion, without any reference to forces. The present chapter discusses these concepts as they apply to Motion in One Dimension, and the next chapter treats two-dimensional motion. Dynamics deals with the effect that forces have on motion, a topic that is considered in Chapter 4. Together, kinematics and dynamics form the branch of physics known as mechanics. We turn now to the first of the kinematics con- cepts to be discussed, which is displacement. To describe the motion of an object, we must be able to specify the location of the object at all times, and Figure 2.1 shows how to do this for one-dimensional motion. In this drawing, the initial position of a car is indicated by the vector labeled x 0 B . The length of x 0 B is the distance of the car from an arbitrarily chosen origin. At a later time the car has moved to a new position, which is indicated by the vector x B .

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  Classical Mechanics

  • A. Douglas Davis(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  2 ONE-DIMENSIONAL MOTION Th e subjec t o f mechanic s ca n b e divide d int o kinematics, dynamics, an d statics. Kinematic s is a stud y o f th e concis e description o f a bod y in motion , wherea s dynamic s seek s t o explai n th e cause o f tha t motion . Static s seek s t o understan d th e detail s o f force s tha t hol d a bod y in equilibrium . W e shal l defe r ou r discussio n o f static s unti l later . Section s 2.1 , 2.2 , an d 2. 3 ma y wel l b e review . If the y are , loo k a t the m quickl y jus t t o refres h th e idea s an d becom e familia r wit h th e notatio n w e us e in thi s book . 2.1 Kinematic s in On e Dimensio n Th e locatio n o f a n objec t ca n ofte n b e full y specifie d b y a singl e number—th e distanc e fro m a n origi n t o tha t object . Fo r example , w e ca n locat e a trai n b y givin g it s distanc e fro m th e station . W e kno w it mus t b e o n th e track , s o th e additiona l informatio n o f it s distanc e fro m th e station , positiv e for on e directio n an d negativ e for th e other , full y locate s th e train . Similarly , a ca r movin g dow n a straigh t road , a bea d slidin g o n a wire , a n an t climbin g a piec e of string , a bucke t bein g lowere d int o a well , an d a mas s attache d t o a sprin g ca n al l b e locate d full y an d completel y b y a singl e number . CHAPTE R 2 / ONE-DIMENSIONA L MOTIO N Thi s specifi c location , x , is th e displacement fro m som e referenc e point , o r origin , t o th e object . An operational definition o f thi s distanc e involve s comparin g it wit h som e standard . W e migh t mov e a meterstic k ove r an d ove r to se e ho w man y time s it fits betwee n th e tw o positions . O r w e migh t pul l ou t a stee l tap e wit h distance s marke d o n it . O r w e migh t stan d a t a n origin , shin e a ligh t o n ou r object , an d se e ho w muc h tim e elapse s befor e th e reflecte d ligh t returns .

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  eBook - ePub

  The Britannica Guide to Heat, Force, and Motion

  • Britannica Educational Publishing, Erik Gregersen(Authors)
  • 2010(Publication Date)
  • Britannica Educational Publishing

    (Publisher)

  Resonances are not uncommon in the world of familiar experience. For example, cars often rattle at certain engine speeds, and windows sometimes rattle when an airplane flies by. Resonance is particularly important in music. For example, the sound box of a violin does its job well if it has a natural frequency of oscillation that responds resonantly to each musical note. Very strong resonances to certain notes—called “wolf notes” by musicians—occur in cheap violins and are much to be avoided. Sometimes, a glass may be broken by a singer as a result of its resonant response to a particular musical note.

  MOTION OF A PARTICLE IN TWO OR MORE DIMENSIONS

  More complex problems in mechanics involve a particle moving in two or more dimensions. Such problems include those of the pendulum and the circular orbit.

  PROJECTILE MOTION

  Galileo pointed out with some detectable pride that none before him had realized that the curved path followed by a missile or projectile is a parabola. He had arrived at his conclusion by realizing that a body undergoing ballistic motion executes, quite independently, the motion of a freely falling body in the vertical direction and inertial motion in the horizontal direction. These considerations, and terms such as ballistic and projectile, apply to a body that, once launched, is acted upon by no force other than Earth’s gravity.

  Projectile motion may be thought of as an example of motion in space—that is to say, of three-dimensional motion rather than motion along a line, or one-dimensional motion. In a suitably defined system of Cartesian coordinates, the position of the projectile at any instant may be specified by giving the values of its three coordinates, x (t ), y (t ), and z (t ). By generally accepted convention, z (t ) is used to describe the vertical direction. To a very good approximation, the motion is confined to a single vertical plane, so that for any single projectile it is possible to choose a coordinate system such that the motion is two-dimensional [say, x (t ) and z (t )] rather than three-dimensional [x (t ), y (t ), and z (t

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