Relative Motion in 2 Dimensions

8 Key excerpts on "Relative Motion in 2 Dimensions"

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  eBook - PDF

  Cutnell & Johnson Physics, P-eBK

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  CHAPTER 3 Kinematics in two dimensions LEARNING OBJECTIVES After reading this module, you should be able to: 3.1 define two‐dimensional kinematic variables 3.2 use two‐dimensional kinematic equations to predict future or past values of variables 3.3 analyse projectile motion to predict future or past values of variables 3.4 apply relative velocity equations. INTRODUCTION Moving the ball forward and scoring is what rugby league and union are all about, but passing the ball forward is illegal. Disgruntled fans sometimes think a pass is forward but experienced referees know to look at the direction of the ball relative to the passing player and not the ground. In this chapter we take a look at the properties of motion in two dimensions, including the relative motion of a football pass. 1 3.1 Displacement, velocity, and acceleration LEARNING OBJECTIVE 3.1 Define two‐dimensional kinematic variables. FIGURE 3.1 The displacement Δ r of the car is a vector that points from the initial position of the car at time t 0 to the final position at time t. The magnitude of Δ r is the shortest distance between the two positions. +y +x r 0 r Δr t 0 t In chapter 2 the concepts of displacement, velocity, and accelera- tion are used to describe an object moving in one dimension. There are also situations in which the motion is along a curved path that lies in a plane. Such two‐dimensional motion can be described using the same concepts. In Grand Prix racing, for example, the course follows a curved road, and figure 3.1 shows a race car at two different positions along it. These positions are identified by the vectors  r and  r 0 , which are drawn from an arbitrary coordinate origin. The displacement Δ r of the car is the vector drawn from the initial position  r 0 at time t 0 to the final position  r at time t. The magnitude of Δ r is the shortest distance between the two positions.

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  Fundamentals of Physics, Extended

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

    (Publisher)

  Thus, the accel- eration relative to Alex is Observers on different frames of reference that move at constant velocity rela- tive to each other will measure the same acceleration for a moving particle. 86 CHAPTER 4 MOTION IN TWO AND THREE DIMENSIONS 4.7 RELATIVE MOTION IN TWO DIMENSIONS Learning Objective After reading this module, you should be able to . . . 4.7.1 Apply the relationship between a particle’s posi- tion, velocity, and acceleration as measured from two reference frames that move relative to each other at constant velocity and in two dimensions. Key Ideas ● When two frames of reference A and B are moving relative to each other at constant velocity, the velocity of a particle P as measured by an observer in frame A usually differs from that measured from frame B. The two measured velocities are related by v → PA = v → PB + v → BA , where v → BA is the velocity of B with respect to A. Both observers measure the same acceleration for the particle: a → PA = a → PB . Relative Motion in Two Dimensions Our two observers are again watching a moving particle P from the origins of refer- ence frames A and B, while B moves at a constant velocity v → BA relative to A. (The corresponding axes of these two frames remain parallel.) Figure 4.7.1 shows a cer- tain instant during the motion. At that instant, the position vector of the origin of B relative to the origin of A is r → BA . Also, the position vectors of particle P are r → PA relative to the origin of A and r → PB relative to the origin of B. From the arrangement of heads and tails of those three position vectors, we can relate the vectors with r → PA = r → PB + r → BA . (4.7.1) By taking the time derivative of this equation, we can relate the velocities v → PA and v → PB of particle P relative to our observers: v → PA = v → PB + v → BA . (4.7.2) By taking the time derivative of this relation, we can relate the accelera- tions a → PA and a → PB of the particle P relative to our observers.

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  Introduction to Physics

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

    (Publisher)

  3.2 Equations of Kinematics in Two Dimensions Motion in two dimensions can be described in terms of the time t and the x and y components of four vectors: the displacement, the acceleration, and the initial and final velocities. The x part of the motion occurs exactly as it would if the y part did not occur at all. Similarly, the y part of the motion occurs exactly as it would if the x part of the motion did not exist. The motion can be analyzed by treating the x and y components of the four vectors separately and realizing that the time t is the same for each component. v B 5 r B 2 r B 0 t 2 t 0 5 D r B Dt (3.1) v B 5 lim Dt B0 D r B D t (1) a B 5 v B 2 v B 0 t 2 t 0 5 Dv B D t (3.2) B a 5 lim Dt B0 D v B Dt (2) Concept Summary 65 66 Chapter 3 | Kinematics in Two Dimensions When the acceleration is constant, the x components of the displacement, the acceleration, and the initial and final velocities are related by the equations of kinematics, and so are the y components: x Component v x 5 v 0x 1 a x t (3.3a) x 5 1 2 (v 0x 1 v x )t (3.4a) x 5 v 0x t 1 1 2 a x t 2 (3.5a) v x 2 5 v 0x 2 1 2a x x (3.6a) y Component v y 5 v 0y 1 a y t (3.3b) y 5 1 2 (v 0y 1 v y )t (3.4b) y 5 v 0y t 1 1 2 a y t 2 (3.5b) v y 2 5 v 0y 2 1 2a y y (3.6b) The directions of the components of the displacement, the acceleration, and the initial and final velocities are conveyed by assigning a plus (1) or minus (2) sign to each one. 3.3 Projectile Motion Projectile motion is an idealized kind of motion that occurs when a moving object (the projectile) experiences only the acceleration due to gravity, which acts vertically down- ward. If the trajectory of the projectile is near the earth’s surface, the vertical component a y of the acceleration has a magnitude of 9.80 m/s 2 . The acceleration has no horizontal component (a x 5 0 m/s 2 ), the effects of air resistance being negligible.

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  University Physics Volume 1

  • 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.

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  68 CHAPTER 3 Kinematics in Two Dimensions 3.4 Relative Velocity To someone hitchhiking along a highway, two cars speeding by in adjacent lanes seem like a blur. But if the cars have the same velocity, each driver sees the other remaining in place, one lane away. The hitchhiker observes a velocity of perhaps 30 m/s, but each driver observes the other’s velocity to be zero. Clearly, the velocity of an object is relative to the observer who is making the measurement. Figure 3.13 illustrates the concept of relative velocity by showing a passenger walking toward the front of a moving train. The people sitting on the train see the passenger walking with a velocity of +2.0 m/s, where the plus sign denotes a direction to the right. Suppose the train is moving with a velocity of +9.0 m/s relative to an observer standing on the ground. Then the ground-based observer would see the passenger moving with a velocity of +11 m/s, due in part to the walking motion and in part to the train’s motion. As an aid in describing relative velocity, let us define the following symbols: v → PT = velocity of the Passenger relative to the Train = +2.0 m/s v → TG = velocity of the Train relative to the Ground = +9.0 m/s v → PG = velocity of the Passenger relative to the Ground = +11 m/s In terms of these symbols, the situation in Figure 3.13 can be summarized as follows: v → PG = v → PT + v → TG (3.7) or v → PG = (2.0 m/s) + (9.0 m/s) = +11 m/s According to Equation 3.7*, v → PG is the vector sum of v → PT and v → TG , and this sum is shown in the drawing. Had the passenger been walking toward the rear of the train, rather than toward the front, the velocity relative to the ground-based observer would have been v → PG = (−2.0 m/s) + (9.0 m/s) = +7.0 m/s. Each velocity symbol in Equation 3.7 contains a two-letter subscript. The first letter in the subscript refers to the body that is moving, while the second letter indicates the object relative to which the velocity is measured.

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  General Physics Mechanics Thermodynamics

  • Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
  • 2022(Publication Date)
  • Società Editrice Esculapio

    (Publisher)

  Relative Motion 5.1 Introduction The choice of the reference system used to describe motion is arbitrary and the form of the equations determined using Newton’s model may depend on this choice. It is therefore sensible to wonder how to compare the description of a body motion according to observers choosing different reference systems. Any two observers placed on the origin of their reference system appear in general to move one with respect to the other. Some kind of reciprocity must exist between the mo- tions of the two observers, that is, if both observers analyse the motion of the same point- like particle, the kind of motions the two find must be related to each other by precise physical laws. The laws we find for a general case, after a short introductory argumentation about the simplest case, are transformation formulae between any two reference systems of the kine- matic quantities: coordinates, velocity and acceleration. 5.2 Relative Translational Motion Consider two observers O and ′ O who choose two different reference systems: observer O chooses a fixed, i.e. time independent, refer- ence system Oxyz, while observer ′ O choos- es a reference system ′ O ′ x ′ y ′ z moving with respect to Oxyz with velocity  v ′ O = v ′ O  u x and acceleration  a ′ O =  a ′ O  u x . The coordinate axes ′ x , ′ y and ′ z are selected parallel respectively to x, y and z and unit vec- tors  u ′ x ,  u ′ y and  u ′ z have invariant di- rection while time goes by. The ′ O ′ x ′ y ′ z reference system therefore translates rigidly along the x axis with respect to the Oxyz reference system. A point-like particle placed in a point P of the space is identified by position vector  r = OP    in the Oxyz reference system and by vector  ′ r = ′ O P    in the ′ O ′ x ′ y z reference sys- tem. Furthermore the origin of the ′ O ′ x ′ y ′ z reference system in the Oxyz reference system is identified by vector  r ′ O = O ′ O    .

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  Introductory Physics

  Summaries, Examples, and Practice Problems

  • Michael Antosh(Author)
  • 2023(Publication Date)
  • CRC Press

    (Publisher)

  Motion in Two and Three Dimensions

  DOI: 10.1201/9781003005049-3

  3.1 Introduction: Three Dimensions Are More Realistic Than One

  Chapter 2 was all about motion in one dimension, which means that all of the motion was along a straight line. Sometimes this works just fine, like a straight road. But in general, we need more than one dimension to accurately describe motion. In this chapter, we’ll use similar strategies to Chapter 2 for situations where the motion is not in a straight line.

  3.2 Dimensions Behave Separately

  From where you are, right now, you can describe the space around you using a combination of three directions:

  • Up/down
  • Left/right
  • Forward/backward

  These directions are what we call dimensions. Left/right is often called “the x direction” in physics. Up/down is often called “the y direction”. Forward/backward is often called “the z direction”. Which directions are called x, y, and z can be different, and they don’t have to be exactly these definitions.

  When we look at motion in three dimensions for this course, the motion actually behaves like three separate motion problems, with each one similar to the problems in Chapter 2 : a problem in the x direction, a problem in the y direction, and a problem in the z direction. This may not seem obvious at first.

  A classic physics class demonstration shows the dimensions behaving separately during motion. The demonstration starts with a contraption holding two identical balls at exactly the same height. When the teacher presses a button, both of the balls start moving at exactly the same time. One of the balls is dropped straight downward (call it the –y direction), while the second ball goes sideways (call it the +x direction). There are two results of this motion:

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  Physics

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

    (Publisher)

  3.3 Projectile Motion Projectile motion is an idealized kind of motion that occurs when a moving object (the projectile) experiences only the acceleration due to gravity, which acts vertically down- ward. If the trajectory of the projectile is near the earth’s surface, the vertical component a y of the acceleration has a magnitude of 9.80 m/s 2 . The acceleration has no horizontal component (a x 5 0 m/s 2 ), the effects of air resistance being negligible. There are several symmetries in projectile motion: (1) The time to reach maximum height from any vertical level is equal to the time spent returning from the maximum height to that level. (2) The speed of a projectile depends only on its height above its launch point, and not on whether it is moving upward or downward. 3.4 Relative Velocity The velocity of object A relative to object B is v B AB , and the velocity of object B relative to object C is v B BC . The velocity of A relative to C is shown in Equation 3 (note the ordering of the subscripts). While the velocity of object A relative to object B is v B AB , the velocity of B relative to A is v B AB 5 2 v B AB . v B 5 r B 2 r B 0 t 2 t 0 5 D r B Dt (3.1) v B 5 lim Dt B0 D r B D t (1) a B 5 v B 2 v B 0 t 2 t 0 5 Dv B D t (3.2) B a 5 lim Dt B0 D v B Dt (2) v B AC 5 v B AB 1 v B BC (3) x Component v x 5 v 0x 1 a x t (3.3a) x 5 1 2 (v 0x 1 v x )t (3.4a) x 5 v 0x t 1 1 2 a x t 2 (3.5a) v x 2 5 v 0x 2 1 2a x x (3.6a) y Component v y 5 v 0y 1 a y t (3.3b) y 5 1 2 (v 0y 1 v y )t (3.4b) y 5 v 0y t 1 1 2 a y t 2 (3.5b) v y 2 5 v 0y 2 1 2a y y (3.6b) FOCUS ON CONCEPTS Note to Instructors: The numbering of the questions shown here reflects the fact that they are only a representative subset of the total number that are available online. However, all of the questions are available for assignment via an online homework management program such as WileyPLUS or WebAssign.

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