4 - Motion in Two Dimensions - 77 Motion in Two Dimensions...

4 - Motion in Two Dimensions
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77 Motion in Two Dimensions CHAPTER OUTLINE 4.1 The Position, Velocity, and Acceleration Vectors 4.2 Two-Dimensional Motion with Constant Acceleration 4.3 Projectile Motion 4.4 Uniform Circular Motion 4.5 Tangential and Radial Acceleration 4.6 Relative Velocity and Relative Acceleration ± Lava spews from a volcanic eruption. Notice the parabolic paths of embers projected into the air. We will find in this chapter that all projectiles follow a parabolic path in the absence of air resistance. (© Arndt/Premium Stock/PictureQuest) Chapter 4
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I n this chapter we explore the kinematics of a particle moving in two dimensions. Know- ing the basics of two-dimensional motion will allow us to examine—in future chapters—a wide variety of motions, ranging from the motion of satellites in orbit to the motion of electrons in a uniform electric field. We begin by studying in greater detail the vector nature of position, velocity, and acceleration. As in the case of one-dimensional motion, we derive the kinematic equations for two-dimensional motion from the fundamental definitions of these three quantities. We then treat projectile motion and uniform circular motion as special cases of motion in two dimensions. We also discuss the concept of relative motion, which shows why observers in different frames of reference may measure different positions, velocities, and accelerations for a given particle. 4.1 The Position, Velocity, and Acceleration Vectors In Chapter 2 we found that the motion of a particle moving along a straight line is completely known if its position is known as a function of time. Now let us extend this idea to motion in the xy plane. We begin by describing the position of a particle by its position vector r , drawn from the origin of some coordinate system to the particle lo- cated in the xy plane, as in Figure 4.1. At time t i the particle is at point ± , described by position vector r i . At some later time t f it is at point ² , described by position vector r f . The path from ± to ² is not necessarily a straight line. As the particle moves from ± to ² in the time interval ± t ² t f ³ t i , its position vector changes from r i to r f . As we learned in Chapter 2, displacement is a vector, and the displacement of the particle is the difference between its final position and its initial position. We now define the dis- placement vector ± r for the particle of Figure 4.1 as being the difference between its final position vector and its initial position vector: (4.1) The direction of ± r is indicated in Figure 4.1. As we see from the figure, the magnitude of ± r is less than the distance traveled along the curved path followed by the particle. As we saw in Chapter 2, it is often useful to quantify motion by looking at the ratio of a displacement divided by the time interval during which that displacement occurs, which gives the rate of change of position. In two-dimensional (or three-dimensional) kinematics, everything is the same as in one-dimensional kinematics except that we
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