Motion In Two Dimensions - Universidad De Sonora

P U Z Z L E RThis airplane is used by NASA for astronaut training. When it flies along a certain curved path, anything inside theplane that is not strapped down begins tofloat. What causes this strange effect?(NASA)webFor more information on microgravity ingeneral and on this airplane, visithttp://microgravity.msfc.nasa.gov/and http://www.jsc.nasa.gov/coop/kc135/kc135.htmlc h a p t e rMotion in Two DimensionsChapter Outline4.1 The Displacement, Velocity, andAcceleration Vectors4.2 Two-Dimensional Motion withConstant Acceleration4.3 Projectile Motion764.4 Uniform Circular Motion4.5 Tangential and Radial Acceleration4.6 Relative Velocity and RelativeAcceleration

4.177The Displacement, Velocity, and Acceleration VectorsIn this chapter we deal with the kinematics of a particle moving in two dimensions. Knowing 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 bystudying in greater detail the vector nature of displacement, velocity, and acceleration. As in the case of one-dimensional motion, we derive the kinematic equationsfor two-dimensional motion from the fundamental definitions of these three quantities. We then treat projectile motion and uniform circular motion as special casesof motion in two dimensions. We also discuss the concept of relative motion,which shows why observers in different frames of reference may measure differentdisplacements, velocities, and accelerations for a given particle.y4.1THE DISPLACEMENT, VELOCITY, ANDACCELERATION VECTORS r훽 tiIn Chapter 2 we found that the motion of a particle moving along a straight line iscompletely known if its position is known as a function of time. Now let us extendthis 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 theparticle located in the xy plane, as in Figure 4.1. At time ti the particle is at point훽, and at some later time tf it is at point 훾. The path from 훽 to 훾 is not necessarily a straight line. As the particle moves from 훽 to 훾 in the time interval t t f t i , its position vector changes from ri to rf . 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 formally define the displacement vector r for the particle of Figure 4.1 as being the difference between its final position vector and its initial position vector: r rf ri(4.1)rfOFigure 4.1Displacement vectorWe define the average velocity of a particle during the time interval t as thedisplacement of the particle divided by that time interval: r t(4.2)Multiplying or dividing a vector quantity by a scalar quantity changes only the magnitude of the vector, not its direction. Because displacement is a vector quantityand the time interval is a scalar quantity, we conclude that the average velocity is avector quantity directed along r.Note that the average velocity between points is independent of the path taken.This is because average velocity is proportional to displacement, which dependsPath ofparticlexA particle moving inthe xy plane is located with the position vector r drawn from the origin to the particle. The displacement of the particle as it movesfrom 훽 to 훾 in the time interval t t f ti is equal to the vector r rf ri .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 bythe particle.As we saw in Chapter 2, it is often useful to quantify motion by looking at theratio of a displacement divided by the time interval during which that displacement occurred. In two-dimensional (or three-dimensional) kinematics, everyt

4.1 The Displacement, 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 Chapter Outline This airplane is used by NASA for astro-naut training. When it flies along a cer-