Chapter%203 - PHYSICS, Part I Chapter 3: Kinematics in Two...

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PHYSICS, Part I Chapter 3: Kinematics in Two Dimensions Displacement, Velocity and Acceleration Equations of Kinematics in Two Dimensions Projectile Motion Relative Velocity in Two Dimensions
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Velocity in Two Dimensions (or in a Plane) Let’s consider different kinds of space. x y x y z 1 -dimensional space 2 -dimensional space 3 -dimensional space (y-axis is perpendicular to the plane of the picture) In addition, physicists are usually working with 4 -dimensional space-time.
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The object’s position in 1-, 2-, and 3-dimensional spaces can be determined by one, two, and three coordinates, respectively. The motion can correspondingly be determined by change in these coordinates. We have already considered position, velocity, and acceleration in 1- dimensioanl space, i.e. along a straight line. Here we do the same in a two-dimensional space. For motion in a plane, the horizontal distance from the origin is the x-coordinate , and the vertical distance from the origin is the y-coordinate . The vector r from the origin to the point P (the object is considered as a particle ) is called the position vector . The x and y axes together make a Cartesian, or an orthogonal, or rectangular coordinate system. These two coordinates are the x and y components of the position vector.
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The distance of point P from the origin is the magnitude of the vector r : r r x y = = + 2 2 The displacement from point P 1 to point P 2 is: r r r = - 2 1 The average velocity is: v r r t t r t av = - - = 2 1 2 1 The displacement and the average velocity do not depend on the actual path. Because x and y are the components of r , the components of the velocity are: v x t av x , = v y t av y , = and (3.1) (3.2) (3.3) 2 2 2 2 2 2 , , 2 2 av av x av y x y x y v v v t t t + ∆ = = + = + (3.4) tan y x θ = From the above Figure and Eqs. 3.3 and 3.4 we have for the magnitude and direction of v av : and (3.5)
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The instantaneous velocity is the limit of the average velocity as t approaches zero. 0 lim t r v t ∆ → = At every point along the path, the instantaneous velocity is tangent to the path. 0 lim x t x v t ∆ → = 0 lim y t y v t ∆ → = The x- and y-components of the instantaneous velocity are: and The instantaneous speed is the magnitude v of the instantaneous velocity 2 2 x y v v v v = = + tan y x v v θ= The direction of v is given by: (3.6) (3.8) (3.7) (3.9)
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This note was uploaded on 09/21/2011 for the course PHY 2053 taught by Professor Vellaris during the Spring '08 term at University of Central Florida.

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Chapter%203 - PHYSICS, Part I Chapter 3: Kinematics in Two...

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