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Unformatted text preview: CHAPTER 3 Motion in Two and Three Dimensions 1* Can the magnitude of the displacement of a particle be less than the distance traveled by the particle along its path? Can its magnitude be more than the distance traveled? Explain. The magnitude of the displacement can be less but never more than the distance traveled. If the path is a semi circle of radius R , the distance traveled is 2 p R , the magnitude of the displacement is 2 R . The maximum magni tude of the displacement occurs when the path is a straight line; then the two quantities are equal. 2 Give an example in which the distance traveled is a significant amount yet the corresponding displacement is zero. The path is a complete circle; Displacement = 0; distance traveled = 2 p R . 3 The magnitude of the displacement of a particle is ___ the distance the object has traveled. ( a ) larger than ( b ) smaller than ( c ) either larger or smaller ( d ) the same as ( e ) smaller than or equal to ( e ) See Problem 1. 4 A bear walks northeast for 12 m and then east for 12 m. Show each displacement graphically, and find the resultant displacement vector. The two vector displacements are shown. The total displacement is 22.2 m at an angle of 22.5 o with respect to the horizontal. 5* ( a ) A man walks along a circular arc from the position x = 5 m, y = 0 to a final position x = 0, y = 5 m. What is his displacement? ( b ) A second man walks from the same initial position along the x axis to the origin and then along the y axis to y = 5 m and x = 0. What is his displacement? ( a ), ( b ) Since the initial and final positions are the same, the displacements are equal. D = (5 i + 5 j ) m. 6 A circle of radius 8 m has its center on the y axis at y = 8 m. You start at the origin and walk along the circle at a steady speed, returning to the origin exactly 1 min after you started. ( a ) Find the magnitude and direction of your displacement from the origin 15, 30, 45, and 60 s after you start. ( b ) Find the magnitude and direction of your Chapter 3 Motion in Two and Three Dimensions displacement for each of the four successive 15s intervals of your walk. ( c ) How is your displacement for the first 15 s related to that for the second 15 s? ( d ) How is your displacement for the second 15s interval related to that for the last 15s interval? The adjacent figure shows the path around the circle. Since the radius of the circle is 8 m, the length of each vector is (8 2 ) m = 11.3 m. ( a ) 1. Find the coordinates for each point 2. Find r ( t 0) and q ( t 0) ( b ) The displacement magnitudes are equal; the angles rotate by 90 o ( c ) See the figure ( d ) See the figure r (0) = r (60) = 0; r (15) = 8 i + 8 j ; r (30) = 8 j ; r ( 45) = 8 i + 8 j r (15  0) = 11.3 m, q = 45 o ; r (30  0) = 8 m, q = 180 o ; r (45  0) = 11.3 m, q = 135 o ; r (60  0 ) = 0 displacement magnitudes = 11.3 m q 1 = 45 o ; q 2 = 135 o ; q 3 = 225 o ; q 4 = 315 o The second is the first rotated 90 o The two are oppositely directed. The two are oppositely directed....
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 Spring '08
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 Acceleration, Velocity, m/s, θ

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