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Unformatted text preview: S E C T I O N 14.5 Motion in ThreeSpace (ET Section 13.5) 543 We now compute T ( ) : T ( ) = r w ( ) w r w ( ) w = w , 1 , w w w , 1 , w w = w , 1 , w Finally we find N = B × T : N ( ) = 1 √ 5 w 2 , , 1 w × w , 1 , w = 1 √ 5 ( 2 i + k ) × j = 1 √ 5 ( 2 i × j + k × j ) = 1 √ 5 ( 2 k i ) = 1 √ 5 w  1 , , 2 w (b) Differentiating r ( t ) = ￥ t 2 , t 1 , t ￥ gives r w ( t ) = ￥ 2 t , t 2 , 1 ￥ r w w ( t ) = ￥ 2 , 2 t 3 , ￥ ⇒ r w ( 1 ) = w 2 , 1 , 1 w r w w ( 1 ) = w 2 , 2 , w We compute the cross product: r w ( 1 ) × r w w ( 1 ) = ￥ ￥ ￥ ￥ ￥ ￥ i j k 2 1 1 2 2 ￥ ￥ ￥ ￥ ￥ ￥ = ￥ ￥ ￥ ￥ 1 1 2 ￥ ￥ ￥ ￥ i ￥ ￥ ￥ ￥ 2 1 2 ￥ ￥ ￥ ￥ j + ￥ ￥ ￥ ￥ 2 1 2 2 ￥ ￥ ￥ ￥ k =  2 i + 2 j + 6 k = w  2 , 2 , 6 w w r w ( 1 ) × r w w ( 1 ) w = ￥ ( 2 ) 2 + 2 2 + 6 2 = √ 44 = 2 √ 11 Substituting in (1) gives: B ( 1 ) = w  2 , 2 , 6 w 2 √ 11 = 1 √ 11 w  1 , 1 , 3 w We now find T ( 1 ) : T ( 1 ) = r w ( 1 ) w r w ( 1 ) w = w 2 , 1 , 1 w √ 4 + 1 + 1 = 1 √ 6 w 2 , 1 , 1 w Finally we find N ( 1 ) by computing the following cross product: N ( 1 ) = B ( 1 ) × T ( 1 ) = 1 √ 11 w  1 , 1 , 3 w × 1 √ 6 w 2 , 1 , 1 w = 1 √ 66 ￥ ￥ ￥ ￥ ￥ ￥ i j k 1 1 3 2 1 1 ￥ ￥ ￥ ￥ ￥ ￥ = 1 √ 66 ￥ ￥ ￥ ￥ ￥ 1 3 1 1 ￥ ￥ ￥ ￥ i ￥ ￥ ￥ ￥ 1 3 2 1 ￥ ￥ ￥ ￥ j + ￥ ￥ ￥ ￥ 1 1 2 1 ￥ ￥ ￥ ￥ k ￥ = 1 √ 66 ( 4 i + 7 j k ) = 1 √ 66 w 4 , 7 , 1 w 14.5 Motion in ThreeSpace (ET Section 13.5) Preliminary Questions 1. If a particle travels with constant speed, must its acceleration vector be zero? Explain. SOLUTION If the speed of the particle is constant, the tangential component, a T ( t ) = v w ( t ) , of the acceleration is zero. However, the normal component, a N ( t ) = κ ( t )v( t ) 2 is not necessarily zero, since the particle may change its direction. 2. For a particle in uniform circular motion around a circle, which of the vectors v ( t ) or a ( t ) always points toward the center of the circle? SOLUTION For a particle in uniform circular motion around a circle, the acceleration vector a ( t ) points towards the center of the circle, whereas v ( t ) is tangent to the circle. 3. Two objects travel to the right along the parabola y = x 2 with nonzero speed. Which of the following must be true? (a) Their velocity vectors point in the same direction. (b) Their velocity vectors have the same length. (c) Their acceleration vectors point in the same direction. SOLUTION (a) The velocity vector points in the direction of motion, hence the velocities of the two objects point in the same direction. (b) The length of the velocity vector is the speed. Since the speeds are not necessarily equal, the velocity vectors may have different lengths. 544 C H A P T E R 14 CALCULUS OF VECTORVALUED FUNCTIONS (ET CHAPTER 13) (c) The acceleration is determined by the tangential component v w ( t ) and the normal component...
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This note was uploaded on 07/15/2009 for the course MATH 210 taught by Professor Hubscher during the Spring '08 term at University of Illinois at Urbana–Champaign.
 Spring '08
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