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blueexam2sol.08w

blueexam2sol.08w - blue MATH 32A Exam 2 LAST NAME FIRST...

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Unformatted text preview: blue MATH 32A Exam 2 February 20, 2008 LAST NAME FIRST NAME ID NO. Your TA: To receive credit, you must write your answer in the space provided . DO NOT WRITE BELOW THIS LINE 1 (20 pts) 5 (20 pts) 2 (10 pts) 6 (20 pts) 3 (20 pts) 7 (20 pts) 4 (20 pts) TOTAL 2 PROBLEM 1 (20 Points) A particle travels along a path parametrized by a vector valued function r ( t ). Assume that at time t = 2, r (2) = h 1 , 2 , 2 i , r 00 (2) = h- 2 , , 10 i Find the curvature of the path at t = 2. Answer : Solution: At t = 2: r 00 (2) = a T T + a N N We have a T T = r 00 (2) · r (2) r (2) · r (2) r (2) = h- 2 , , 10 i · h 1 , 2 , 2 i h 1 , 2 , 2 i · h 1 , 2 , 2 i h 1 , 2 , 2 i = 18 9 h 1 , 2 , 2 i = h 2 , 4 , 4 i and κv 2 = a N = || a N N || = || r 00 (2)- a T T || = ||h- 2 , , 10 i - h 2 , 4 , 4 i|| = ||h- 4 ,- 4 , 6 i|| = √ 68 The square of the speed is v 2 = || r (2) || 2 = 9 and thus κv 2 = 9 κ = √ 68 It follows that κ = √ 68 9 3 PROBLEM 2 (20 Points) A particle travels along the curve y = x 3 . Assume units of meters and seconds. At a certain moment, the particle is located at the point...
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blueexam2sol.08w - blue MATH 32A Exam 2 LAST NAME FIRST...

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