MATH 317 T2 SOLUTIONS TO MIDTERM

# MATH 317 T2 SOLUTIONS TO MIDTERM - MATH 317 T2 SOLUTIONS TO...

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MATH 317 T2 SOLUTIONS TO MIDTERM #1 1. As shown in class, the curvature . | ) ( | | ) ( ) ( | ) ( 3 t t t t r r r × = κ In this question . ) ( ) ( ) ( , ) ( ) ( ) ( ) ( ) ( ) ( j i r j i r j i r t g t f t t g t f t t g t f t = = = Hence | ) ( ) ( ) ( ) ( | | ) ( ) ( | . t f t g t g t f t t - = × r r and 2 2 )] ( [ )] ( [ | ) ( | . t g t f t + = r . In turn this yields the given answer. 2. (a) The velocity is given by , 2 ) ( ) ( k j i r v + - = = - t t e e t t the acceleration by . ) ( ) ( ) ( j i r v a t t e e t t t - + = = = (b) The unit tangent vector to the path is given by = = | ) ( | ) ( t t r r T . 2 2 2 2 + + + - - - t t t t e e e e k j i (c) The length of the path is given by . ) ( ) ( 2 | ) ( | 1 1 0 1 0 2 1 0 2 2 1 0 - - - - - = + = + = + + = e e dt e e dt e e dt e e dt t t t t t t t r (d) The curvature of the path at time t is given by . | ) ( | | ) ( ) ( | ) ( 3 t t t t r r r × = κ Here ). ( 2 2 2 4 | ) ( ) ( | 2 2 2 ) ( ) ( 2 2 t t t t t t e e e e t t e e t t - - - + = + + = × + - - = × r r i j k r r Then . ) ( | ) ( | 3 3 t t e e t - + = r Consequently, .

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• Spring '10
• BLUMAN
• 2k, Inverse-square law, inverse square law, equal areas

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