a4 solu - MATH 251-3, Fall 2007 Simon Fraser University...

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Unformatted text preview: MATH 251-3, Fall 2007 Simon Fraser University Assignment 4: Solutions Additional Question Due: 4:30pm, Monday 15 October 2007 1. The position of a particle at time t is given by r ( t ) = t, t 2 , 2 3 t 3 . (a) Find the velocity v , speed v = | v | and acceleration a of the particle, and the curvature κ of its path, as a function of t . Solution: We compute: v ( t ) = r ( t ) = 1 , 2 t, 2 t 2 , v ( t ) = | v ( t ) | = √ 1 + 4 t 2 + 4 t 4 = p (1 + 2 t 2 ) 2 = 1 + 2 t 2 , a ( t ) = v ( t ) = h , 2 , 4 t i . We can compute r × r 00 = v × a = h 1 , 2 t, 2 t 2 i × h , 2 , 4 t i = h 4 t 2 ,- 4 t, 2 i , so the curvature is κ = | v × a | v 3 = √ 16 t 4 + 16 t 2 + 4 (1 + 2 t 2 ) 3 = 2(1 + 2 t 2 ) (1 + 2 t 2 ) 3 = 2 (1 + 2 t 2 ) 2 . (b) Find the arc length function s ( t ), and the arc length of the curve as t increases from t = 0 to t = 3. Solution: The arc length function satisfies ds/dt = | r ( t ) | = v ( t ) = 1 + 2 t 2 ; assuming that s = 0 when t = 0 , we have s ( t ) = Z t v ( τ...
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This note was uploaded on 10/09/2009 for the course MATH macm 101 taught by Professor Jcliu during the Spring '09 term at Simon Fraser.

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a4 solu - MATH 251-3, Fall 2007 Simon Fraser University...

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