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TangentialAndNormalComponents

# TangentialAndNormalComponents - Example Tangential and...

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Example: Tangential and Normal Components of Accelera- tion Find the decomposition of the acceleration at t = 1 where r ( t ) = ( t - 1 , ln t, t 2 ) Also, find the unit vectors T and N . Solution We must decompose a (1) = r primeprime (1) in the form a (1) = v prime (1) T (1) + κ (1) v 2 (1) N (1) Step 1 : Compute velocity and acceleration: v ( t ) = (- t - 2 , t - 1 , 2 t ) , a ( t ) = ( 2 t - 3 , - t - 1 , 2 ) , Therefore v (1) = (- 1 , 1 , 2 ) , T (1) = 1 6 (- 1 , 1 , 2 ) , a (1) = ( 2 , - 1 , 2 ) Step 2 : Compute the speed and its derivative: v ( t ) = || r prime ( t ) || = ( t - 4 + t - 2 + 4 t 2 ) 1 / 2 , v (1) = 6 v prime ( t ) = - 4 t - 5 - 2 t - 3 + 8 t 2( t - 4 + t - 2 + 4 t 2 ) 1 / 2 , v prime (1) = - 4 - 2 + 8 2 6 = 1 6 Therefore a (1) = 1 6 parenleftbigg 1 6 (- 1 , 1 , 2 ) parenrightbigg + κv 2 N (1) = 1 6 (- 1 , 1 , 2 ) + κv 2 N (1) Step 3 : Compute the normal part κv 2 N (1): κv 2 N = a (1) - 1 6 (- 1 , 1 , 2
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