Explanation Acceleration vector a ( t ) can be decompose into tangential component (tangent to the path of motion) and normal component (perpendicular to the path of motion) as a ( t ) = a T T ( t ) + a N N ( t ) ------------------------(1) where unit tangent vector, T ( t ) = r ′ ( t ) ∥ r ′ ( t ) ∥ = v ∥ v ∥ and unit normal vector, N ( t ) = T ′ ( t ) ∥ T ′ ( t ) ∥ a T = a · T = a · v ∥ v ∥ a N = a · N = ∥ a × v ∥ ∥ v ∥ Given: Position vector r ( t ) = 〈 t , e t , t e t 〉 Formula used: Decomposition of a ( t ) into tangential component T ( t ) is, a T = a · T = a · v ∥ v ∥ ----------------(2) Decomposition of a ( t ) into normal component N ( t ) is, a N = a · N = ∥ a × v ∥ ∥ v ∥ ---------------(3) Calculation: Since position vector r ( t ) = 〈 t , e t , t e t 〉 ----------------(4) Therefore, velocity, v ( t ) = r ′ ( t ) = 〈 1 , e t , ( 1 + t ) e t 〉 ------------(5) and acceleration, a ( t ) = v ′ ( t ) = r ″ ( t ) = 〈 0 , e t , ( 2 + t ) e t 〉 ---------(6) also, ∥ v ∥ = 1 + e 2 t + ( 1 + t ) 2 e 2 t --------------------------------------(7) Using equations (2), (5), (6) and (7) we determine the decomposition of a ( t ) into tangential component as a T ( t ) = 〈 0 , e t , ( 2 + t ) e t 〉 〈 1 , e t , ( 1 + t ) e t 〉 1 + e 2 t + ( 1 + t ) 2 e 2 t

4th Edition

ISBN: 9781319434540

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 14.5, Problem 41E

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Decomposition of acceleration vector $a (t)$ of $r (t) = 〈 t, e^{t}, t e^{t} 〉$ into tangential and normal components at the point $t = 0$ .

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Chapter 14.5, Problem 40E

Chapter 14.5, Problem 42E

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Chapter 14 Solutions

ACHIEVE STANDALONE ACCESS F/ CALC 4E

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

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Decomposition of acceleration vector a ( t ) of r ( t ) = 〈 t , e t , t e t 〉 into tangential and normal components at the point t = 0 .

Solution Summary: The author explains the decomposition of acceleration vector a (t) into tangential and normal components at the point t = 0.

Decomposition of acceleration vector $a (t)$ of $r (t) = 〈 t, e^{t}, t e^{t} 〉$ into tangential and normal components at the point $t = 0$ .

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Chapter 14 Solutions

Decomposition of acceleration vector a ( t ) of r ( t ) = 〈 t , e t , t e t 〉 into tangential and normal components at the point t = 0 .

Solution Summary: The author explains the decomposition of acceleration vector a (t) into tangential and normal components at the point t = 0.

Decomposition of acceleration vector a(t) of r(t)=〈t,et,tet〉 into tangential and normal components at the point t=0.

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Chapter 14 Solutions

Decomposition of acceleration vector $a (t)$ of $r (t) = 〈 t, e^{t}, t e^{t} 〉$ into tangential and normal components at the point $t = 0$ .