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 2 , 4 t − 3 〉 Formula used: Unit tangent vector, T ( t ) = r ′ ( t ) ∥ r ′ ( t ) ∥ = v ∥ v ∥ -----------------(2) Unit normal vector, N ( t ) = T ′ ( t ) ∥ T ′ ( t ) ∥ ------------------------(3) Decomposition of a ( t ) into tangential component T ( t ) is, a T = a · T = a · v ∥ v ∥ ----------------(4) Decomposition of a ( t ) into normal component N ( t ) is, a N = a · N = ∥ a × v ∥ ∥ v ∥ ---------------(5) Calculation: Since position vector r ( t ) = 〈 t 2 , 4 t − 3 〉 ----------------(6) Therefore, velocity, v ( t ) = r ′ ( t ) = 〈 2 t , − 4 〉 ------------(7) and acceleration, a ( t ) = v ′ ( t ) = r ″ ( t ) = 〈 2 , 0 〉 ---------(8) also, ∥ v ∥ = 4 t 2 + 16 = 2 t 2 + 4 --------------------------------------(9) Using equation (2), the unit tangential vector is, ...

4th Edition

ISBN: 9781319434526

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 14.5, Problem 44E

To determine

(a) Find the tangential vector $T (t)$ and normal vector $N (t)$ of the acceleration vector.

(b) Decomposition of acceleration vector $a (t)$ into tangential and normal components.

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Chapter 14.5, Problem 43E

Chapter 14.5, Problem 45E

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Chapter 14 Solutions

CALCULUS 4E (LL) W/ ACHIEVE ACCESS

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

Knowledge Booster

(a) Find the tangential vector T ( t ) and normal vector N ( t ) of the acceleration vector. (b) Decomposition of acceleration vector a ( t ) into tangential and normal components.

Solution Summary: The author explains how acceleration vector a ( t ) can be decomposed into tangential and normal components.

(a) Find the tangential vector $T (t)$ and normal vector $N (t)$ of the acceleration vector.

(b) Decomposition of acceleration vector $a (t)$ into tangential and normal components.

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Chapter 14 Solutions

(a) Find the tangential vector T ( t ) and normal vector N ( t ) of the acceleration vector. (b) Decomposition of acceleration vector a ( t ) into tangential and normal components.

Solution Summary: The author explains how acceleration vector a ( t ) can be decomposed into tangential and normal components.

(a) Find the tangential vector T(t) and normal vector N(t) of the acceleration vector. (b) Decomposition of acceleration vector a(t) into tangential and normal components.

Want to see the full answer?

Chapter 14 Solutions

(a) Find the tangential vector $T (t)$ and normal vector $N (t)$ of the acceleration vector.

(b) Decomposition of acceleration vector $a (t)$ into tangential and normal components.