Explanation Given information: The given vector is, r ( t ) = ( 2 + t ) i + ( t + 2 t 2 ) j + ( 1 + t 2 ) k , t = 0 (1) Formula used: Write the general expression to calculate the velocity. v = d r d t (2) Here, d r d t is the rate of change in displacement with respect to time. Write the general expression to calculate the acceleration. a = d v d t (3) Here, d v d t is the rate of change in velocity with respect to time. Write the general expression for the acceleration vector. a = a T T + a N N (4) Here, a T is the tangential scalar component of acceleration, and a N is the normal scalar component of acceleration. Take magnitude on both sides of the equation (4). | a | = a T 2 + a N 2 (5) Write the general expression to calculate the tangential scalar component of acceleration. a T = d d t | v | (6) Calculation: Substitute the equation (1) in equation (2) to find v . v = d d t ( ( 2 + t ) i + ( t + 2 t 2 ) j + ( 1 + t 2 ) k ) = ( 0 + 1 ) i + ( 1 + 2 ( 2 t ) ) j + ( 0 + 2 t ) k v = 1 i + ( 1 + 4 t ) j + 2 t k (7) Take magnitude on both sides of the equation (7). | v | = | 1 i + ( 1 + 4 t ) j + 2 t k | = ( 1 ) 2 + ( 1 + 4 t ) 2 + ( 2 t ) 2 = 1 + 1 + 16 t 2 + 8 t + 4 t 2 | v | = 2 + 8 t + 20 t 2 (8) Substitute equation (8) in equation (6) to find a T . a T = d d t ( 2 + 8 t + 20 t 2 ) = d d t ( 2 + 8 t + 20 t 2 ) 1 2 = 1 2 ( 2 + 8 t + 20 t 2 ) 1 2 − 1 ( 0 + 8 ( 1 ) + 20 ( 2 t ) ) = 1 2 ( 2 + 8 t + 20 t 2 ) − 1 2 ( 8 + 40 t ) Substitute 0 for t in above equation to find a T ( 0 ) . a T ( 0 ) = 1 2 ( 2 + 8 ( 0 ) + 20 ( 0 ) 2 ) − 1 2 ( 8 + 40 ( 0 ) ) = 1 2 ( 2 + 0 + 0 ) − 1 2 ( 8 + 0 ) = 2 2 Substitute equation (7) in equation (3) to find a

Pearson eText for Thomas' Calculus: Early Transcendentals -- Instant Access (Pearson+)

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Chapter 13, Problem 22PE

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Find the acceleration vector a for the given vector in the form a=aTT+aNN at the value of t without finding T and N.

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Chapter 13, Problem 21PE

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Pearson eText for Thomas' Calculus: Early Transcendentals -- Instant Access (Pearson+)

Ch. 13.1 - In Exercises 1–4, find the given limits. 1. Ch. 13.1 - In Exercises 1–4, find the given limits. 2. Ch. 13.1 - In Exercises 1–4, find the given limits. 3. Ch. 13.1 - In Exercises 1–4, find the given limits. 4. Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

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Find the acceleration vector a for the given vector in the form a = a T T + a N N at the value of t without finding T and N .

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Find the acceleration vector a for the given vector in the form a=aTT+aNN at the value of t without finding T and N.

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Chapter 13 Solutions