To determine Find the acceleration vector a in the form a = a T T + a N N at the value of t without finding T and N for the given vector r ( t ) = t 2 i + ( t + ( 1 3 ) t 3 ) j + ( t − ( 1 3 ) t 3 ) k , t = 0 . Explanation Given information: r ( t ) = t 2 i + ( t + ( 1 3 ) t 3 ) j + ( t − ( 1 3 ) t 3 ) k (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) 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 equation (1) in equation (2). v = d d t ( t 2 i + ( t + ( 1 3 ) t 3 ) j + ( t − ( 1 3 ) t 3 ) k ) v = 2 t i + ( 1 + t 2 ) j + ( 1 − t 2 ) k (7) Take magnitude on both sides of the above equation. | v | = ( 2 t ) 2 + ( 1 + t 2 ) 2 + ( 1 − t 2 ) 2 = 4 t 2 + 1 + t 4 + 2 t 2 + 1 + t 4 − 2 t 2 = 4 t 2 + 2 t 4 + 2 = 2 ( t 4 + 2 t 2 + 1 ) | v | = 2 ( t 2 + 1 ) 2 | v | = 2 ( t 2 + 1 ) (8) Substitute equation (8) in equation (6). a T = d d t ( 2 ( t 2 + 1 ) ) = 2 ( 2 t ) = 2 t 2 At t = 0 , the tangential scalar component of acceleration a T is, a T ( 0 ) = 2 ( 0 ) 2 = 0 Substitute equation (7) in equation (3)

University Calculus: Early Transcendentals (4th Edition)

4th Edition

ISBN: 9780134995540

Author: Joel R. Hass, Christopher E. Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 12.5, Problem 5E

To determine

Find the acceleration vector a in the form a=aTT+aNN at the value of t without finding T and N for the given vector r(t)=t2i+(t+(13)t3)j+(t−(13)t3)k, t=0.

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Chapter 12.5, Problem 4E

Chapter 12.5, Problem 6E

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Chapter 12 Solutions

University Calculus: Early Transcendentals (4th Edition)

Ch. 12.1 - In Exercises 1–4, find the given limits. 1. Ch. 12.1 - In Exercises 1–4, find the given limits. 2. Ch. 12.1 - In Exercises 1–4, find the given limits. 3. Ch. 12.1 - In Exercises 1–4, find the given limits. 4. Ch. 12.1 - Motion in the Plane In Exercises 58, r(t) is the...Ch. 12.1 - Motion in the Plane In Exercises 5–8, r(t) is the...Ch. 12.1 - In Exercises 58, r(t) is the position of a...Ch. 12.1 - In Exercises 5–8, r(t) is the position of a...Ch. 12.1 - Prob. 9E Ch. 12.1 - Prob. 10E

Ch. 12.1 - Exercises 9–12 give the position vectors of...Ch. 12.1 - Prob. 12E Ch. 12.1 - In Exercises 13–18, r(t) is the position of a...Ch. 12.1 - Prob. 14E Ch. 12.1 - In Exercises 13–18, r(t) is the position of a...Ch. 12.1 - Prob. 16E Ch. 12.1 - Prob. 17E Ch. 12.1 - In Exercises 13–18, r(t) is the position of a...Ch. 12.1 - In Exercises 1922, r(t) is the position of a...Ch. 12.1 - In Exercises 19–22, r(t) is the position of a...Ch. 12.1 - In Exercises 19–22, r(t) is the position of a...Ch. 12.1 - Prob. 22E Ch. 12.1 - As mentioned in the text, the tangent line to a...Ch. 12.1 - Prob. 24E Ch. 12.1 - Tangents to Curves As mentioned in the text, the...Ch. 12.1 - Prob. 26E Ch. 12.1 - Prob. 27E Ch. 12.1 - Prob. 28E Ch. 12.1 - Prob. 29E Ch. 12.1 - Prob. 30E Ch. 12.1 - Prob. 31E Ch. 12.1 - Prob. 32E Ch. 12.1 - Prob. 33E Ch. 12.1 - Prob. 34E Ch. 12.1 - Prob. 35E Ch. 12.1 - Prob. 36E Ch. 12.1 - Motion along a circle Each of the following...Ch. 12.1 - Motion along a circle Show that the vector-valued...Ch. 12.1 - Prob. 39E Ch. 12.1 - Motion along a cycloid A particle moves in the...Ch. 12.1 - Prob. 41E Ch. 12.1 - Prob. 42E Ch. 12.1 - Prob. 43E Ch. 12.1 - Prob. 44E Ch. 12.1 - Component test for continuity at a point Show that...Ch. 12.1 - Limits of cross products of vector functions...Ch. 12.1 - Differentiable vector functions are continuous...Ch. 12.1 - Constant Function Rule Prove that if u is the...Ch. 12.2 - Evaluate the integrals in Exercises 1–10. 1. Ch. 12.2 - Evaluate the integrals in Exercises 1–10. 2. Ch. 12.2 - Evaluate the integrals in Exercises 1–10. 3. Ch. 12.2 - Evaluate the integrals in Exercises 1–10. 4. Ch. 12.2 - Evaluate the integrals in Exercises 1–10. 5. Ch. 12.2 - Evaluate the integrals in Exercises 1–10. 6. Ch. 12.2 - Evaluate the integrals in Exercises 110. 7....Ch. 12.2 - Evaluate the integrals in Exercises 1–10. 8. Ch. 12.2 - Prob. 9E Ch. 12.2 - Prob. 10E Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Prob. 15E Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - Prob. 18E Ch. 12.2 - Prob. 19E Ch. 12.2 - Solve the initial value problems in Exercises...Ch. 12.2 - At time t = 0, a particle is located at the point...Ch. 12.2 - Prob. 22E Ch. 12.2 - Prob. 23E Ch. 12.2 - Range and height versus speed Show that doubling a...Ch. 12.2 - Flight time and height A projectile is fired with...Ch. 12.2 - Prob. 26E Ch. 12.2 - Prob. 27E Ch. 12.2 - Beaming electrons An electron in a TV tube is...Ch. 12.2 - Prob. 29E Ch. 12.2 - Finding muzzle speed Find the muzzle speed of a...Ch. 12.2 - Prob. 31E Ch. 12.2 - Colliding marbles The accompanying figure shows an...Ch. 12.2 - 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