Explanation Given information: The given vector is, r ( t ) = ( 3 sec t ) i + ( 3 tan t ) j , 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) Write the expression to calculate the curvature of the function. κ = a N | v | 2 (7) Calculation: Substitute the equation (1) in equation (2) to find v . v = d d t ( ( 3 sec t ) i + ( 3 tan t ) j ) = ( 3 ( sec t tan t ) ) i + ( 3 ( sec 2 t ) ) j v = ( 3 sec t tan t ) i + ( 3 sec 2 t ) j (8) Take magnitude on both sides of the equation (8). | v | = | ( 3 sec t tan t ) i + ( 3 sec 2 t ) j | = ( 3 sec t tan t ) 2 + ( 3 sec 2 t ) 2 | v | = 3 sec 2 t tan 2 t + 3 sec 4 t (9) Substitute equation (9) in equation (6) to find a T . a T = d d t ( 3 sec 2 t tan 2 t + 3 sec 4 t ) = d d t ( 3 sec 2 t tan 2 t + 3 sec 4 t ) 1 2 = 1 2 ( 3 sec 2 t tan 2 t + 3 sec 4 t ) 1 2 − 1 ( ( 3 ( sec 2 t ( 2 tan t ) ( sec 2 t ) + tan 2 t ( 2 sec t ) ( sec t tan t ) ) ) + 3 ( 4 sec 3 t ) ( sec t tan t ) ) = 1 2 ( 3 sec 2 t tan 2 t + 3 sec 4 t ) − 1 2 ( 6 tan t sec 4 t + 6 sec 2 t tan 3 t + 12 sec 4 t tan t ) Simplify the above equation to find a T . a T = ( 1 2 ( 6 sec 2 t tan 3 t + 18 sec 4 t tan t ) ) ( 3 sec 2 t tan 2 t + 3 sec 4 t ) 1 2 = 6 sec 2 t tan 3 t + 18 sec 4 t tan t 2 ( 3 sec 2 t tan 2 t + 3 sec 4 t ) Substitute 0 for t in above equation to find a T ( 0 ) . a T ( 0 ) = 6 sec 2 ( 0 ) tan 3 ( 0 ) + 18 sec 4 ( 0 ) tan ( 0 ) 2 ( 3 sec 2 ( 0 ) tan 2 ( 0 ) + 3 sec 4 ( 0 ) ) = 6 ( 1 ) ( 0 ) + 18 ( 1 ) ( 0 ) 2 ( 3 ( 1 ) ( 0 ) + 3 ( 1 ) ) = 0 Substitute the equation (8) in equation (3) to find a . a = d d t ( ( 3 sec t tan t ) i + ( 3 sec 2 t ) j ) = ( 3 ( sec t ( sec 2 t ) + tan t ( sec t tan t ) ) ) i + ( 3 ( 2 sec t ( sec t tan t ) ) ) j = ( 3 ( sec 3 t + sec t tan 2 t ) ) i + ( 2 3 sec 2 t tan t ) j = ( 3 sec 3 t + 3 sec t tan 2 t ) i + ( 2 3 sec 2 t tan t ) j Take magnitude on both sides of the above equation

Thomas' Calculus: Early Transcendentals in SI Units

14th Edition

ISBN: 9781292253114

Author: Hass, Joel R., Heil, Christopher E., WEIR, Maurice D.

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 13, Problem 2PE

To determine

Find the acceleration vector a for the given position vector in the form a=aTT+aNN at the given value of t without finding T and N and graph the curves with their velocity and acceleration vectors.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13, Problem 1PE

Chapter 13, Problem 3PE

Students have asked these similar questions

3 The total profit P(X) (in thousands of dollars) from the sale of x hundred thousand automobile tires is approximated by P(x) = -x³ + 12x² + 60x - 200, x≥5. Find the number of hundred thousands of tires that must be sold to maximize profit. Find the maximum profit. The maximum profit is $ when hundred thousand tires are sold.

A fence must be built to enclose a rectangular area of 5000 ft². Fencing material costs $4 per foot for the two sides facing north and south and $8 per foot for the other two sides. Find the cost of the least expensive fence. The cost of the least expensive fence is $ (Simplify your answer.)

The number of fish swimming upstream to spawn is approximated by the function given below, where x represents the temperature of the water in degrees Celsius. Find the water temperature that produces the maximum number of fish swimming upstream. F(x) = x3 + 3x² + 360x + 5017, 5≤x≤18

Chapter 13 Solutions

Thomas' Calculus: Early Transcendentals in SI Units

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

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Find the acceleration vector a for the given position vector in the form a = a T T + a N N at the given value of t without finding T and N and graph the curves with their velocity and acceleration vectors .

Videos

Find the acceleration vector a for the given position vector in the form a=aTT+aNN at the given value of t without finding T and N and graph the curves with their velocity and acceleration vectors.

Want to see the full answer?

Chapter 13 Solutions