Explanation Given Info : Acceleration a ( t ) = cos t k Initial velocity v ( 0 ) = 1 i − 1 j Initial position r ( 0 ) = 1 i Formula Used: (i) Apply the formula for velocity vector v ( t ) = ∫ a ( t ) d t + C 1 (iii) Apply the formula for position vector r ( t ) = ∫ v ( t ) d t + C 2 Here, all alphabets are in their usual meanings. Calculation: Apply the formula for velocity v ( t ) = ∫ a ( t ) d t + C 1 v ( t ) = ∫ cos t k d t + C 1 v ( t ) = − sin t k + C 1 − − − − − ( i ) Now, we shall apply given condition for initial velocity vector in equation (i) to calculated the integral constant ‘C 1 ’ 1 i − 1 j + 0 k = v ( 0 ) = 0 i + 0 j + 0 k + C 1 o r , C 1 = 1 i − 1 j − − − − − − ( i i ) From equation (i) and (ii), the required expression for velocity vector v ( t ) can be calculated as v ( t ) = − sin t k + ( 1 i − 1 j ) v ( t ) = 1 i − 1 j − sin t k − − − − ( a ) v ( t ) = ( 1 , − 1 , − sin t ) Apply the formula for position vector r ( t ) = ∫ v

4th Edition

ISBN: 9781319055844

Author: FRANZOSA

Publisher: VST

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 14.5, Problem 18E

To determine

To calculate position vector $r (t)$ and velocity $v (t)$ from the given conditions.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.5, Problem 17E

Chapter 14.5, Problem 19E

Students have asked these similar questions

Consider the following position function. r(t) = t sin(t) i + t cos(e) j + 4t2 k (a) Find the velocity of a particle with the given position function. v(t) = (sin(t) + t cos(t))i+ (cos(1) – t sin(t))i+ 81k (b) Find the acceleration of a particle with the given position function. a(t) = (2 cos(1) – 1 sin(1))i – (tcos(1) + 2 sin(1))i + 8k| (c) Find the speed of a particle with the given position function. |v(t)| %3D

The functions cos r| and cos are linearly dependent on 0,

dw Use the Chain Rule to find , where w = sin 8x cos 4y, x= dt , and y = t. dw dt (Type an expression using t as the variable.)

Chapter 14 Solutions

EBK CALCULUS

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

$Calculus$

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.

To calculate position vector r ( t ) and velocity v ( t ) from the given conditions.

Solution Summary: The author explains how to calculate position vector r ( t ) and velocity v from the given conditions.

Concept explainers

Videos

To calculate position vector $r (t)$ and velocity $v (t)$ from the given conditions.

Want to see the full answer?

Chapter 14 Solutions

To calculate position vector r ( t ) and velocity v ( t ) from the given conditions.

Solution Summary: The author explains how to calculate position vector r ( t ) and velocity v from the given conditions.

Concept explainers

Videos

To calculate position vector r(t) and velocity v(t) from the given conditions.

Want to see the full answer?

Chapter 14 Solutions

To calculate position vector $r (t)$ and velocity $v (t)$ from the given conditions.