Chapter 11: Q. 3 (page 891)

A decomposition of the acceleration vector: Find $c o m p_{v (t)} a (t)$ , where v and a are the velocity and acceleration vectors, respectively, of the following functions.
$r (t) = <e^{t} \sin t, e^{t} \cos t, e^{t}> .$

Short Answer

Expert verified

$T h e c o m p o n e n t o f a (t) a l o n g v (t) i s \sqrt{3} e^{t} .$

Step by step solution

Step 1. Given Information.

$G i v e n, x - c o o r d i n a t e : e^{t} \sin t, y - c o o r d i n a t e : e^{t} \cos t, z - c o o r d i n a t e : e^{t} .$

Step 2. Calculating the velocity and acceleration vector.

$r (t) = <e^{t} \sin t, e^{t} \cos t, e^{t}>, D i f f e r e n t i a t i n g w e g e t, r' (t) = v (t) = <e^{t} \cos t + e^{t} \sin t, e^{t} \cos t - e^{t} \sin t, e^{t}>, - - - - (i) D i f f e r e n t i a t i n g o n c e a g a i n w e g e t, r'' (t) = v' (t) = a (t) = <e^{t} \cos t - e^{t} \sin t + e^{t} \cos t + e^{t} \sin t, e^{t} \cos t - e^{t} \sin t - (e^{t} \cos t + e^{t} \sin t), e^{t}> a (t) = <2 e^{t} \cos t, - 2 e^{t} \sin t, e^{t}> . - - - - (i i)$

Step 3. Finally calculating the component of a(t) along v(t).

$F r o m s t e p 2 w e g e t v (t) a n d a (t) . N o w c o m p_{v (t)} a (t) = \frac{|a (t) . v (t)|}{||v (t)||} N o w f r o m (i) a n d (i i) w e g e t c o m p_{v (t)} a (t) = \frac{2 e^{t} \cos t (e^{t} \cos t + e^{t} \sin t) + (- 2 e^{t} \sin t (e^{t} \cos t - e^{t} \sin t)) + e^{t} e^{t}}{\sqrt{{(e^{t} \cos t + e^{t} \sin t)}^{2} + {(e^{t} \cos t - e^{t} \sin t)}^{2} + {(e^{t})}^{2}}} S o l v i n g t h e a b o v e e x p r e s s i o n w e g e t, = \frac{e^{2 t} (2 \cos^{2} t + 2 \cos t \sin t - 2 \cos t \sin t + 2 \sin^{2} t + 1)}{e^{t} \sqrt{\cos^{2} t + \sin^{2} t + 2 \cos t \sin t + \cos^{2} t + \sin^{2} t - 2 \cos t \sin t + 1}} = \frac{3 e^{2 t}}{e^{t} \sqrt{3}} = \sqrt{3} e^{t} . S o t h e c o m p o n e n t o f a (t) a l o n g t h e d i r e c t i o n o f v (t) w i l l b e \sqrt{3} e^{t} .$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

A decomposition of the acceleration vector: Find $c o m p_{v (t)} a (t)$ , where v and a are the velocity and acceleration vectors, respectively, of the following functions.
$r (t) = <e^{t} \sin t, e^{t} \cos t, e^{t}> .$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Calculating the velocity and acceleration vector.

Step 3. Finally calculating the component of a(t) along v(t).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Probability and Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

A decomposition of the acceleration vector: Find compv(t)a(t), where v and a are the velocity and acceleration vectors, respectively, of the following functions.r(t)=etsint,etcost,et.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Calculating the velocity and acceleration vector.

Step 3. Finally calculating the component of a(t) along v(t).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Probability and Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

A decomposition of the acceleration vector: Find $c o m p_{v (t)} a (t)$ , where v and a are the velocity and acceleration vectors, respectively, of the following functions.
$r (t) = <e^{t} \sin t, e^{t} \cos t, e^{t}> .$