Q. 3 A decomposition of the accelerat... [FREE SOLUTION]

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).

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

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91影视

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.

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91影视

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

Discrete Mathematics

Theoretical and Mathematical Physics

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Applied Mathematics

Statistics

Logic and Functions

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.
$r (t) = <e^{t} \sin t, e^{t} \cos t, e^{t}> .$