Chapter 11: Q. 16 (page 898)

Find
(a) the displacement vectors from r(a) tor(b),
(b) the magnitude of the displacement vector, and
(c) the distance travelled by a particle on the curve from a to b.
r(t) = e^t sint, e^t cost, e^ta = 0, b = 1

Short Answer

Expert verified

$(a) The displacement vector from r (a) t o r (b) i s <e \sin 1, ecos 1 - 1, e - 1> (b) Hence the magnitude of the displacement vector is = \sqrt{e^{2} - 2 e (\cos 1 + 1) + e^{2} + 2} (c) Thus the distance travelled by the particle from a to b is \sqrt{3} (e - 1)$

Step by step solution

Part (a) Step 1. Given information

The given function is r(t) = e^t sint, e^t cost, e^ta = 0, b = 1.

Part (a) Step 2. Finding the displacement vector

$= r (b) - r a (a) The displacement vector from r (a) t o r (b) = r (b) - r a (a) = r (1) - r (0) = <e \sin 1, e \cos 1, e> - <\sin 0, \cos 0, 1> = <e \sin 1, e \cos 1, e> - <0, 1, 1> = <e \sin 1, e \cos 1 - 1, e - 1> The displacement vector from r (a) t o r (b) i s <e \sin 1, ecos 1 - 1, e - 1> .$

Part (b) Step 1. Finding the magnitude of the displacement vector,

$The magnitude of displacement vector ||<e \sin 1, ecos 1 - 1, e - 1>|| = \sqrt{<e^{2} \sin^{2} 1, {(ecos 1 - 1)}^{2}, {(e - 1)}^{2}>} = \sqrt{e^{2} - 2 e (\cos 1 + 1) + e^{2} + 2} Hence the magnitude of the displacement vector is = \sqrt{e^{2} - 2 e (\cos 1 + 1) + e^{2} + 2}$

Part (c) Step 1. Finding the displacement vector

$r (t) = <e^{t} \sin t, e^{t} \cos t, e^{t}> r' (t) = <e^{t} \sin t, e^{t} \cos t, e^{t}> ||r' (t)|| = \sqrt{{(e^{t} \sin t + e^{t} \cos t)}^{2} + {(e^{t} \sin t - e^{t} \cos t),}^{2} + {(e^{t})}^{2}} The distance travelled by the particle from a to b is \int_{a}^{b} ||r' (t)|| d t \int \sqrt{3 e^{2 t}} = \sqrt{3} (e^{t} - e^{0}) = \sqrt{3} e^{t} {|_{0}}^{1} Thus the distance travelled by the particle from a to b is \sqrt{3} (e - 1)$

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.

Find
(a) the displacement vectors from r(a) tor(b),
(b) the magnitude of the displacement vector, and
(c) the distance travelled by a particle on the curve from a to b.
r(t) = e^t sint, e^t cost, e^ta = 0, b = 1

Short Answer

Step by step solution

Part (a) Step 1. Given information

Part (a) Step 2. Finding the displacement vector

Part (b) Step 1. Finding the magnitude of the displacement vector,

Part (c) Step 1. Finding the displacement vector

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Decision Maths

Applied Mathematics

Probability and Statistics

Mechanics Maths

Calculus

Study anywhere. Anytime. Across all devices.

Find(a) the displacement vectors from r(a) tor(b),(b) the magnitude of the displacement vector, and(c) the distance travelled by a particle on the curve from a to b.r(t) = et sint, et cost, eta = 0, b = 1

Short Answer

Step by step solution

Part (a) Step 1. Given information

Part (a) Step 2. Finding the displacement vector

Part (b) Step 1. Finding the magnitude of the displacement vector,

Part (c) Step 1. Finding the displacement vector

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Decision Maths

Applied Mathematics

Probability and Statistics

Mechanics Maths

Calculus

Study anywhere. Anytime. Across all devices.

Find
(a) the displacement vectors from r(a) tor(b),
(b) the magnitude of the displacement vector, and
(c) the distance travelled by a particle on the curve from a to b.
r(t) = e^t sint, e^t cost, e^ta = 0, b = 1