Q. 15 Find聽(a) the displacement vecto... [FREE SOLUTION]

Chapter 11: Q. 15 (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) = sin t , t cost, a = 0, b = 蟺

Short Answer

Expert verified

$(a) Thus the magnitude of the displacement vector = 蟺 (b) The displacement vector from r (a) t o r (b) i s <0, - 蟺> . (c) Thus the distance travelled by the particle from a to b is \frac{1}{2} (蟺 \sqrt{蟺^{2} + 1} + \ln (蟺 \sqrt{蟺^{2} + 1}))$

Step by step solution

Part (a) Step 1. Given Information

The given function is r(t) = sin t , t cost, a = 0, b = 蟺

Part (a) Step 2. Finding the displacement vector

r(t) = sin t , t cost, a = 0, b = 蟺

$= r (b) - r a (a) The displacement vector from r (a) t o r (b) = r (b) - r a (a) = r (蟺) - r (0) = <蟺蝉颈苍蟺, 蟺肠辞蝉蟺> - <0, 0> = <0, - 蟺> - <0, 0> = <0, - 蟺> The displacement vector from r (a) t o r (b) i s <0, - 蟺> .$

Part (b) Step 1. Magnitude of displacement vector

Magnitude of displacement vector = $||0, - 蟺||$

$\sqrt{0^{2} + {(- 蟺)}^{2}} = \sqrt{蟺^{2}} = 蟺 Thus the magnitude of the displacement vector = 蟺$

Part (c) Step 1. The distance travelled by a particle on the curve from a to b.

$r (t) = <t \sin t, t \cos t^{}> r' (t) = <t \cos t + \sin t, - t \sin t + \cos t> ||r' (t)|| = \sqrt{t^{2} + 1} The distance travelled by the particle from a to b is {鈭�}_{a}^{b} ||r' (t)|| d t 鈭� \sqrt{t^{2} + 1} d t = 鈭� \sqrt{a^{2} + x^{2}} d x = \frac{x}{2} \sqrt{a^{2} + x^{2}} + \frac{a^{2}}{2} \sin h^{- 1} (\frac{x}{a}) = \frac{t}{2} \sqrt{t^{2} + 1} + {\frac{1}{2} \sinh^{- 1} (t)|}_{0}^{蟺} = (\frac{蟺}{2} \sqrt{蟺^{2} + 1} + \frac{1}{2} \sinh^{- 1} (蟺)) - (\frac{1}{2} \sqrt{0} + \frac{1}{2} (0)) = \frac{蟺}{2} \sqrt{蟺^{2} + 1} + \frac{1}{2} \sinh^{- 1} (蟺) = \frac{蟺}{2} \sqrt{蟺^{2} + 1} + \frac{1}{2} \ln (蟺 + \sqrt{蟺^{2} + 1}) since \sinh^{- 1} x = \ln (x + \sqrt{x^{2} + 1}) \frac{1}{2} (蟺 \sqrt{蟺^{2} + 1} + \ln (蟺 \sqrt{蟺^{2} + 1})) Thus the distance travelled by the particle from a to b is \frac{1}{2} (蟺 \sqrt{蟺^{2} + 1} + \ln (蟺 \sqrt{蟺^{2} + 1}))$

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91影视

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) = sin t , t cost, a = 0, b = 蟺

Short Answer

Step by step solution

Part (a) Step 1. Given Information

Part (a) Step 2. Finding the displacement vector

Part (b) Step 1. Magnitude of displacement vector

Part (c) Step 1. The distance travelled by a particle on the curve from a to b.

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91影视

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) = sin t , t cost, a = 0, b = 蟺

Short Answer

Step by step solution

Part (a) Step 1. Given Information

Part (a) Step 2. Finding the displacement vector

Part (b) Step 1. Magnitude of displacement vector

Part (c) Step 1. The distance travelled by a particle on the curve from a to b.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Applied Mathematics

Probability and Statistics

Logic and Functions

Discrete Mathematics

Geometry

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) = sin t , t cost, a = 0, b = 蟺