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 = π

Short Answer

Expert verified

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

Step by step solution

Part (a) Step 1. Given Information

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

Part (a) Step 2. Finding the displacement vector

r(t) = sin t , t cost, a = 0, b = π

$= r (b) - r a (a) The displacement vector from r (a) t o r (b) = r (b) - r a (a) = r (π) - r (0) = <πsinπ, πcosπ> - <0, 0> = <0, - π> - <0, 0> = <0, - π> The displacement vector from r (a) t o r (b) i s <0, - π> .$

Part (b) Step 1. Magnitude of displacement vector

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

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

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

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

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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 = π

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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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 = π

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

Pure Maths

Applied Mathematics

Logic and Functions

Calculus

Decision Maths

Probability and Statistics

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 = π