The positions of a particle at times t = 1,2 and 3 are given below. What is the average velocity of the particle in the time interval [1,3] ? %3D r(1) = (2,0, 3) r(2) = (1, 1, 1) r(3) = (-2, 4, 1) Select one: а. 5 b. (1, 1, 1) с. 3 d. (-2, 2, -1) e. (-1,3, 1)
The positions of a particle at times t = 1,2 and 3 are given below. What is the average velocity of the particle in the time interval [1,3] ? %3D r(1) = (2,0, 3) r(2) = (1, 1, 1) r(3) = (-2, 4, 1) Select one: а. 5 b. (1, 1, 1) с. 3 d. (-2, 2, -1) e. (-1,3, 1)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Educational Content: Understanding Average Velocity**
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**Problem Statement:**
The positions of a particle at times \( t = 1, 2, \) and \( 3 \) are given below. What is the average velocity of the particle in the time interval \([1, 3]\)?
- \( r(1) = \langle 2, 0, 3 \rangle \)
- \( r(2) = \langle 1, 1, 1 \rangle \)
- \( r(3) = \langle -2, 4, 1 \rangle \)
**Select one:**
- a. 5
- b. \( \langle 1, 1, 1 \rangle \)
- c. 3
- d. \( \langle -2, 2, -1 \rangle \)
- e. \( \langle -1, 3, 1 \rangle \)
---
**Solution Explanation:**
1. **Understanding Vectors:**
- Each \( r(t) \) represents the position of a particle in three-dimensional space at a given time \( t \).
2. **Average Velocity Calculation:**
- Average velocity is defined as the change in position divided by the change in time.
- Calculate the change in position:
\[
\Delta r = r(3) - r(1) = \langle -2, 4, 1 \rangle - \langle 2, 0, 3 \rangle = \langle -4, 4, -2 \rangle
\]
- Calculate the change in time:
\[
\Delta t = 3 - 1 = 2
\]
- Compute the average velocity:
\[
\text{Average Velocity} = \frac{\Delta r}{\Delta t} = \left\langle \frac{-4}{2}, \frac{4}{2}, \frac{-2}{2} \right\rangle = \langle -2, 2, -1 \rangle
\]
3. **Answer Identification:**
- The correct choice is d. \( \langle -2, 2, -1 \rangle \).
This problem explores the concept of average velocity in vector form,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F50589fea-07bb-4f50-a34c-95e031a700c5%2Fcd18178b-aab6-48e0-86f2-f46a7d5557fd%2Fvf7qbk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content: Understanding Average Velocity**
---
**Problem Statement:**
The positions of a particle at times \( t = 1, 2, \) and \( 3 \) are given below. What is the average velocity of the particle in the time interval \([1, 3]\)?
- \( r(1) = \langle 2, 0, 3 \rangle \)
- \( r(2) = \langle 1, 1, 1 \rangle \)
- \( r(3) = \langle -2, 4, 1 \rangle \)
**Select one:**
- a. 5
- b. \( \langle 1, 1, 1 \rangle \)
- c. 3
- d. \( \langle -2, 2, -1 \rangle \)
- e. \( \langle -1, 3, 1 \rangle \)
---
**Solution Explanation:**
1. **Understanding Vectors:**
- Each \( r(t) \) represents the position of a particle in three-dimensional space at a given time \( t \).
2. **Average Velocity Calculation:**
- Average velocity is defined as the change in position divided by the change in time.
- Calculate the change in position:
\[
\Delta r = r(3) - r(1) = \langle -2, 4, 1 \rangle - \langle 2, 0, 3 \rangle = \langle -4, 4, -2 \rangle
\]
- Calculate the change in time:
\[
\Delta t = 3 - 1 = 2
\]
- Compute the average velocity:
\[
\text{Average Velocity} = \frac{\Delta r}{\Delta t} = \left\langle \frac{-4}{2}, \frac{4}{2}, \frac{-2}{2} \right\rangle = \langle -2, 2, -1 \rangle
\]
3. **Answer Identification:**
- The correct choice is d. \( \langle -2, 2, -1 \rangle \).
This problem explores the concept of average velocity in vector form,
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