Find the decomposition of a(t) into tangential and normal components at the point indicated. r(t) = = (3-t,t +5,1²), t = 2
Find the decomposition of a(t) into tangential and normal components at the point indicated. r(t) = = (3-t,t +5,1²), t = 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem Statement:**
Find the decomposition of \( \mathbf{a}(t) \) into tangential and normal components at the point indicated.
\[ \mathbf{r}(t) = \langle 3 - t, t + 5, t^2 \rangle, \quad t = 2 \]
*(Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.)*
**Answers:**
1. **Tangential Component \( \mathbf{a_T} \):**
Input: \( \langle 0, 0, \frac{16t}{\sqrt{18}} \rangle \)
Status: Incorrect
2. **Normal Component \( \mathbf{a_N} \):**
Input: \( \langle \, \, \rangle \)
Status: Incorrect
**Explanation:**
This problem involves vector calculus. To decompose the acceleration vector into tangential and normal components, one needs to follow these steps:
- Compute the velocity vector \( \mathbf{v}(t) = \mathbf{r}'(t) \).
- Calculate the acceleration vector \( \mathbf{a}(t) = \mathbf{v}'(t) \).
- Find the tangential component \( \mathbf{a_T} \) using the formula \( \mathbf{a_T} = \frac{\mathbf{a} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v} \).
- Determine the normal component \( \mathbf{a_N} = \mathbf{a} - \mathbf{a_T} \).
The student needs to recompute these values, ensuring the components are expressed accurately in the exact form requested.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4ff74c58-eecb-4627-bed6-d22e34f963af%2Fb997af3f-43e0-4769-b6f3-98b0ce79c091%2F0obn9sh_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the decomposition of \( \mathbf{a}(t) \) into tangential and normal components at the point indicated.
\[ \mathbf{r}(t) = \langle 3 - t, t + 5, t^2 \rangle, \quad t = 2 \]
*(Give your answer using component form or standard basis vectors. Express numbers in exact form. Use symbolic notation and fractions where needed.)*
**Answers:**
1. **Tangential Component \( \mathbf{a_T} \):**
Input: \( \langle 0, 0, \frac{16t}{\sqrt{18}} \rangle \)
Status: Incorrect
2. **Normal Component \( \mathbf{a_N} \):**
Input: \( \langle \, \, \rangle \)
Status: Incorrect
**Explanation:**
This problem involves vector calculus. To decompose the acceleration vector into tangential and normal components, one needs to follow these steps:
- Compute the velocity vector \( \mathbf{v}(t) = \mathbf{r}'(t) \).
- Calculate the acceleration vector \( \mathbf{a}(t) = \mathbf{v}'(t) \).
- Find the tangential component \( \mathbf{a_T} \) using the formula \( \mathbf{a_T} = \frac{\mathbf{a} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v} \).
- Determine the normal component \( \mathbf{a_N} = \mathbf{a} - \mathbf{a_T} \).
The student needs to recompute these values, ensuring the components are expressed accurately in the exact form requested.
Expert Solution
Step 1
Given a position vector .
To Find : Decomposition of in terms of tangential and normal component.
Step by step
Solved in 2 steps
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