Given the vector function r(t) = (t², 9t — 2, 5 — ²), find the tangential and normal components of acceleration aT and aN, at the point when t = 1. aT = aN = Round your answers to two decimal places.
Given the vector function r(t) = (t², 9t — 2, 5 — ²), find the tangential and normal components of acceleration aT and aN, at the point when t = 1. aT = aN = Round your answers to two decimal places.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 39RE
Related questions
Question
![### Problem Statement
Given the vector function \(\vec{r}(t) = \langle t^2, 9t - 2, 5 - t^2 \rangle\), find the tangential and normal components of acceleration \(\alpha_T\) and \(\alpha_N\), at the point when \(t = 1\).
#### Calculation
- **Tangential Component of Acceleration, \(\alpha_T\):**
\[
\alpha_T = \quad \_\_\_\_\_\_\_\_\_\_
\]
- **Normal Component of Acceleration, \(\alpha_N\):**
\[
\alpha_N = \quad \_\_\_\_\_\_\_\_
\]
Round your answers to two decimal places.
### Explanation:
1. **Tangential Component of Acceleration (\(\alpha_T\)):**
This is the component of acceleration along the direction of the velocity vector. It is calculated using the derivative of the speed with respect to time.
2. **Normal Component of Acceleration (\(\alpha_N\)):**
This is the component of acceleration perpendicular to the velocity vector. It is associated with changes in the direction of the velocity vector.
### Steps to Solve:
1. Compute the velocity vector \(\vec{v}(t)\) by differentiating the position vector \(\vec{r}(t)\).
2. Compute the speed \(||\vec{v}(t)||\).
3. Differentiate the speed to find the tangential acceleration \(\alpha_T\).
4. Compute the acceleration vector \(\vec{a}(t)\) by differentiating the velocity vector \(\vec{v}(t)\).
5. Use the velocity and acceleration vectors to find the normal acceleration \(\alpha_N\).
#### Note:
Values for \(\alpha_T\) and \(\alpha_N\) should be computed when \(t = 1\) and rounded to two decimal places.
### Additional Resources:
- Vector Calculus
- Tangential and Normal Components of Acceleration in Curvilinear Motion](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe4962ae7-1cb2-4276-8f6a-b9f851b67289%2F2c83370f-a4ef-4959-ab9e-f50133a79039%2Fp8br8aj_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Given the vector function \(\vec{r}(t) = \langle t^2, 9t - 2, 5 - t^2 \rangle\), find the tangential and normal components of acceleration \(\alpha_T\) and \(\alpha_N\), at the point when \(t = 1\).
#### Calculation
- **Tangential Component of Acceleration, \(\alpha_T\):**
\[
\alpha_T = \quad \_\_\_\_\_\_\_\_\_\_
\]
- **Normal Component of Acceleration, \(\alpha_N\):**
\[
\alpha_N = \quad \_\_\_\_\_\_\_\_
\]
Round your answers to two decimal places.
### Explanation:
1. **Tangential Component of Acceleration (\(\alpha_T\)):**
This is the component of acceleration along the direction of the velocity vector. It is calculated using the derivative of the speed with respect to time.
2. **Normal Component of Acceleration (\(\alpha_N\)):**
This is the component of acceleration perpendicular to the velocity vector. It is associated with changes in the direction of the velocity vector.
### Steps to Solve:
1. Compute the velocity vector \(\vec{v}(t)\) by differentiating the position vector \(\vec{r}(t)\).
2. Compute the speed \(||\vec{v}(t)||\).
3. Differentiate the speed to find the tangential acceleration \(\alpha_T\).
4. Compute the acceleration vector \(\vec{a}(t)\) by differentiating the velocity vector \(\vec{v}(t)\).
5. Use the velocity and acceleration vectors to find the normal acceleration \(\alpha_N\).
#### Note:
Values for \(\alpha_T\) and \(\alpha_N\) should be computed when \(t = 1\) and rounded to two decimal places.
### Additional Resources:
- Vector Calculus
- Tangential and Normal Components of Acceleration in Curvilinear Motion
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