A roller coaster travels upward along the path r(t) = (t, t², ³) (p n meters) (a) Compute the acceleration at t = 4 seconds (include units) (b) poloration into ita compo tial omponents
A roller coaster travels upward along the path r(t) = (t, t², ³) (p n meters) (a) Compute the acceleration at t = 4 seconds (include units) (b) poloration into ita compo tial omponents
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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Can you please do this step by step and don't miss any steps please so I can follow along and can you label them
![### Problem
A roller coaster travels upward along the path \(\vec{r}(t) = \langle t, \frac{1}{2}t^2, \frac{1}{6} t^3 \rangle \) (position here is measured in meters)
#### Questions
(a) Compute the acceleration at \( t = 4 \) seconds (include units).
(b) Decompose the acceleration into its tangential and normal components.
### Solution Approach
To solve this problem, we'll need to take the following steps:
1. **Compute the position function \(\vec{r}(t)\):**
\[
\vec{r}(t) = \langle t, \frac{1}{2} t^2, \frac{1}{6} t^3 \rangle
\]
2. **Find the velocity vector \(\vec{v}(t)\) by differentiating \(\vec{r}(t)\):**
\[
\vec{v}(t) = \frac{d}{dt} \vec{r}(t) = \left\langle 1, t, \frac{1}{2} t^2 \right\rangle
\]
3. **Find the acceleration vector \(\vec{a}(t)\) by differentiating \(\vec{v}(t)\):**
\[
\vec{a}(t) = \frac{d}{dt} \vec{v}(t) = \left\langle 0, 1, t \right\rangle
\]
4. **Evaluate \(\vec{a}(t)\) at \( t = 4 \) seconds:**
\[
\vec{a}(4) = \left\langle 0, 1, 4 \right\rangle \, \text{meters per second squared (m/s\(^2\))}
\]
5. **Decompose the acceleration into tangential and normal components:**
- **Tangential Component (a\(_T\)):** This is the component of the acceleration in the direction of the velocity vector.
\[
a_T = \frac{\vec{a}(t) \cdot \vec{v}(t)}{||\vec{v}(t)||}
\]
- **Normal Component (a\(_N\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb937826-6739-4e54-9ce3-419aba968b7a%2Feb7e903c-f80a-4848-9ccd-37b6f593200d%2Flip89ae_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem
A roller coaster travels upward along the path \(\vec{r}(t) = \langle t, \frac{1}{2}t^2, \frac{1}{6} t^3 \rangle \) (position here is measured in meters)
#### Questions
(a) Compute the acceleration at \( t = 4 \) seconds (include units).
(b) Decompose the acceleration into its tangential and normal components.
### Solution Approach
To solve this problem, we'll need to take the following steps:
1. **Compute the position function \(\vec{r}(t)\):**
\[
\vec{r}(t) = \langle t, \frac{1}{2} t^2, \frac{1}{6} t^3 \rangle
\]
2. **Find the velocity vector \(\vec{v}(t)\) by differentiating \(\vec{r}(t)\):**
\[
\vec{v}(t) = \frac{d}{dt} \vec{r}(t) = \left\langle 1, t, \frac{1}{2} t^2 \right\rangle
\]
3. **Find the acceleration vector \(\vec{a}(t)\) by differentiating \(\vec{v}(t)\):**
\[
\vec{a}(t) = \frac{d}{dt} \vec{v}(t) = \left\langle 0, 1, t \right\rangle
\]
4. **Evaluate \(\vec{a}(t)\) at \( t = 4 \) seconds:**
\[
\vec{a}(4) = \left\langle 0, 1, 4 \right\rangle \, \text{meters per second squared (m/s\(^2\))}
\]
5. **Decompose the acceleration into tangential and normal components:**
- **Tangential Component (a\(_T\)):** This is the component of the acceleration in the direction of the velocity vector.
\[
a_T = \frac{\vec{a}(t) \cdot \vec{v}(t)}{||\vec{v}(t)||}
\]
- **Normal Component (a\(_N\
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