A roller coaster travels upward along the path r(t) = (t, t², ³) (position here is measured in meters) (a) Compute the acceleration at t = 4 seconds (include units) (b) Decompose the acceleration into its tangential and normal components.
A roller coaster travels upward along the path r(t) = (t, t², ³) (position here is measured in meters) (a) Compute the acceleration at t = 4 seconds (include units) (b) Decompose the acceleration into its tangential and normal components.
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Can you help me with this problem, can you do this step by step so I can understand it better and not skip any steps and can you label them
![**Problem Description:**
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).
(a) Compute the acceleration at \( t = 4 \) seconds (include units).
(b) Decompose the acceleration into its tangential and normal components.
**Solution:**
To approach this problem, we will first calculate the acceleration vector by differentiating the position vector twice with respect to time.
**Step-by-step Process:**
1. **Position Vector:**
\(\vec{r}(t) = \langle t, \frac{1}{2}t^2, \frac{1}{6}t^3 \rangle\)
2. **Velocity Vector:**
The velocity vector \(\vec{v}(t)\) is obtained by differentiating the position vector \(\vec{r}(t)\) with respect to time \(t\):
\[\vec{v}(t) = \frac{d}{dt} [\vec{r}(t)] = \left\langle \frac{d}{dt}[t], \frac{d}{dt}\left[\frac{1}{2}t^2\right], \frac{d}{dt}\left[\frac{1}{6}t^3\right] \right\rangle\]
\[\vec{v}(t) = \langle 1, t, \frac{1}{2}t^2 \rangle\]
3. **Acceleration Vector:**
The acceleration vector \(\vec{a}(t)\) is obtained by differentiating the velocity vector \(\vec{v}(t)\) with respect to time \(t\):
\[\vec{a}(t) = \frac{d}{dt} [\vec{v}(t)] = \left\langle \frac{d}{dt}[1], \frac{d}{dt}[t], \frac{d}{dt}\left[\frac{1}{2}t^2\right] \right\rangle\]
\[\vec{a}(t) = \langle 0, 1, t \rangle\]
4. **Acceleration at \(t = 4\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7f4d9a86-377e-4713-8398-ed20a45cabed%2Fc69b120f-6632-4144-b1d8-56cb69b77ad4%2Feelp5jj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
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).
(a) Compute the acceleration at \( t = 4 \) seconds (include units).
(b) Decompose the acceleration into its tangential and normal components.
**Solution:**
To approach this problem, we will first calculate the acceleration vector by differentiating the position vector twice with respect to time.
**Step-by-step Process:**
1. **Position Vector:**
\(\vec{r}(t) = \langle t, \frac{1}{2}t^2, \frac{1}{6}t^3 \rangle\)
2. **Velocity Vector:**
The velocity vector \(\vec{v}(t)\) is obtained by differentiating the position vector \(\vec{r}(t)\) with respect to time \(t\):
\[\vec{v}(t) = \frac{d}{dt} [\vec{r}(t)] = \left\langle \frac{d}{dt}[t], \frac{d}{dt}\left[\frac{1}{2}t^2\right], \frac{d}{dt}\left[\frac{1}{6}t^3\right] \right\rangle\]
\[\vec{v}(t) = \langle 1, t, \frac{1}{2}t^2 \rangle\]
3. **Acceleration Vector:**
The acceleration vector \(\vec{a}(t)\) is obtained by differentiating the velocity vector \(\vec{v}(t)\) with respect to time \(t\):
\[\vec{a}(t) = \frac{d}{dt} [\vec{v}(t)] = \left\langle \frac{d}{dt}[1], \frac{d}{dt}[t], \frac{d}{dt}\left[\frac{1}{2}t^2\right] \right\rangle\]
\[\vec{a}(t) = \langle 0, 1, t \rangle\]
4. **Acceleration at \(t = 4\)
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