The motion of a point on the circumference of a rolling wheel of radius 4 feet is described by the vector function r(t) = 4(12t - sin(12t))i + 4(1 − cos(12t))] Find the velocity vector of the point. v(t) Find the acceleration vector of the point. ä(t) Find the speed of the point. s(t) =
The motion of a point on the circumference of a rolling wheel of radius 4 feet is described by the vector function r(t) = 4(12t - sin(12t))i + 4(1 − cos(12t))] Find the velocity vector of the point. v(t) Find the acceleration vector of the point. ä(t) Find the speed of the point. s(t) =
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![### Vector Calculus Problem: Motion on the Circumference of a Rolling Wheel
The motion of a point on the circumference of a rolling wheel of radius 4 feet is described by the vector function:
\[ \vec{r}(t) = 4(12t - \sin(12t))\hat{i} + 4(1 - \cos(12t))\hat{j} \]
#### Find the Velocity Vector of the Point
\[ \vec{v}(t) = \]
#### Find the Acceleration Vector of the Point
\[ \vec{a}(t) = \]
#### Find the Speed of the Point
\[ s(t) = \]
### Explanation:
- \(\vec{r}(t)\) represents the position vector as a function of time \(t\).
- The term \(4(12t - \sin(12t))\) is the x-component of the vector, denoted with \(\hat{i}\).
- The term \(4(1 - \cos(12t))\) is the y-component of the vector, denoted with \(\hat{j}\).
### Steps to Solve:
1. **Calculate the Velocity Vector \(\vec{v}(t)\):**
\[
\vec{v}(t) = \frac{d\vec{r}(t)}{dt}
\]
- Differentiate the x-component \(4(12t - \sin(12t))\) with respect to \(t\).
- Differentiate the y-component \(4(1 - \cos(12t))\) with respect to \(t\).
2. **Calculate the Acceleration Vector \(\vec{a}(t)\):**
\[
\vec{a}(t) = \frac{d\vec{v}(t)}{dt}
\]
- Differentiate the x-component of \(\vec{v}(t)\) with respect to \(t\).
- Differentiate the y-component of \(\vec{v}(t)\) with respect to \(t\).
3. **Calculate the Speed \(s(t)\):**
\[
s(t) = \|\vec{v}(t)\|
\]
- Find the magnitude of the velocity vector \(\vec{v}(t)\).
Please input your answers in the designated fields to complete the vector functions](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74b6836e-7075-4604-8b7e-b1441272473c%2Fe17a8c25-cfcf-45bf-b586-b051e0a6d2f7%2Fk4com1_processed.png&w=3840&q=75)
Transcribed Image Text:### Vector Calculus Problem: Motion on the Circumference of a Rolling Wheel
The motion of a point on the circumference of a rolling wheel of radius 4 feet is described by the vector function:
\[ \vec{r}(t) = 4(12t - \sin(12t))\hat{i} + 4(1 - \cos(12t))\hat{j} \]
#### Find the Velocity Vector of the Point
\[ \vec{v}(t) = \]
#### Find the Acceleration Vector of the Point
\[ \vec{a}(t) = \]
#### Find the Speed of the Point
\[ s(t) = \]
### Explanation:
- \(\vec{r}(t)\) represents the position vector as a function of time \(t\).
- The term \(4(12t - \sin(12t))\) is the x-component of the vector, denoted with \(\hat{i}\).
- The term \(4(1 - \cos(12t))\) is the y-component of the vector, denoted with \(\hat{j}\).
### Steps to Solve:
1. **Calculate the Velocity Vector \(\vec{v}(t)\):**
\[
\vec{v}(t) = \frac{d\vec{r}(t)}{dt}
\]
- Differentiate the x-component \(4(12t - \sin(12t))\) with respect to \(t\).
- Differentiate the y-component \(4(1 - \cos(12t))\) with respect to \(t\).
2. **Calculate the Acceleration Vector \(\vec{a}(t)\):**
\[
\vec{a}(t) = \frac{d\vec{v}(t)}{dt}
\]
- Differentiate the x-component of \(\vec{v}(t)\) with respect to \(t\).
- Differentiate the y-component of \(\vec{v}(t)\) with respect to \(t\).
3. **Calculate the Speed \(s(t)\):**
\[
s(t) = \|\vec{v}(t)\|
\]
- Find the magnitude of the velocity vector \(\vec{v}(t)\).
Please input your answers in the designated fields to complete the vector functions
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