The position of an object at time tt is given by r(t)r(t). Find the velocity, speed, and acceleration both as functions of tt and at the indicated value of tt. r(t)=⟨3sin(t),4cos(t)⟩ , t=π4 2) Find the position function for the object with the given acceleration, initial velocity, and initial position. a(t)=6ti+cos(2t)j−sin(4t)k, v(0)=j−k, r(0)=i, a(t)=6ti+cos(2t)j−sin(4t)k, v(0)=j−k, r(0)=i, Position Function
The position of an object at time tt is given by r(t)r(t). Find the velocity, speed, and acceleration both as functions of tt and at the indicated value of tt. r(t)=⟨3sin(t),4cos(t)⟩ , t=π4 2) Find the position function for the object with the given acceleration, initial velocity, and initial position. a(t)=6ti+cos(2t)j−sin(4t)k, v(0)=j−k, r(0)=i, a(t)=6ti+cos(2t)j−sin(4t)k, v(0)=j−k, r(0)=i, Position Function
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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1) The position of an object at time tt is given by r(t)r(t). Find the velocity, speed, and acceleration both as functions of tt and at the indicated value of tt.
r(t)=⟨3sin(t),4cos(t)⟩ , t=π4
2) Find the position function for the object with the given acceleration, initial velocity, and initial position.
a(t)=6ti+cos(2t)j−sin(4t)k, v(0)=j−k, r(0)=i, a(t)=6ti+cos(2t)j−sin(4t)k, v(0)=j−k, r(0)=i,
Position Function
![### Problem: Finding the Position Function
**Question: (1 point)**
Find the position function for the object with the given acceleration, initial velocity, and initial position.
Given data:
\[ \mathbf{a}(t) = 6t\mathbf{i} + \cos(2t)\mathbf{j} - \sin(4t)\mathbf{k} \]
\[ \mathbf{v}(0) = \mathbf{j} - \mathbf{k} \]
\[ \mathbf{r}(0) = \mathbf{i} \]
**Input Required:**
- Position Function
**Solution Placeholder:**
\[ \text{Position Function} \] (Text box for input without value filled in)
**Explanation:**
To solve this, you need to integrate the acceleration function to find the velocity function, and then integrate the velocity function to find the position function. Make sure to use the given initial conditions to find any constants of integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb9845818-362e-4d49-93db-7dbcd6513608%2Fea93a8e4-413e-4c0f-80c4-cfd6931047dc%2Fwgbag2_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem: Finding the Position Function
**Question: (1 point)**
Find the position function for the object with the given acceleration, initial velocity, and initial position.
Given data:
\[ \mathbf{a}(t) = 6t\mathbf{i} + \cos(2t)\mathbf{j} - \sin(4t)\mathbf{k} \]
\[ \mathbf{v}(0) = \mathbf{j} - \mathbf{k} \]
\[ \mathbf{r}(0) = \mathbf{i} \]
**Input Required:**
- Position Function
**Solution Placeholder:**
\[ \text{Position Function} \] (Text box for input without value filled in)
**Explanation:**
To solve this, you need to integrate the acceleration function to find the velocity function, and then integrate the velocity function to find the position function. Make sure to use the given initial conditions to find any constants of integration.
![**(1 point)** The position of an object at time \( t \) is given by \( \mathbf{r}(t) \). Find the velocity, speed, and acceleration both as functions of \( t \) and at the indicated value of \( t \).
\[
\mathbf{r}(t) = \big( 3\sin(t), 4\cos(t) \big) , \quad t = \frac{\pi}{4}
\]
**Velocity Function** \(\longrightarrow\) \_\_\_\_\_\_\_\_\_\_\_
**Velocity at** \( t = \frac{\pi}{4} \) \(\longrightarrow\) \_\_\_\_\_\_\_\_\_\_\_
**Speed Function** \(\longrightarrow\) \_\_\_\_\_\_\_\_\_\_\_
**Speed at** \( t = \frac{\pi}{4} \) \(\longrightarrow\) \_\_\_\_\_\_\_\_\_\_\_
**Acceleration Function** \(\longrightarrow\) \_\_\_\_\_\_\_\_\_\_\_
**Acceleration at** \( t = \frac{\pi}{4} \) \(\longrightarrow\) \_\_\_\_\_\_\_\_\_\_\_
[link to help page on vectors](#)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb9845818-362e-4d49-93db-7dbcd6513608%2Fea93a8e4-413e-4c0f-80c4-cfd6931047dc%2Fce20jtt_processed.png&w=3840&q=75)
Transcribed Image Text:**(1 point)** The position of an object at time \( t \) is given by \( \mathbf{r}(t) \). Find the velocity, speed, and acceleration both as functions of \( t \) and at the indicated value of \( t \).
\[
\mathbf{r}(t) = \big( 3\sin(t), 4\cos(t) \big) , \quad t = \frac{\pi}{4}
\]
**Velocity Function** \(\longrightarrow\) \_\_\_\_\_\_\_\_\_\_\_
**Velocity at** \( t = \frac{\pi}{4} \) \(\longrightarrow\) \_\_\_\_\_\_\_\_\_\_\_
**Speed Function** \(\longrightarrow\) \_\_\_\_\_\_\_\_\_\_\_
**Speed at** \( t = \frac{\pi}{4} \) \(\longrightarrow\) \_\_\_\_\_\_\_\_\_\_\_
**Acceleration Function** \(\longrightarrow\) \_\_\_\_\_\_\_\_\_\_\_
**Acceleration at** \( t = \frac{\pi}{4} \) \(\longrightarrow\) \_\_\_\_\_\_\_\_\_\_\_
[link to help page on vectors](#)
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