A particle moves according to a law of motionst), ta 0, where t is measured in seconds and s in feet. (If an answer does not exist, enter DNE.) ) =- 9r+ 24 (a) Find the velocity (in Us) at time t. vt) - 3(1-4)(1-2) n/s (b) What is the velocity (in f/s) after 1 second? 1)=9 (c) When is the partide at rest? (Enter your answers as a comma-separated list.) t=2,4 (d) When is the particle moving in the positive direction? (Enter your answer using interval notation.) (0,2)u (4,0) 4, (e) Draw a diagram to llustrate the motion of the particle and use it to find the total distance (in n) traveled during the first 6 seconds. (t) Find the acceleration (in ts) at time t.
A particle moves according to a law of motionst), ta 0, where t is measured in seconds and s in feet. (If an answer does not exist, enter DNE.) ) =- 9r+ 24 (a) Find the velocity (in Us) at time t. vt) - 3(1-4)(1-2) n/s (b) What is the velocity (in f/s) after 1 second? 1)=9 (c) When is the partide at rest? (Enter your answers as a comma-separated list.) t=2,4 (d) When is the particle moving in the positive direction? (Enter your answer using interval notation.) (0,2)u (4,0) 4, (e) Draw a diagram to llustrate the motion of the particle and use it to find the total distance (in n) traveled during the first 6 seconds. (t) Find the acceleration (in ts) at time t.
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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The question is in the image.
![```markdown
### Particle Motion Analysis
A particle moves according to a law of motion \( s = f(t) \), where \( t \) is measured in seconds and \( s \) in feet. If an answer does not exist, enter DNE.
**(a)** Given \( s(t) = 3t^2 - 2t^3 + 24 \), find the velocity \( v(t) \) at time \( t \).
\[
v(t) = \frac{d}{dt}[3t^2 - 2t^3 + 24] = 6t - 6t^2
\]
**(b)** Find the velocity \( v(1) \) after 1 second.
\[
v(1) = 6(1) - 6(1)^2 = 0
\]
**(c)** When is the particle at rest? (Enter your answers as a comma-separated list.)
To find when the particle is at rest, solve:
\[
6t - 6t^2 = 0
\]
\[
6t(1 - t) = 0
\]
\[
t = 0, 1
\]
**(d)** When is the particle moving in the positive direction? (Enter your answer using interval notation.)
The particle is moving in the positive direction when \( v(t) > 0 \).
\[
(0, 1) \cup (4, \infty)
\]
**(e)** Draw a diagram to illustrate the motion of the particle and use it to find the total distance (in ft) traveled during the first 6 seconds.
**(f)** Find the acceleration \( a(t) \) at time \( t \).
\[
a(t) = \frac{d}{dt}[6t - 6t^2] = 6 - 12t
\]
Find the acceleration \( a(1) \) after 1 second.
\[
a(1) = 6 - 12(1) = -6
\]
**(g)** Graph the position, velocity, and acceleration functions for \( 0 \leq t \leq 6 \).
#### Explanation of Graph
- **Position Function**: The blue curve represents the position over time, showing how the particle's location changes.
- **Velocity Function**: The red curve](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F453241d9-c72e-47dc-8569-5d82992f1349%2Fb5986104-131d-419f-9751-6ab3d04455f4%2F1yw1ryj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:```markdown
### Particle Motion Analysis
A particle moves according to a law of motion \( s = f(t) \), where \( t \) is measured in seconds and \( s \) in feet. If an answer does not exist, enter DNE.
**(a)** Given \( s(t) = 3t^2 - 2t^3 + 24 \), find the velocity \( v(t) \) at time \( t \).
\[
v(t) = \frac{d}{dt}[3t^2 - 2t^3 + 24] = 6t - 6t^2
\]
**(b)** Find the velocity \( v(1) \) after 1 second.
\[
v(1) = 6(1) - 6(1)^2 = 0
\]
**(c)** When is the particle at rest? (Enter your answers as a comma-separated list.)
To find when the particle is at rest, solve:
\[
6t - 6t^2 = 0
\]
\[
6t(1 - t) = 0
\]
\[
t = 0, 1
\]
**(d)** When is the particle moving in the positive direction? (Enter your answer using interval notation.)
The particle is moving in the positive direction when \( v(t) > 0 \).
\[
(0, 1) \cup (4, \infty)
\]
**(e)** Draw a diagram to illustrate the motion of the particle and use it to find the total distance (in ft) traveled during the first 6 seconds.
**(f)** Find the acceleration \( a(t) \) at time \( t \).
\[
a(t) = \frac{d}{dt}[6t - 6t^2] = 6 - 12t
\]
Find the acceleration \( a(1) \) after 1 second.
\[
a(1) = 6 - 12(1) = -6
\]
**(g)** Graph the position, velocity, and acceleration functions for \( 0 \leq t \leq 6 \).
#### Explanation of Graph
- **Position Function**: The blue curve represents the position over time, showing how the particle's location changes.
- **Velocity Function**: The red curve
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