Suppose that S(t) = t³ – 12t m. is the position of a particle at time t (in seconds) on a line. (a) Is the position increasing or decreasing at time t = 1? (b) What is the displacement from t = 0 to t = 3. (c)What is the total distance travelled between t = 0 and t = 3.
Suppose that S(t) = t³ – 12t m. is the position of a particle at time t (in seconds) on a line. (a) Is the position increasing or decreasing at time t = 1? (b) What is the displacement from t = 0 to t = 3. (c)What is the total distance travelled between t = 0 and t = 3.
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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**Particle Motion Analysis**
Consider the function \( S(t) = t^3 - 12t \) meters, representing the position \( S \) of a particle at time \( t \) (in seconds) on a line.
**Tasks:**
1. **Determine Velocity at \( t = 1 \):**
- **Question:** Is the position increasing or decreasing at time \( t = 1 \)?
- **Method:** Calculate the derivative \( S'(t) \) to find the velocity at \( t = 1 \).
2. **Calculate Displacement from \( t = 0 \) to \( t = 3 \):**
- **Question:** What is the displacement from \( t = 0 \) to \( t = 3 \)?
- **Method:** Evaluate \( S(3) - S(0) \).
3. **Determine Total Distance Traveled from \( t = 0 \) to \( t = 3 \):**
- **Question:** What is the total distance traveled between \( t = 0 \) and \( t = 3 \)?
- **Method:** Identify intervals of motion direction changes by solving \( S'(t) = 0 \), then evaluate and sum the absolute values of position changes over each interval.
**Detailed Steps and Explanations:**
(a) **Is the position increasing or decreasing at time \( t = 1 \)?**
- To determine whether the position is increasing or decreasing, calculate the velocity:
\[
S'(t) = 3t^2 - 12
\]
Evaluate \( S'(1) \):
\[
S'(1) = 3(1)^2 - 12 = 3 - 12 = -9
\]
Since \( S'(1) < 0 \), the position is decreasing at \( t = 1 \).
(b) **What is the displacement from \( t = 0 \) to \( t = 3 \)?**
- Displacement is the difference in position from the start to the end time.
\[
S(0) = (0)^3 - 12(0) = 0
\]
\[
S(3) = (3)^3 - 12(3) = 27 - 36 = -9
\]
Displacement](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc438c34-6e26-4f6a-9067-f367a43a4982%2F348cd5e6-bc5b-4668-9e58-befc7d55029a%2Fwie8n4.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Particle Motion Analysis**
Consider the function \( S(t) = t^3 - 12t \) meters, representing the position \( S \) of a particle at time \( t \) (in seconds) on a line.
**Tasks:**
1. **Determine Velocity at \( t = 1 \):**
- **Question:** Is the position increasing or decreasing at time \( t = 1 \)?
- **Method:** Calculate the derivative \( S'(t) \) to find the velocity at \( t = 1 \).
2. **Calculate Displacement from \( t = 0 \) to \( t = 3 \):**
- **Question:** What is the displacement from \( t = 0 \) to \( t = 3 \)?
- **Method:** Evaluate \( S(3) - S(0) \).
3. **Determine Total Distance Traveled from \( t = 0 \) to \( t = 3 \):**
- **Question:** What is the total distance traveled between \( t = 0 \) and \( t = 3 \)?
- **Method:** Identify intervals of motion direction changes by solving \( S'(t) = 0 \), then evaluate and sum the absolute values of position changes over each interval.
**Detailed Steps and Explanations:**
(a) **Is the position increasing or decreasing at time \( t = 1 \)?**
- To determine whether the position is increasing or decreasing, calculate the velocity:
\[
S'(t) = 3t^2 - 12
\]
Evaluate \( S'(1) \):
\[
S'(1) = 3(1)^2 - 12 = 3 - 12 = -9
\]
Since \( S'(1) < 0 \), the position is decreasing at \( t = 1 \).
(b) **What is the displacement from \( t = 0 \) to \( t = 3 \)?**
- Displacement is the difference in position from the start to the end time.
\[
S(0) = (0)^3 - 12(0) = 0
\]
\[
S(3) = (3)^3 - 12(3) = 27 - 36 = -9
\]
Displacement
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