Problem 4. A particle moves back and forth along a straight line with a velocity of v(t) = 1-² miles/hour. (a) How far is the particle from its original position 2 hours later? (b) What is the average velocity during the first 2 hours.
Problem 4. A particle moves back and forth along a straight line with a velocity of v(t) = 1-² miles/hour. (a) How far is the particle from its original position 2 hours later? (b) What is the average velocity during the first 2 hours.
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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![### Problem 4 Description
In this problem, we examine the motion of a particle that travels back and forth along a straight line. The velocity function of the particle is given by:
\[ v(t) = 1 - t^2 \, \text{miles/hour}. \]
We need to solve the following:
**(a)** Determine how far the particle is from its original position after 2 hours.
**(b)** Calculate the average velocity of the particle during the first 2 hours.
### Detailed Analysis
**Part (a): Distance from the Original Position**
To determine how far the particle is from its starting position, we will integrate the velocity function \( v(t) \) over the interval from \( t = 0 \) to \( t = 2 \) hours. This integral will give us the displacement.
**Part (b): Average Velocity Calculation**
The average velocity over a time interval can be found by dividing the total displacement by the time duration. This involves integrating the velocity function over the specified range and then dividing by the length of the interval (which is 2 hours in this case).
By understanding and solving these two parts, we can grasp both the particle's displacement and its average behavior over the given time period.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4526b347-f4c2-4959-b519-c79fc40f5b52%2F4d508cb3-7189-4502-a5d8-9b2c61d8e33b%2F5k6cc1j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 4 Description
In this problem, we examine the motion of a particle that travels back and forth along a straight line. The velocity function of the particle is given by:
\[ v(t) = 1 - t^2 \, \text{miles/hour}. \]
We need to solve the following:
**(a)** Determine how far the particle is from its original position after 2 hours.
**(b)** Calculate the average velocity of the particle during the first 2 hours.
### Detailed Analysis
**Part (a): Distance from the Original Position**
To determine how far the particle is from its starting position, we will integrate the velocity function \( v(t) \) over the interval from \( t = 0 \) to \( t = 2 \) hours. This integral will give us the displacement.
**Part (b): Average Velocity Calculation**
The average velocity over a time interval can be found by dividing the total displacement by the time duration. This involves integrating the velocity function over the specified range and then dividing by the length of the interval (which is 2 hours in this case).
By understanding and solving these two parts, we can grasp both the particle's displacement and its average behavior over the given time period.
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