Culator. A particle is moving with velocity v(t) = 1²- 9t+18 with distance, s measured in meters, left or right of zero, and t measured in seconds, with t between 0 and 8 seconds inclusive. The position at time t = 0 sec is 1 meter right of zero, that is, s(0) = 1. 5. Find the total distance the particle has traveled between 0 and 8 seconds
Culator. A particle is moving with velocity v(t) = 1²- 9t+18 with distance, s measured in meters, left or right of zero, and t measured in seconds, with t between 0 and 8 seconds inclusive. The position at time t = 0 sec is 1 meter right of zero, that is, s(0) = 1. 5. Find the total distance the particle has traveled between 0 and 8 seconds
Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
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![### Motion and Distance Calculation
A particle is moving with velocity \( v(t) = t^2 - 8t + 18 \), where \( v(t) \) is the velocity in meters per second (m/s), and \( t \) is the time in seconds (s). We are tasked with finding the total distance traveled by the particle between time \( t = 0 \) seconds and \( t = 8 \) seconds inclusive.
The position at time \( t = 0 \) seconds is 1 meter to the right of the origin. Thus, the initial position \( s(0) \) is given by:
\[ s(0) = 1 \]
The position function \( s(t) \), which gives the distance in meters (m), left or right of zero, is derived from the velocity function by integration:
\[ s(t) = \int v(t) \, dt \]
Substituting the given velocity function:
\[ v(t) = t^2 - 8t + 18 \]
we integrate:
\[ s(t) = \int (t^2 - 8t + 18) \, dt \]
Solving this integral will give us the position function \( s(t) \). To find the total distance traveled, we also need to consider the magnitude of movement. It involves calculating the definite integral of the absolute value of the velocity function over the interval from \( t = 0 \) to \( t = 8 \), and adding the absolute changes in position.
### Detailed Steps
1. Integrate the velocity function \( v(t) \) to get the position function \( s(t) \).
2. Evaluate the position function at critical points and end points \( t = 0 \) and \( t = 8 \).
3. Calculate the total distance traveled, ensuring all distances are considered in absolute values to account for changes in direction.
Please refer to additional mathematical resources or textbooks for the detailed calculations and integration steps necessary to derive the position function and the total distance traveled.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4bf8d9f8-9f3c-4f11-9f6a-4feeeac00eb3%2Fe01f4fce-dc19-45ad-8d6d-1c9d72541478%2Fla7plvf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Motion and Distance Calculation
A particle is moving with velocity \( v(t) = t^2 - 8t + 18 \), where \( v(t) \) is the velocity in meters per second (m/s), and \( t \) is the time in seconds (s). We are tasked with finding the total distance traveled by the particle between time \( t = 0 \) seconds and \( t = 8 \) seconds inclusive.
The position at time \( t = 0 \) seconds is 1 meter to the right of the origin. Thus, the initial position \( s(0) \) is given by:
\[ s(0) = 1 \]
The position function \( s(t) \), which gives the distance in meters (m), left or right of zero, is derived from the velocity function by integration:
\[ s(t) = \int v(t) \, dt \]
Substituting the given velocity function:
\[ v(t) = t^2 - 8t + 18 \]
we integrate:
\[ s(t) = \int (t^2 - 8t + 18) \, dt \]
Solving this integral will give us the position function \( s(t) \). To find the total distance traveled, we also need to consider the magnitude of movement. It involves calculating the definite integral of the absolute value of the velocity function over the interval from \( t = 0 \) to \( t = 8 \), and adding the absolute changes in position.
### Detailed Steps
1. Integrate the velocity function \( v(t) \) to get the position function \( s(t) \).
2. Evaluate the position function at critical points and end points \( t = 0 \) and \( t = 8 \).
3. Calculate the total distance traveled, ensuring all distances are considered in absolute values to account for changes in direction.
Please refer to additional mathematical resources or textbooks for the detailed calculations and integration steps necessary to derive the position function and the total distance traveled.
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