The velocity of a (fast) automobile on a straight highway is given by the function to the right, where t is measured in seconds and v has units of m/s. 3t if 0st< 25 v(t) = { 75 if 25 st < 50 225 - 3t t250
The velocity of a (fast) automobile on a straight highway is given by the function to the right, where t is measured in seconds and v has units of m/s. 3t if 0st< 25 v(t) = { 75 if 25 st < 50 225 - 3t t250
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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![**Velocity of an Automobile on a Highway**
The velocity of a fast automobile on a straight highway is given by the following function, where \( t \) is measured in seconds and \( v \) has units of meters per second (m/s):
\[ v(t) =
\begin{cases}
3t & \text{if } 0 \leq t < 25 \\
75 & \text{if } 25 \leq t < 50 \\
225 - 3t & \text{if } t \geq 50
\end{cases} \]
### Explanation of the Velocity Function
The function \( v(t) \) describes the velocity of the automobile in different intervals of time:
1. **For \( 0 \leq t < 25 \):** The velocity \( v(t) \) increases linearly with time according to the equation \( v(t) = 3t \). This means that for every second, the velocity increases by 3 meters per second.
2. **For \( 25 \leq t < 50 \):** The velocity \( v(t) \) is constant at 75 meters per second. This indicates that the automobile maintains a steady speed of 75 m/s during this time interval.
3. **For \( t \geq 50 \):** The velocity \( v(t) \) decreases linearly with time according to the equation \( v(t) = 225 - 3t \). This indicates that the velocity decreases by 3 meters per second every second.
This piecewise function effectively describes how the automobile accelerates, maintains a constant speed, and then decelerates over successive intervals of time.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F777e0514-8160-4940-b57c-96884b4e2915%2F0fa49394-bef3-4cf3-81da-8b80971738f9%2Ffcpyadd_processed.png&w=3840&q=75)
Transcribed Image Text:**Velocity of an Automobile on a Highway**
The velocity of a fast automobile on a straight highway is given by the following function, where \( t \) is measured in seconds and \( v \) has units of meters per second (m/s):
\[ v(t) =
\begin{cases}
3t & \text{if } 0 \leq t < 25 \\
75 & \text{if } 25 \leq t < 50 \\
225 - 3t & \text{if } t \geq 50
\end{cases} \]
### Explanation of the Velocity Function
The function \( v(t) \) describes the velocity of the automobile in different intervals of time:
1. **For \( 0 \leq t < 25 \):** The velocity \( v(t) \) increases linearly with time according to the equation \( v(t) = 3t \). This means that for every second, the velocity increases by 3 meters per second.
2. **For \( 25 \leq t < 50 \):** The velocity \( v(t) \) is constant at 75 meters per second. This indicates that the automobile maintains a steady speed of 75 m/s during this time interval.
3. **For \( t \geq 50 \):** The velocity \( v(t) \) decreases linearly with time according to the equation \( v(t) = 225 - 3t \). This indicates that the velocity decreases by 3 meters per second every second.
This piecewise function effectively describes how the automobile accelerates, maintains a constant speed, and then decelerates over successive intervals of time.
![### Questions for Automotive Motion Analysis
#### c. Distance Traveled Calculation
**Question:** What is the distance traveled by the automobile in the first 75 seconds?
#### d. Position Determination
**Question:** What is the position of the automobile when \( t = 90 \) seconds?
**Notes:** The above questions are aimed at analyzing the motion of an automobile over a given time interval. To address these questions, consider using kinematic equations or any provided velocity-time graphs or data tables related to the automobile's motion. Calculations should account for initial velocities, accelerations, and any pertinent information described in the broader problem set.
##### Example Approach:
- For distance calculation, integrate the velocity over the given time period.
- For position determination, calculate the position using a known initial position and integrating velocity as a function of time up to the specified time of 90 seconds.
These calculations play a crucial role in understanding the motion dynamics of an automobile and are fundamental in physics and engineering contexts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F777e0514-8160-4940-b57c-96884b4e2915%2F0fa49394-bef3-4cf3-81da-8b80971738f9%2Fhcs3koe_processed.png&w=3840&q=75)
Transcribed Image Text:### Questions for Automotive Motion Analysis
#### c. Distance Traveled Calculation
**Question:** What is the distance traveled by the automobile in the first 75 seconds?
#### d. Position Determination
**Question:** What is the position of the automobile when \( t = 90 \) seconds?
**Notes:** The above questions are aimed at analyzing the motion of an automobile over a given time interval. To address these questions, consider using kinematic equations or any provided velocity-time graphs or data tables related to the automobile's motion. Calculations should account for initial velocities, accelerations, and any pertinent information described in the broader problem set.
##### Example Approach:
- For distance calculation, integrate the velocity over the given time period.
- For position determination, calculate the position using a known initial position and integrating velocity as a function of time up to the specified time of 90 seconds.
These calculations play a crucial role in understanding the motion dynamics of an automobile and are fundamental in physics and engineering contexts.
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