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Need help with parts d and e.
![**High-Speed Train Construction for Maximum Comfort**
A billionaire plans to build a high-speed train that will travel between Seattle and Los Angeles (approximately 2000 km) in 2 hours. The goal is to ensure maximum comfort during the ride.
Some physicists suggest that the least uncomfortable way to travel a distance, \( D \), in time, \( T \), follows a specific mathematical model. The position \( s(t) \) at time \( t \) is given by:
\[ s(t) = 3 \cdot A \cdot t^2 - 2 \cdot B \cdot t^3 \]
### Tasks:
(a) **Dimensional Analysis:**
- The expression for \( s(t) \) must have dimensions of length. Constants \( A \) and \( B \) are different combinations of distance \( D \) and time \( T \).
- Use dimensional analysis to determine these combinations (numerical factors are already correct). Determine the appropriate units for \( A \) and \( B \).
(b) **Velocity and Acceleration:**
- Find symbolic expressions for velocity \( v(t) \) and acceleration \( a(t) \) as functions of time using \( s(t) \).
- Confirm that all units align correctly using results from part (a).
(c) **Determine Constants:**
- Utilize results from part (a) to calculate constants \( A \) and \( B \) for the journey.
(d) **Graphing:**
- Plot \( s(t) \), \( v(t) \), and \( a(t) \).
- Analyze and compare plots. Ensure velocity is positive while the distance is increasing, and observe acceleration when the train decelerates. Identify key regions for insights.
(e) **Maxima Analysis:**
- Find the maximum speed and acceleration during the journey.
- Determine maximum "G" forces. One "G" equals 9.8 m/s², representing Earth's gravitational acceleration. Analyze if the experienced forces during the train ride are comfortable.
These tasks involve mathematical reasoning and graphical analysis to optimize the commuter experience on this high-speed train journey.](https://content.bartleby.com/qna-images/question/5c23085c-d041-4baf-a967-24cb96374560/2f7d34da-a754-463a-8bf0-1b213961f659/ryqjnuj_processed.jpeg)
Transcribed Image Text:**High-Speed Train Construction for Maximum Comfort**
A billionaire plans to build a high-speed train that will travel between Seattle and Los Angeles (approximately 2000 km) in 2 hours. The goal is to ensure maximum comfort during the ride.
Some physicists suggest that the least uncomfortable way to travel a distance, \( D \), in time, \( T \), follows a specific mathematical model. The position \( s(t) \) at time \( t \) is given by:
\[ s(t) = 3 \cdot A \cdot t^2 - 2 \cdot B \cdot t^3 \]
### Tasks:
(a) **Dimensional Analysis:**
- The expression for \( s(t) \) must have dimensions of length. Constants \( A \) and \( B \) are different combinations of distance \( D \) and time \( T \).
- Use dimensional analysis to determine these combinations (numerical factors are already correct). Determine the appropriate units for \( A \) and \( B \).
(b) **Velocity and Acceleration:**
- Find symbolic expressions for velocity \( v(t) \) and acceleration \( a(t) \) as functions of time using \( s(t) \).
- Confirm that all units align correctly using results from part (a).
(c) **Determine Constants:**
- Utilize results from part (a) to calculate constants \( A \) and \( B \) for the journey.
(d) **Graphing:**
- Plot \( s(t) \), \( v(t) \), and \( a(t) \).
- Analyze and compare plots. Ensure velocity is positive while the distance is increasing, and observe acceleration when the train decelerates. Identify key regions for insights.
(e) **Maxima Analysis:**
- Find the maximum speed and acceleration during the journey.
- Determine maximum "G" forces. One "G" equals 9.8 m/s², representing Earth's gravitational acceleration. Analyze if the experienced forces during the train ride are comfortable.
These tasks involve mathematical reasoning and graphical analysis to optimize the commuter experience on this high-speed train journey.
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