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A billionaire wants to construct a high speed train that will carry him between Seattle and Los Angeles (about 2000 km) in 2 hours. He wants the most comfortable ride possible. Some physicists have proposed that the least uncomfortable way to travel a distance D in time T follows a specific math- ematical form. * The position s(t) at time t is given by s(t) = 3 - A ·- 2. B (a) The expression for s(t) must have dimensions of length. The constants A and B are two different combinations of the distance D and time T. Use dimensional analysis to determine these combinations. (The numerical factors are already correct. All that remains is to determine the combination of D and T with the appropriate units.) (b) Find symbolic expressions for velocity u(t) and acceleration a(t) as functions of time for the journey described by s(t). Use your results from part (a) to show that all the units work out. (c) Use your result from part (a) to determine the constants A and B for the billionaire's journey. (d) Make separate plots of s(t), v(t), and a(t). Neatness counts! Compare the plots and try to make sense of them. For example, the velocity should be positive while distance is increasing, acceleration should be negative when the train is slowing down, etc. Identify specific regions of your plot and make such comparisons. (e) Find the maximum speed and maximum acceleration during the journey. Keep going after you find the maxima. "One G" is the approximate acceleration due to gravity at the surface of the earth: 9.8 m/s?. How many G's must the billionaire endure during the train ride? Does this sound comfortable?

A billionaire wants to construct a high speed train that will carry him between Seattle and Los Angeles (about 2000 km) in 2 hours. He wants the most comfortable ride possible. Some physicists have proposed that the least uncomfortable way to travel a distance D in time T follows a specific math- ematical form. * The position s(t) at time t is given by s(t) = 3 - A ·- 2. B (a) The expression for s(t) must have dimensions of length. The constants A and B are two different combinations of the distance D and time T. Use dimensional analysis to determine these combinations. (The numerical factors are already correct. All that remains is to determine the combination of D and T with the appropriate units.) (b) Find symbolic expressions for velocity u(t) and acceleration a(t) as functions of time for the journey described by s(t). Use your results from part (a) to show that all the units work out. (c) Use your result from part (a) to determine the constants A and B for the billionaire's journey. (d) Make separate plots of s(t), v(t), and a(t). Neatness counts! Compare the plots and try to make sense of them. For example, the velocity should be positive while distance is increasing, acceleration should be negative when the train is slowing down, etc. Identify specific regions of your plot and make such comparisons. (e) Find the maximum speed and maximum acceleration during the journey. Keep going after you find the maxima. "One G" is the approximate acceleration due to gravity at the surface of the earth: 9.8 m/s?. How many G's must the billionaire endure during the train ride? Does this sound comfortable?

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**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.
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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