A particular bullet train accelerates and decelerates at the...

A particular bullet train accelerates and decelerates at the rate of $1.2 \mathrm{~m} / \mathrm{s}^{2}$. Its maximum cruising speed is $145 \mathrm{~km} / \mathrm{h}$. (a) What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for 20 minutes? (b) Suppose that the train starts from rest and must come to a complete stop in 20 minutes. What is the maximum distance it can travel under these conditions? (c) Find the minimum time that the train takes to travel between two consecutive stations that are $72 \mathrm{~km}$ apart. (d) The trip from one station to the next takes $37.5$ minutes. How far apart are the stations?

A particular bullet train accelerates and decelerates at the rate of $1.2 \mathrm{~m} / \mathrm{s}^{2}$. Its maximum cruising speed is $145 \mathrm{~km} / \mathrm{h}$.
(a) What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for 20 minutes?
(b) Suppose that the train starts from rest and must come to a complete stop in 20 minutes. What is the maximum distance it can travel under these conditions?
(c) Find the minimum time that the train takes to travel between two consecutive stations that are $72 \mathrm{~km}$ apart.
(d) The trip from one station to the next takes $37.5$ minutes. How far apart are the stations?

Key Concepts

Uniformly Accelerated Motion

This concept involves motion with constant acceleration or deceleration. It describes how an object's speed changes in a predictable manner, allowing the use of simple kinematic equations to determine quantities like final speed, time, and distance during acceleration or deceleration phases.

Kinematic Equations

Kinematic equations relate displacement, initial and final velocities, acceleration, and time in motion scenarios. These equations are essential for calculating distances covered during acceleration or deceleration periods and are used to solve problems involving segmented journeys with different phases of motion.

Unit Conversion

Converting between units, such as from kilometers per hour to meters per second, is crucial in solving motion problems where constants like acceleration are given in specific units. Consistency in units ensures that the computed distances and times are accurate.

Phases of Motion

Many motion problems are solved by breaking down the journey into distinct phases such as acceleration, steady cruising, and deceleration. Understanding how to analyze each phase independently and then combine the results is key to accurately calculating overall distance or time.

Motion Planning Optimization

This concept involves determining the conditions that minimize travel time or maximize distance under given acceleration limits and constraints. It requires a strategic division of the journey into acceleration, constant speed, and deceleration segments to optimize performance based on time or distance constraints.

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