A commuter train travels between two downtown stations....

A commuter train travels between two downtown stations. Because the stations are only $1.00 \mathrm{~km}$ apart, the train never reaches its maximum possible cruising speed. The engineer minimizes the time $t$ between the two stations by accelerating at a rate $a_{1}=0.100 \mathrm{~m} / \mathrm{s}^{2}$ for a time $t_{1}$ and then by braking with acceleration $a_{2}=-0.500 \mathrm{~m} / \mathrm{s}^{2}$ for a time $t_{2} .$ Find the minimum time of travel $t$ and the time $t_{1}$.

Key Concepts

Continuity of Velocity at Phase Transition

At the junction between the acceleration and deceleration phases, the vehicle's velocity must be continuous. This means the final velocity of the acceleration phase is equal to the initial velocity of the deceleration phase. This condition is essential for accurately connecting the separate pieces of the motion analysis.

Optimization Under Kinematic Constraints

Optimization in kinematics entails selecting the acceleration and deceleration durations to minimize or maximize a quantity such as time. The problem requires finding the precise point at which to transition from acceleration to deceleration such that the overall travel time is minimized, subject to given acceleration limits and total distance.

Constant Acceleration Equations

These equations relate displacement, initial and final velocities, acceleration, and time when acceleration is uniform. They are foundational in analyzing motion problems such as this one, allowing the determination of distances traveled during acceleration and deceleration phases.

Piecewise Motion Analysis

This concept involves breaking the overall motion into separate intervals, each with its own constant acceleration. In this problem, the journey is divided into an acceleration phase and a deceleration phase, which must be analyzed individually and then connected through their boundary conditions.

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