The motors of an electric train can give it a uniform...

The motors of an electric train can give it a uniform acceleration of $1 \mathrm{~m} \mathrm{~s}^{-2}$ and the brakes can give it a uniform negative acceleration of $3 \mathrm{~m} \mathrm{~s}^{-2}$. The shortest time in which the train can make a trip, starting from a station and stopping at another station that is separated from the starting station by a distance of $1350 \mathrm{~m}$ on straight track, is (a) $15 \mathrm{~s}$ (b) $30 \mathrm{~s}$ (c) $60 \mathrm{~s}$ (d) $120 \mathrm{~s}$

The motors of an electric train can give it a uniform acceleration of $1 \mathrm{~m} \mathrm{~s}^{-2}$ and the brakes can give it a uniform negative acceleration of $3 \mathrm{~m} \mathrm{~s}^{-2}$. The shortest time in which the train can make a trip, starting from a station and stopping at another station that is separated from the starting station by a distance of $1350 \mathrm{~m}$ on straight track, is
(a) $15 \mathrm{~s}$
(b) $30 \mathrm{~s}$
(c) $60 \mathrm{~s}$
(d) $120 \mathrm{~s}$

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Key Concepts

Optimization under Acceleration Constraints

This concept focuses on determining the minimum time or best strategy for completing a trip when acceleration and deceleration are capped. It involves using maximum allowable accelerations and decelerations, sometimes with a bang-bang control approach, to ensure that the object reaches its destination in the least possible time while adhering to the constraints.

Piecewise Analysis of Motion

When an object experiences different phases of motion – such as acceleration and then deceleration – the overall motion is analyzed by breaking it into segments. Each segment is treated independently with its acceleration, and the results are combined to determine total distance and time.

Uniformly Accelerated Motion

This concept involves motion where an object is subject to a constant acceleration or deceleration. It utilizes key kinematic equations such as v = u + at and s = ut + (1/2)at², which help calculate final velocity and displacement over a given time interval.

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