As two trains move along a track, their conduetors suddenly...

As two trains move along a track, their conduetors suddenly notice that they arc headed toward cach other. Figure $2-31$ gives their velocities $v$ as functions of time $t$ as the conductors slow the trains. The figure's vertical scaling is set by $v_{1}=40.0 \mathrm{~m} / \mathrm{s} .$ The slowing pro- cesses begin when the trains are $200 \mathrm{~m}$ apart. What is their separation when both trains have stopped?

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Key Concepts

Kinematics

Kinematics involves studying the motion of objects without considering the causes of motion. In problems like this, it is used to analyze variation in velocity, displacement, and acceleration over time, particularly when objects are decelerating or accelerating. Here, determining the distance each train covers while slowing down is a direct application of kinematics principles.

Integration of Velocity

Integration of the velocity function with respect to time provides the displacement of an object. In this context, integrating the decelerating velocity functions of each train over the time during which they slow to a stop yields the distances they travel. The sum of these distances subtracted from the initial separation provides the final separation when both trains have stopped.

Relative Motion Analysis

Relative motion analysis focuses on the motion of one object as observed from another. In the given problem, the separation between the two trains is affected by their individual motions, and the combined displacement needs to be considered to determine their final distance apart. This technique ensures that the net change in separation due to both motion contributions is correctly computed.

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