Two-color Train

A train consists of a locomotive, pulling 100 yellow cars, which in turn pull 200 blue cars. The train travels on a level track, accelerating forward at rate \(a\).

Each yellow car has 3 times as much mass as a blue car.

The cars experience rolling friction opposing the motion: \(F_f = \mu m g\), where \(m\) is the mass of the car, \(\mu = 0.10\) for yellow cars, and \(\mu = 0.20\) for blue cars.

Let \(F_{LY}\) be the force exerted by the locomotive on the first yellow car, and \(F_{YB}\) the force by the last yellow car on the first blue car.

If \(F_{YB} = \tfrac12\ F_{LY}\), how fast does the train accelerate? Give your answer as the ratio \(\frac{a}{g}\).


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