# Green waves

A green wave is a synchronized pattern of traffic lights that helps reduce gridlock. When there is a major thoroughfare with a lot of traffic lights, often the lights turn green sequentially so that cars can travel a long way before they have to stop. This is a "wave of green lights," or green wave.

Consider a horizontal road with such a set of traffic lights, each spaced $$100~\mbox{m}$$ apart. A car sits at the first traffic light. At $$t=0$$ the light turns green and the car accelerates. What is the maximum time $$t$$ in seconds the next light must turn green by so the driver of the car does not start braking?

Details and assumptions

• The car has a total mass of $$1000~\mbox{kg}$$ and the acceleration of gravity is $$-9.8~\mbox{m/s}^2$$.
• There is no delay between when a light changes color and the driver starts to accelerate/brake (ignore human reaction time).
• The car accelerates/brakes at $$4~\mbox{m/s}^2$$.
• The driver tries to go as fast as possible, while staying at or below the speed limit of $$60~\mbox{km/hr}$$.
• If the driver sees a red light, they will brake (at $$4~\mbox{m/s}^2$$) such that they stop directly under the light.
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