Green waves

Classical Mechanics Level 3

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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