The amber traffic light

Many real-life situations can be used as a basis for doing some mathematics, but very often these pseudo-real?life contexts are not very plausible. In many cases, the mathematics isn’t really needed, and feels bolted on for the sake of doing mathematics. In real life, often it’s enough to use common sense and informal judgments, and calculations aren’t actually needed. So, I’m always interested in finding examples where the answer isn’t obvious without some calculation, and mathematics really helps. In these kinds of mathematical modelling situations, doing the calculations is always the easy bit: the challenge is deciding what information you need and how to manipulate it.

Someone was recently talking about driving through traffic lights. In the UK, traffic lights show a single amber light as a warning that a red (stop) light is about to come on. According to the Highway Code “AMBER means ‘Stop’ at the stop line. You may go on only if the AMBER appears after you have crossed the stop line or are so close to it that to pull up might cause an accident”(www.highwaycodeuk.co.uk/light-signals?controlling-traffic.html). Inexperienced drivers will sometimes jam on the brakes hard as soon as they see an amber light, but this can be more dangerous to following traffic than just continuing through the junction, if they are sure that they can make it through safely before the red light comes on.

This decision requires quite a bit of judgment. Should you slow down when approaching a green traffic light, in case it turns to red before you arrive? Or can you proceed as normal until the moment that you see an amber light, and only slow down then if necessary? Presumably the answer depends on the speed at which you are travelling. You might like to pause here to think about what mathematics might help to answer these questions and what information you might require to do so? (cont.)