# optimum way of slowing down at a red traffic light

Suppose we are driving a car and we see a red light at a distance $d$. We don't know when it will turn green again but we know the period of red lights, that is, an upper limit for waiting. We want to minimize the expected loss of energy due to braking. What should be our slowing curve? (assuming we're driving at adequately high speeds)

If we knew the remaining time $t$ exactly, then the solution would simply be constant speed. We would adjust our speed as quickly as possible to the value $d/t$. (actually $d/t$ is possible only when we are able to decrease speed instantly.)

However the probabilistic version involves continuous Bayesian updates I think which I'm not really good at. Anyway, I follow some reasoning and end up with a parabolic displacement-time curve which ends up with zero speed at the worst case scenario(which we see the red light just after it has turned from green).

• let's ignore the information from car queue in front of the lights. Commented Jan 8, 2018 at 11:01
• – Dap
Commented Jan 8, 2018 at 11:05
• oh so many people thought the same thing before :) Commented Jan 8, 2018 at 11:13
• @AhmedBilâl I've thought about this exact same thing for awhile now. I was formulating the same question and it looks frighteningly similar to the way you wrote it. Glad to see an answer!
– Ryan
Commented Jan 30, 2019 at 19:55