Red Light Optimisation

Bengt is cruising around in his car, when suddenly he comes to a red light. The annoys him very much. In order to minimise his annoyance, he would like to make sure he gets through as quickly as possible. The thing is, if he drives all the way up to the light and stops, it will take several seconds to accelerate, and we can't have that, now can we?

We can safely assume that Bengt is driving at the maximum allowed speed, v, when he is not taking the traffic light into account. As a simplification, we'll assume that his maximum acceleration (forward) is a constant m, and that he can break (i.e. accelerate backward) without limit. The question is of course, what speed should he drive at, as a function of something suitable, like perhaps the distance to the traffic light, or some kind of time?

1. First, let's simplify a little further, by assuming that Bengt just wants to maximise his speed at the moment the light turns green. In a sense, we are saying that m is close to zero, which is somewhat insulting to Bengts car, but might make the calculations easier.
   Suppose now that we have no idea when the light will turn green. Can the problem be solved? If so, please do.
2. Suppose instead that the light turns green after an average time T, and the probability that it will happen in the next moment remains constant - that is, the red light has a "half-life".
3. Suppose instead that Bengt sees the light turn red at a particular time (or distance, if you like) and that it will turn green at any time between then and another time, which, for simplicity, we might call "one minute later".
4. Now do a), b) and c) again, but remove that simplification about the acceleration being close to zero.
5. Pick the option that you think is the most realistic, and find how much time he would save by doing all this rather than driving as fast as possible up to the light.