Was solving some undergrad PDE traffic flow problems and we have a nasty. The set up is as follows.
If you are looking at the correct allocation of green light time to red light time for busy traffic conditions, we need to determine the length $L$ of a queue of cars that will pass through in a time interval.
Assume that a long line of cars are at rest behind a red light and the light turns green at $t=0$ and it will stay that way for time $T$.
What I don't know how to do is determine is:
How long the car stays at rest for?
I believe that once the car is moving that $X(t)$ satisfies $$ \frac{dX}{dt} = 0.5V_{max}[1+\frac{X}{V_{max}}]$$ but I have struggled to find it. I tried applying some of the basics we learnt but I ended with a minus in the above equation
What is the appropriate initial condition for the above equation and what is the condition that the car to reach the lights at time $T$ (This can be done by figuring out the length of queue $L_{1}$ that can be cleared.