In a model of traffic flow on a single lane road, it is assumed that each vehicle is \(\quantity{4}{m}\) long, travelling at constant speed and separated from the one in front by the typical stopping distance for that speed.

Find an expression for the rate of traffic flow, \(R_s\), in vehicles per hour, as a function of the speed, \(v\). Plot a graph of this function for speeds up to \(\quantity{70}{mph}\). Use your graph or the algebra to find the minimum or maximum value of this function and at what speed(s) it occurs?

If instead of the typical stopping distance, the vehicles are separated by the two-second rule, what is the flow rate, \(R_t\)? What would the maximum or minimum flow rate be and when does it occur?

Why might these models be unrealistic in practice?