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7.2.2 Forbes’ Model

Forbes et al. (1958) produced exactly the same form of model as did Pipes by considering that the time gap between the front of the following vehicle and the rear of the vehicle in front should never be less than the following driver’s reaction time,\(\Delta t\) . This meant that the minimum (time) headway, hmin, should be equal to the reaction time plus the time taken for the vehicle in front to travel its own length, that is:

\[h_{\min }     =    \Delta t   +   \frac{L_{n} }{\text{ v }_{n} (t)} \]7.3

Given that the general relationship between headway, spacing and speed is s = h.v (see Section 2.2.1), this can be written as:

\[s_{\min }     =    \Delta t . \text{ v }_{n} (t)   +   L_{n} \]7.4

which is the same as Pipes’ model, with the constant K equal to the reaction time \(\Delta t\).