The speed-density relation on a one-way, single lane road is shown in the figure, where speed \( u \) is in km/hour and density \( k \) is in vehicles/km. The maximum flow (in vehicles/hour) on this road is
Show Hint
In traffic flow analysis, the maximum flow occurs at the point where the product of speed and density is maximized. This is typically at the point where the density is half of the jam density and the speed is half of the free-flow speed.
From the given speed-density curve, we can determine the maximum flow by finding the point where the product of speed \( u \) and density \( k \) is maximized. This corresponds to the point where the slope of the curve is zero.
The formula for the maximum flow \( q_{{max}} \) is given by:
\[
q_{{max}} = \frac{k_j}{2} \times v_f \times \left( \frac{k_{{max}}}{2} \right),
\]
where:
- \( k_j = 100 \, {veh/km} \) (jam density),
- \( v_f = 100 \, {km/h} \) (free flow speed),
- \( k_{{max}} = 100 \, {veh/km} \) (maximum density).
Substituting these values into the formula:
\[
q_{{max}} = \frac{100}{2} \times 100 \times \left( \frac{100}{2} \right) = 2500 \, {veh/hr}.
\]
Thus, the maximum flow on this road is 2500 vehicles/hour, which corresponds to option (A).
