# On a road, the speed-density (u - k) relationship relationship on a single lane road with unidirectional flow is u = 70 - 0.7 k (where speed u, is in km/h and density k, is in veh/km). The capacity of the road (in veh/hr) is ___________

## Answer (Detailed Solution Below) 1750

__Concept:__

The relationship between speed ( u ) and density ( k ) is given by the Greenshield macroscopic model

\({\rm{v}} = {{\rm{v}}_{\rm{f}}} - \left( {\frac{{{{\rm{v}}_{\rm{f}}}}}{{{{\rm{k}}_{\rm{j}}}}}} \right) × {\rm{k}}\)

u - mean speed at density k ( m/s )

k- density of stream ( veh/km ), uf - free mean speed

kj – jam density

q – Flow ( veh/hr )

__Calculation:__

We know Flow q = k × u

Given u = 70 – 0.7 k

q = k × ( 70 – 0.7 k )

q = 70 k – 0.7 k2

Capacity condition:

The capacity condition denotes maximum flow

For q to be maximum

\(\left( {\frac{{{\rm{dq}}}}{{{\rm{dk}}}}} \right) = 0\)

\(\left( {\frac{{dq}}{{dk}}} \right) = 70 - 0.14\;k = 0\)

k = 50 veh/km

∴ Capacity of road (C) = q = 70 k – 0.7 k2

= 70 × 50 - 0.7 × 502

**C = 1750 veh/hr**