Traffic Flow (Greenshields model) – Linear speed–density relation Free-flow speed = 80 km/h; jam density = 100 vehicles/km. Compute the maximum flow (vehicles/h) and the speed at that flow.

Introduction / Context:
The Greenshields linear model assumes speed v decreases linearly with density k: v = v_f * (1 − k/k_j), where v_f is free-flow speed and k_j is jam density. Flow q is the product of speed and density. The model predicts a parabolic flow–density curve with a distinct maximum (capacity) at an intermediate density.

Given Data / Assumptions:

• v_f = 80 km/h.
• k_j = 100 veh/km.
• Uniform, uninterrupted flow; single stream.

Concept / Approach:

Flow q(k) = k * v = k * v_f * (1 − k/k_j) = v_f * (k − k^2/k_j). The parabola attains its maximum at dq/dk = 0 → k = k_j / 2. At that density, speed is v = v_f / 2 and capacity is q_max = v_f * k_j / 4.

Step-by-Step Solution:

Verification / Alternative check:

Using q_max = v_f * k_j / 4 directly: 80 * 100 / 4 = 2000 veh/h; v* = v_f / 2 = 40 km/h. Both methods agree.

Why Other Options Are Wrong:

Options A and B exaggerate capacity by a factor of 4 (confusing km with m or omitting the 1/4 factor). Option C keeps free-flow speed at capacity (incorrect). Option E picks arbitrary values inconsistent with the linear model.

Common Pitfalls:

Multiplying speed by jam density directly, or assuming capacity occurs at free-flow speed rather than at half of both v_f and k_j in this model.

Final Answer:

2000 vehicles per hour and 40 km per hour