Question

For a traffic stream, $v$ is the space mean speed, $k$ is the density, $q$ is the flow, $v_f$ is the free flow speed, and $k_j$ is the jam density. Assume that the speed decreases linearly with density.
Which of the following relation(s) is/are correct?

• A. $q = k_j k - \left( \frac{k_j}{v_f} \right) k^2$
• B. $q = v_f k \left( \frac{v_f}{k_j} \right) k^2$
• C. $q = v_f v \left( \frac{v_f}{k_j} \right) v^2$
• D. $q = k_j v - \left( \frac{k_j}{v_f} \right) v^2$

Hint:

In traffic flow analysis, flow is a product of speed and density, and the speed typically decreases linearly with increasing density.

Answer 1

The Correct Option is B, D

The flow rate \( q \) in a traffic stream can be expressed as the product of the speed \( v \) and the density \( k \). The flow is dependent on both the density and the free flow speed. Since the speed decreases linearly with the density, we get the following relations:
- For (A), the expression is not correct as it doesn't match the linear relationship between speed and density.
- For (B), we can express flow as: \[ q = v_f k \left( \frac{v_f}{k_j} \right) k^2, \] which matches the expected form.
- For (C), the dimensions of speed and density don't align correctly for this expression.
- For (D), this expression is correct as it represents the general relation between flow and speed. The form is consistent with traffic flow behavior: \[ q = k_j v - \left( \frac{k_j}{v_f} \right) v^2. \] Thus, the correct answers are:
\[ \boxed{\text{(B) and (D)}}. \]