Question

Based on Greenshield's model, a speed-density relationship is developed on the data of a traffic stream. This relationship is represented as \(v = 75 - 0.03k\), where \(v\) (in km/hr) is the mean speed at density \(k\) (in vehicle/km). The jam density (in vehicle/km) of this traffic stream is (in integer).

Hint:

Jam density is the density at which speed becomes zero. Put \(v=0\) in the given equation and solve for \(k\).

Answer 1

Correct Answer: 2500

Step 1: Recall the standard form of Greenshield's model.
Greenshield proposed a straight line relation between speed and density: $$ v = v_f\left(1 - \frac{k}{k_j}\right) $$
Here $v_f$ is the free flow speed (the speed on an empty road) and $k_j$ is the jam density (the density at which speed falls to zero).

Step 2: Expand the standard form and line it up with the given equation.
Expanding the bracket gives $$ v = v_f - \frac{v_f}{k_j}k $$ so the constant term is $v_f$ and the coefficient of $k$ is $-\frac{v_f}{k_j}$. Comparing this term by term with the given $v = 75 - 0.03k$: $$ v_f = 75 \ \text{km/hr}, \qquad \frac{v_f}{k_j} = 0.03 $$

Step 3: Solve for the jam density using the free flow speed.
$$ k_j = \frac{v_f}{0.03} = \frac{75}{0.03} = 2500 $$

Step 4: Check the answer.
Put $k = 2500$ back into the given relation: $v = 75 - 0.03(2500) = 75 - 75 = 0$. Speed does fall to zero exactly at this density, confirming the value.
$$ \boxed{k_j = 2500 \ \text{vehicle/km}} $$