Traffic Density Measurement
1. Identify Relative Velocities
Let the linear traffic density (vehicles per km) be $\lambda$.
Speed of vehicles, $v = 40$ km/h.
Speed of sensor, $u = 5$ km/h (moving opposite to traffic).
2. Flux Calculation
The sensor measures a length $L = 1$ km. The time taken for the sensor to cover this length is $t = L/u$.
During this time, the number of vehicles $N$ encountered is determined by the relative velocity between the sensor and the traffic.
$$ v_{rel} = v + u = 40 + 5 = 45 \text{ km/h} $$
The rate at which vehicles pass the sensor (flux) is $R = \lambda v_{rel}$.
Total count $N = R \cdot t$.
Substituting the expressions:
$$ N = \lambda (v+u) \times \frac{L}{u} $$3. Solving for Density ($\lambda$)
Rearranging for $\lambda$:
4. Calculate for Specific Length
The question asks for the number of vehicles in a length $l = 100 \text{ m} = 0.1 \text{ km}$.
Answer: 4 vehicles