All India Civil Engineering (CE) Group

Understanding the Traffic Flow Equation
The traffic flow equation given is:
u = 80 - 0.7K
where:
- u = speed in km/h
- K = density in vehicles per km (vpkm)
Finding the Flow (Q)
Flow can be expressed as:
Q = u * K
Substituting the equation for u:
Q = (80 - 0.7K) * K
This simplifies to:
Q = 80K - 0.7K²
Maxi... moremizing the Flow
To find the maximum flow, we need to take the derivative of Q with respect to K and set it to zero:
dQ/dK = 80 - 1.4K = 0
Solving for K gives:
1.4K = 80
K = 80 / 1.4
K ≈ 57.14 vpkm
Calculating Maximum Flow
Now, substituting K back into the flow equation:
Q = 80K - 0.7K²
Q = 80(57.14) - 0.7(57.14)²
Calculating each part:
- 80 * 57.14 ≈ 4571.2
- 0.7 * (57.14)² ≈ 0.7 * 3265.62 ≈ 2286
Thus, the maximum expected flow is:
Q ≈ 4571.2 - 2286 ≈ 2285.2 vph
Conclusion
The maximum expected flow is approximately 2286 vph, confirming that the correct answer is option 'B'.

Understanding Stoke's Theorem
Stoke's Theorem relates the surface integral of a vector field over a surface to the line integral of the field around the boundary of that surface. For a vector field A, it can be expressed as:
∮C A · dr = ∬S (curl A) · dS
Where C is the boundary curve, S is the surface, and curl A represents the curl of vector field A.
Analyzing the Vec... moretor Field A
Given the vector field A = x i + y j + z k:
- Components:
- A has three components: x, y, z, which correspond to the x, y, and z directions, respectively.
Calculating the Curl of A
To determine if A is curl-free (or rotational), we calculate the curl:
- Curl A = ∇ × A:
- For A = x i + y j + z k, the curl results in:
- curl A = (∂z/∂y - ∂y/∂z) i + (∂x/∂z - ∂z/∂x) j + (∂y/∂x - ∂x/∂y) k
- All partial derivatives are zero, leading to:
curl A = 0 i + 0 j + 0 k = 0
Conclusion: Curl Free
Since curl A = 0, it indicates that the vector field A is:
- Curl Free: No rotation or circulation around any point in space.
- This aligns with option 'D' provided in the question.
Understanding the Correct Answer
Thus, the correct answer is indeed option 'D' - the vector field A is curl-free, meaning it has no rotational component. This is a crucial concept in fluid dynamics and electromagnetism, where curl-free fields often correspond to conservative forces.

Understanding Spot Speed Studies
Spot speed studies are essential for assessing traffic flow and safety on freeways. However, there are specific conditions under which these studies are conducted, particularly concerning traffic volumes.
Traffic Volume Thresholds
- Spot speed studies are generally not conducted when traffic volumes exceed 750-1000 vehicles per hour ... moreper lane.
- This threshold is important because high traffic volumes can result in congested conditions, which may skew the results of speed measurements.
Reasons for the Threshold
- Safety Concerns: High traffic volumes can create unsafe conditions for conducting speed studies, as the interactions between vehicles become more complex.
- Data Reliability: At elevated volumes, speeds may be artificially reduced due to congestion, making the data less reliable for assessing the free flow of traffic.
- Operational Conditions: Spot speed studies aim to measure speeds under free-flow conditions. High volumes often lead to stop-and-go traffic, which is not representative of typical freeway operation.
Conclusion
The traffic volume threshold of 750-1000 veh/hr/ln is critical for ensuring that spot speed studies provide accurate and meaningful data. Conducting these studies under lower volume conditions allows for better insights into traffic behavior and aids in effective roadway design and management.