Question 3 of 7: Definitions and discussion

Question 3: Definitions and discussion (5 × 4 = 20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Part (a) — leading vs. lagging protected phase. A protected left-turn phase gives left-turning drivers a green arrow while the conflicting through movement is held at red, so the turn is made without accepting gaps. The distinction between leading and lagging is simply where that protected interval sits relative to the adjacent through green. A leading protected phase runs before the through green: the left arrow and the parallel through movement start together, the arrow terminates, and the opposing through movement is then released. A lagging protected phase runs after the through green: both directions run through movements first, the opposing through is stopped, and the left arrow is displayed at the tail of the phase.

The choice is not cosmetic. Leading lefts are the North American default because they match driver expectancy — a queued left-turner sees the arrow first and clears the intersection before the opposing platoon arrives — and because they discharge the left-turn queue before it can spill out of a short bay. Lagging lefts are used where a leading arrow would strand the opposing platoon (progression along an arterial often runs better with the arrow at the end of the band), where an exclusive bay is absent, or where opposite approaches are timed to overlap. The safety trade-off is the yellow trap: when one direction lags and the other leads, a permitted left-turner facing a circular yellow may believe the opposing through movement is also stopping when in fact it retains green, and turns into it. That is why a lagging left is normally paired with a flashing-yellow-arrow display or with matched lagging in both directions.

Part (b) — time mean speed vs. space mean speed. The two averages differ in what is being averaged over. Time mean speed \(u_t\) is the arithmetic mean of the spot speeds of the vehicles that pass a fixed point during a period; it is what a radar gun or a loop detector at one cross-section reports. Space mean speed \(u_s\) is the harmonic mean of the speeds of the vehicles occupying a length of road at an instant, equivalently the section length divided by the mean travel time; it is what a floating-car or licence-plate-matching study reports.

Formally, for \(n\) observations, \(u_t = \frac{1}{n}\sum u_i\) while \(u_s = n / \sum (1/u_i)\). The two are linked by \(u_t = u_s + \sigma_s^{2}/u_s\), where \(\sigma_s^{2}\) is the variance of the speeds about the space mean. Take the five spot speeds 88, 92, 96, 100 and 104 km/h: the time mean speed is 96.00 km/h and the space mean speed is 95.67 km/h, and the identity reproduces the time mean to within rounding. The space mean speed is therefore always the smaller of the two, and equal only when every vehicle travels at the same speed, because faster vehicles spend less time in the section and so are under-represented in a space-based average while being fully counted at a fixed point.

The distinction is not academic. The fundamental relation \(q = u k\) — flow equals speed times density — holds only for the space mean speed, so travel-time studies, density estimates and every Greenshields-type flow model must use \(u_s\). Using a radar-measured time mean speed in \(q = uk\) systematically under-estimates density and over-estimates level of service. Conversely, speed-limit and enforcement studies, and the 85th-percentile speed used to set posted limits, are properly based on spot (time mean) data.

Part (c) — circular vs. spiral curves. A circular (simple) horizontal curve is an arc of constant radius \(R\) inserted between two tangents. Its curvature jumps discontinuously from zero on the tangent to \(1/R\) at the point of curvature, so the lateral acceleration a vehicle experiences, and the steering input required to produce it, also change instantaneously. A spiral (transition, or clothoid) curve is inserted between the tangent and the circular arc, and has the property that its curvature increases linearly with distance along it, from zero at the tangent end to \(1/R\) where it meets the arc.

Three practical consequences follow. First, the spiral lets the driver steer at a constant rate of change of curvature rather than making a step input, which is what drivers naturally do anyway — without a spiral they encroach on the adjacent lane or the shoulder tracing their own transition. Second, the spiral provides the length over which superelevation can be run off: the pavement rotates from normal crown to full superelevation at a uniform rate as curvature builds, so the superelevation applied always matches the curvature present. Third, the spiral gives a longer, more natural sight line into the curve and a better appearance. The cost is a more elaborate layout — the circular arc is shifted inward by the throw \(p \approx L_s^{2}/(24R)\) and the tangent length increases. TAC and AASHTO practice is to use spirals on higher-speed alignments and on sharp curves, and to omit them on flat curves where the required superelevation runoff can be accommodated on the tangent.

Part (d) — HOV lanes. A high-occupancy-vehicle lane is a lane reserved, permanently or during stated periods, for vehicles carrying at least a specified number of occupants — typically 2+ or 3+ — together with buses, and in many Canadian jurisdictions taxis, motorcycles and permitted low-emission vehicles. Physically it may be a concurrent-flow lane on the left of the general-purpose lanes (the common form on Highway 401, Highway 1 and the Ontario 400-series), a barrier-separated roadway, a contraflow lane borrowed from the off-peak direction, or a queue-jump lane at a ramp or intersection.

The rationale is that a freeway lane is capacity-limited in vehicles but the trip demand is in people. A general-purpose lane at capacity carries roughly 2,000 veh/h at about 1.15 persons per vehicle, i.e. some 2,300 persons/h; the same lane carrying only 2+ vehicles and buses can move considerably more people at a far higher and more reliable speed. The travel-time saving is the incentive that induces mode shift, so the lane must be demonstrably faster than the adjacent lanes to work at all. The classic failure modes are symmetrical: set the occupancy threshold too high or enforce too weakly and the lane runs empty, which is politically indefensible and wastes real capacity; set it too low and the lane degrades to general-purpose conditions and the incentive vanishes. Enforcement, access management (where vehicles may weave in and out) and clear MUTCDC signing and diamond marking are therefore as important to an HOV facility as its geometry.

Part (e) — pedestrian clearance time. Pedestrian clearance time is the interval, displayed as the flashing DON’T WALK (flashing hand), during which a pedestrian who has already left the curb may finish crossing but no new pedestrian may start. It is distinct from the WALK interval, which is the time during which entry to the crosswalk is permitted, and the two together with the phase’s vehicle clearance make up the pedestrian’s share of the cycle.

It is sized from the crossing distance and an assumed walking speed: \(\text{clearance} = L_c / S_p\), with \(L_c\) measured from the curb (or from the near edge of the travelled way) to the far curb or to a median refuge. Canadian practice uses \(S_p = 1.2\) m/s for a normal population, reduced to about 1.0 m/s where older pedestrians or people with mobility limitations predominate — near seniors’ residences, hospitals or transit interchanges. For this paper’s own 15 m crosswalk at 1.2 m/s the clearance is \(15.0/1.2 = 12.50\) s, and adding the 3.2 s start-up allowance used in the HCM minimum-green expression gives 15.70 s of pedestrian time. The interval matters because it is often the binding constraint on cycle length at wide urban intersections: it is a fixed number of seconds that each phase must contain, so a wide crossing served by a short phase forces the whole cycle longer regardless of what the vehicle demand wants — exactly the calculation carried out in Questions 2 and 5.