Posted speed limit at head-on crash

Given: $g_1 = +4\%$, $g_2 = -3.5\%$, posted speed $V = 70$ km/h, highest point at 120 m from the beginning of the vertical curve (BVC), eye height $h_1 = 1.08$ m, reaction time $t = 2.5$ s, $f = 0.35$.

Assumption: only the driver's eye height is given, so the object (the other car or an obstruction) is taken at road level, $h_2 = 0$. The sight distance to be checked is the stopping sight distance.

Step 1: Length of the vertical curve

$$N = g_1 - g_2 = 0.04 + 0.035 = 0.075$$

Highest point from BVC: $x = \dfrac{g_1 L}{N}$

$$120 = \frac{0.04\,L}{0.075}\ \Rightarrow\ L = 225\ \text{m}$$

Step 2: SSD required at the posted speed

$$\begin{aligned} SSD &= 0.278\,V\,t + \frac{V^2}{254 f} \\ &= 0.278(70)(2.5) + \frac{70^2}{254\times0.35} \\ &= 48.65 + 55.12 = 103.8\ \text{m} \end{aligned}$$

Step 3: Sight distance available on the curve

For a summit curve with $S < L$ and $h_2 = 0$, $H = 1.08$ m:

$$L = \frac{N S^2}{2H}\ \Rightarrow\ S = \sqrt{\frac{2HL}{N}} = \sqrt{\frac{2(1.08)(225)}{0.075}} = 80.5\ \text{m}$$

$S = 80.5\ \text{m} < L = 225$ m, so the assumption $S < L$ is valid.

Step 4: Check

Available sight distance 80.5 m $<$ required SSD 103.8 m. The sight distance is not sufficient at 70 km/h, so the posted speed sign is faulty.

Step 5: Safe posted speed

Find $V$ for which SSD = 80.5 m:

$$\frac{V^2}{88.9} + 0.695V - 80.5 = 0\ \Rightarrow\ V = 59.2\ \text{km/h}$$

Check: at 60 km/h, SSD = 82.2 m (slightly greater than 80.5 m); at 55 km/h, SSD = 72.3 m (< 80.5 m).

Answer: The posted 70 km/h is unsafe (SSD 103.8 m > available 80.5 m). The posted speed should be 55 km/h (the calculated limit is about 59 km/h, rounded down to a practical value of 55 km/h).

Note: the head-on reading gives the same result. With both drivers' eyes at 1.08 m, the available sight distance is $\sqrt{8HL/N} = 161$ m, and two opposing cars need $2\times SSD$, so each SSD must not exceed 80.5 m, i.e. $V \approx 59$ km/h. If the IRC object height of 0.15 m is used instead, the available distance becomes 110.5 m and 70 km/h would be acceptable (maximum 72.9 km/h); the answer above uses the data as given.