# speed-density relationship transportation question

I have the following questions:

Would the capacity be the area under the curve? Or can I use this formula: $$q_c = \frac{k_j \times v_f}{4}$$ from the greenshields model, where the capacity flow is related to the free flow speed and jam density. If so then $$q_c = 6600$$

The following question is :

I am really not sure how to approach this question. AADT = Annual average daily traffic.

• So what happens if you use the formula? You need to show what you have done so far. Commented Aug 13, 2022 at 9:59

This is one of those vocabulary problems. If you know what they mean, it should be a matter of applying unit conversions.

I'll make some sense of the words. And then I will do my utmost to forget it, because it's pretty useless to learn some model when you don't even know if reality fits it...

Capacity is the maximum possible veh/hr possible.

The formula you have works if the curve is linear (which we can believe from the graph).

More generally it is max(speed*density). And here's a derivation of that qc formula.

speed = 110 km/h*(1-density/(120 veh/km)).
speed = vf*(1-density/kj)
speed*density=vf*density-vf*density^2/kj
max of such a parabola occurs where derivative is 0 so:
0=vf-2*vf/kj*density
density = kj/2
speed=vf/2
max(speed*density) = qc = kj*vf/4


Or 3300 veh/hr (which is probably per direction with an implicit 2 directions based on your mention of 6600)

Next question is irksome in how useless it is. If someone knows 12% of AADT occurs... Then that person could have counted what AADT was when taking measurements to obtain that 12%.

Anyway, it mentions a 55/45 directional split, so 3300 corresponds to the 55%. Word trap is whether that's 55% split of veh/hr or veh/km. Traffic in the measurement sense (rather than "stuck in traffic" sense where it means a section that actually lacks traffic per units of time) is something per time so it is the former (veh/hr). That means the other direction contributes 3300/55*45=2700 So both dirs it's 3300+2700=6000 veh in that hr. Divide by .12 aka 12% and you get 50000.

• Hi Abel, thanks for the explanation. I really appreciate this. $6600$ is actually my typo, I meant to type $3300$instead. So would the first question be $3300$ or $6600$? The question asks for 'What is the capacity of this section?'. Does 'section' imply both directions? Commented Aug 14, 2022 at 6:01
• That is a question that probably only the writer of the question can answer. If it were a real highway, I could go measure if I really had to. It isn't unheard of for textbooks and questions in them to be incorrect, especially if someone is going about changing numbers to require new students to purchase a new edition. All I can do is guess what someone means to say with their words.
– Abel
Commented Aug 14, 2022 at 10:47
• Perhaps the meaning of section changes with mention of directional split, meaning the answer to Q14 remains 3300. There is no reason given after all that a density speed relation should be the same in both directions...
– Abel
Commented Aug 14, 2022 at 11:00