Merge!

When traffic merges from two lanes to one, will cars be moving faster in the wide or narrow part?

Assume that there is a steady flow of bumper-to-bumper traffic approaching the merge point.

Both are equal Wide part Narrow part

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13 solutions

Geoff Pilling
Nov 8, 2018

The same number of cars are getting through but in a smaller space.

Therefore, each must be moving faster in the narrower section. It's similar to water moving faster in the section of hose that has a kink in it or is being pinched.

"Can anyone guess what I was doing when I thought of this problem?" Sitting in traffic and contemplating math, of course!

David Vreken - 2 years, 7 months ago

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Hahahahaha... Yup! :)

Geoff Pilling - 2 years, 7 months ago

That's good, David. :-)

Jesse Otis - 2 years, 6 months ago

What did you observe in reality? I’ll bet money it wasn’t this! I guess what I’m saying is in reality I believe this only holds for the traffic very far upstream and downstream of the merge point.

Eric Roberts - 2 years, 6 months ago

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This is observable in reality. Where do you get frustrated in traffic? Waiting to merge or after you've merged? Isn't it always before you've merged that cars move slowly?

I note, however, that when cars go from one lane to two, cars tend to speed up, somewhat contrary to fluid dynamics.

Chris Maitland - 2 years, 6 months ago

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You said it... cars slow down as they approach the merge point. For steady incompressible flows they would necessarily speed up. In reality the flow of traffic is unsteady and compressible.

The observation of acceleration at the onset of divergence in traffic follows the fluid mechanics of compressible flows for a Mach number greater than 1; the speed of traffic is greater than the propagation of the wave that communicates the merge.

Eric Roberts - 2 years, 6 months ago

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@Eric Roberts Eric -- I concur with your viewpoint - and I'm sure that viewpoint is borne of experience. Vehicle traffic flow does not match incompressible fluid flow behavior (e.g. water) because of the possibility of wrecks. I imagine that Geoff is giving an ideal physics parallel in his problem - and that is why the answer is as it is.

Jesse Otis - 2 years, 6 months ago

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@Jesse Otis Jesse, thanks for your input. Just to be clear; I’m fine with the problem/answer given the constraints as a math problem.

I just think it’s not a very realistic model. I would say most people on the site with driving experience know all to well what this model predicts doesn’t happen. Traffic slows for some distance before the merge, and then speeds up once through. So I don’t know that it’s good for explaining the basis on how traffic actually behaves, which is usually the point of a physical model. To accurately and simply ( if possible) describe the phenomenon.

Eric Roberts - 2 years, 6 months ago

Bernoulli's principal of fluid particle flow...

Vijay Simha - 2 years, 6 months ago

But they can easier drive if its wide coz if its narow they need to be more carefull and they are more stressed

LordJellyfish Pavic - 2 years, 6 months ago

Except it never works this way for real traffic...

Chris Nabholz - 2 years, 6 months ago

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Yeah... A lot of times you have the "overly polite driver" issue where the person at the merge spot of one lane just lets everyone through... :-/

Geoff Pilling - 2 years, 6 months ago

wooooooooooooooooooooooo but I don't know if it would work in real traffic...

Qiao Qiao - 2 years, 6 months ago

That's what I thought at first but then I thought that we can't relate cars to fluids since there are not enough respective values for the other variables of fluid movements.

George Giokits - 2 years, 6 months ago

I want to experiment in real life

Spriha Basir - 2 years, 5 months ago

I think we can relate this to fluid flow.

Equation of continuity states, A 1 v 1 = A 2 v 2 A_1v_1 = A_2v_2 i.e, Velocity of a fluid is inversely proportional to the area of cross section.

Since, Area of narrow path is small compared to wider one, the cars should move faster there.

Is this even right way of answering this?

Yes. This is correct!

Geoff Pilling - 2 years, 6 months ago

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Thank You!!

A Former Brilliant Member - 2 years, 6 months ago

That’s what I used to model this. However, I doubt it is an accurate model.

Krishna Karthik - 2 years, 6 months ago

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It's what they teach in civil engineering in some states of the US. Note here v is the following distance.

Jerry McKenzie - 2 years, 6 months ago

Good solution.

Krishna Karthik - 2 years, 6 months ago

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I had solved such questions in past using same method, its indeed a nice way. Thanks!

A Former Brilliant Member - 2 years, 6 months ago

Yes, I think this is a very logical approach to this problem

Asong Randy - 2 years, 6 months ago

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If you were preparing for JEE then this is practice example coaching classes give you after they teach you Equation Of Continuity in Fluid Mechanics.

A Former Brilliant Member - 2 years, 6 months ago

I think it’s not correct model because Av is not fixed.

병무 임 - 2 years, 6 months ago

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I am not fixing a particular value for A v . Av. I am just comparing it in both cases of Narrow Road and Wide Road.

A Former Brilliant Member - 2 years, 6 months ago
Jerica Washington
Nov 18, 2018

The vehicles merging are anticipating the single lane of traffic, therefore they are accelerating slowly. If this precaution is ignored by a speeding driver, the driver runs the risk of missing the merge. Thus it is important to the imagination: the vehicles merging are significantly slower than the vehicles already merged into the single lane of traffic.

this is wrong! i see every morning the opposite: The speed of the cars stays th same in the wider and narrow area, but in the wider area is double space for the cars so the speed is higher.

Mel Koch - 2 years, 6 months ago

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I must agree the solution does not reflect a real life situation. If this were a problem of fluid dynamics the solution would be correct. Bad problem.

David Kline - 2 years, 6 months ago

It depends on the level of traffic. In a bumper-to-bumper situation, the cars in the multiple lanes will have to do more waiting than the cars in the single lane. But if traffic is not as dense as that, we often see a lot more movement in the lanes before the choke point, as there is more space to maneuver.

Richard Desper - 2 years, 6 months ago

I agree - I find it utterly bizarre that people try to apply fluid dynamics to a fundamentally human problem. I think some people are obviously trying to be too clever :-)

Chris Clark - 2 years, 6 months ago

car traffic defer from fluid because cars practically decrease its speed to enter the narrow lane

Khalid Eldesoky - 2 years, 6 months ago

Human behavior is not bound, nor subject to, fluid dynamics. Human operators of vehicles like space, are able to make a choice and in (in the scenario) possess an accelerator pedal. Fluids do not have choice, are indifferent to space (and every thing else) and certainly possess no accelerator pedal. This is behavioral science vs physics. Human operators of vehicles who like to move quickly will find a way to move their vehicle around slower operators by changing lanes in a multi-lane path (even in a bumper to bumper traffic, an empirical truth). Human operators in a single lane are restricted to the speed of the slowest operator. Speed increases on the opposite side of the lane restriction. Why? Because the constraints of the slowest operator are removed. The initial question for the brain teaser is flawed because vehicles would not be in the lane with out a human operator (Driver-less car, you ask? Sorry, human programming. The sensors would allow for an increase in speed when the safety parameters allow it.) The initial question attempts to conflate the human operation of a vehicle in a traffic lane and fluid dynamics. So, the correct answer is "Vehicles go faster in multiple lanes and slower in single lanes".

Tom Ulsh - 2 years, 6 months ago

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Couldn’t agree more ! I see this every day during my morning ride to work. The instant there is a lane merger traffic backs up .

KD MacChambers - 2 years, 6 months ago

Fluid dynamic equations are fine in their place, but telling me that X number of cars will move faster on a one-lane road than a two land road is bizarre.

David Storrs - 2 years, 6 months ago
Aqiba Bashir
Nov 23, 2018

In the wider space, people have room to change lanes and do things like that so that should slow down the traffic. So in a narrower space the cars are moving in a more uniform way

Nilabja Das
Nov 21, 2018

The space is narrower but the number of cars is the same. So the cars will have to move faster.

This was an easy one for me because of my fluid dynamics course and my experience merging into the Queens-Midtown Tunnel in New York!

Harish Sasikumar
Nov 18, 2018

The assumption is that the cars do not wait (in contrast to the regular traffic). In such a case, the average velocity in the narrow section will have to be not just higher, but exactly double that of in the wider section, as double the number of cars have to pass through bumper-to-bumper

The cars in the wider lane have to wait their turn, so their average speed goes down.

Vasudha Uddavan - 2 years, 6 months ago

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They are talking about the speed of moving cars. If waiting is allowed, the velocity in narrow section could be anything, even less than the velocity in the wide section.

Harish Sasikumar - 2 years, 6 months ago
Andy Jiang
Nov 24, 2018

Cars have to slow dowm when merging together so there is traffic jam in the merging part.

Dharik Anwar
Nov 22, 2018

In the wider path the cars can move slow because it has more space.But in the narrow path the cars need to move faster than in wider path.

This is similar to fluids mechanics

Alliana Pagulayan
Nov 20, 2018

In order to do a zipper merge (the proper way to merge lanes- one car from each lane in a back and forth pattern), the flow of traffic must slow down to accommodate the car in the next lane going into the narrow lane. That's how I thought of it at least!

When you’re waiting to merge, you are waiting for someone else beside you to take their turn, and every person who gets to the bottleneck has to spend half their time waiting, which cuts your speed in half. When the car gets to the bottleneck they’re free and clear because they can go as fast as they want under the bridge (with the limitation being the driver in front of them and of course the law) and the single file line of people coming from the bottleneck are the only traffic on the other side. Especially if there is not a stoplight ahead.

Bottleneck? Half the time? Half speed? Bridge? Stoplight? How dost thou bring in imagined things?

Jesse Otis - 2 years, 6 months ago
Vishrut Nalamwar
Nov 20, 2018

I feel that, (assuming normal situation ) as there can be more than one car at the wider section so the chances of collision are more and to avoid it driver has to slow down the car and at narrower section , due to single lane , probability of the occurance of second car is negligible and psychologically driver feels it as no risk situation...

Ayo Abioye
Nov 19, 2018

For the two lane traffic merging into one lane as shown in the figure, cars on the wide part would be waiting half the time in the queue for cars on either lanes to merge, but once merged, the traffic should flow twice as before the merge, hence one would expect that cars should be moving faster in the narrow part.

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