Monday, February 25, 2013

Stuck in Traffic? Pass the Time with Physics

When I find myself stuck in traffic, my mind first turns to the shortcomings of public transit for my typical commute - a part of me wishes I still lived downtown.  The next place my mind often wanders to is physics; a surprising number of seemingly abstract scenarios actually describe the motion (or lack thereof) of one's car in a system of interconnected streets.

One such analogy is that of a system of interconnected springs and masses:

Imagine that you are the sixth car waiting, single file, at a red light.  The moment that the light turns green, the first car begins to accelerate, but you do not move.  Each car must wait for the car in front of them to displace in order to proceed forward.  It is the same for the system of masses.

If a force is imparted on the first mass, there will be a time delay before the effect of this force is felt by the tenth mass.  We can think of the springs like spaces between cars, and the masses as the cars themselves.  This analogy is far from perfect.  For one thing, cars are more independent than this model would suggest.  Car 1 is unaffected by all those behind it, whereas the motion of mass 1 is greatly affected by the motion of those behind it.

Where this analogy succeeds is in the idea that a disturbance takes time to propagate through a medium.  As the number of masses and springs tend to infinity, it actually represents a simple uniform medium, like a string, which has some inertia due to its density, and some stiffness due to its elastic modulus.  These properties, combined with how much tension the string is in, determine how quickly a wave travels across it.  If you are an element of string sitting a distance x from the tip, where someone shakes it, you can know exactly how long it will take to reach you: t = x/v, where v is wave speed. 

It can be shown that the wave speed for a string under stress is about (σ/ρ)1/2, where σ is the stress and ρ is the volumetric density.  So, the time that it takes for information to travel from the source to some point along the string that is x away is x(ρ/σ)1/2.

Now, let us say that you are the nth automobile in line at a red light.  How much time after the light turns green must you wait before moving?  By analogy, the amount of time should be n(ρ/σ)1/2.  But, what are σ and ρ in the car scenario?  As with the uniform string, we assume that each car (string element) is equivalent: one kind of car.  For the string, σ represents how tightly it is held but also points to the force that restores equilibrium if a string element is displaced.  For a car, this parameter is a measure of the alertness of the driver.  Correspondingly, ρ, which measures the density of the string (inertia per unit volume) must relate to the mass of each car (its resistance to accelerate).

The amount of time you wait is then dependent on how many cars are in front of you, the alertness of the drivers in front of you, and the mass of their cars.  Ten Honda Civics with alert drivers ahead of you is better than ten Honda Odysseys with exhausted drivers.  This knowledge can help one choose which lane to wait in at a red light.

I know what you're thinking: "This guy must be in traffic a lot to think of such things."  Well, it gets worse...

The best analogy for a complex system of interconnected roads with different numbers of lanes may be found in fluids.  We can imagine, instead, a complex system of interconnected pipes with various cross-sections.  Instead of transporting cars, the pipes transport water molecules.  The key parameter associated with fluids in pipes is the mass flow rate, measured in kg/s.  For cars, it is the car throughput, measured in cars per second.  One can then apply the continuity theorem to cars, noting that the number of cars that enter a roadway must also leave it.

We sometimes see that the flow rate of traffic decreases as we approach a section that reduces from three lanes to two.  It is similar to water in a pipe: if we decrease the cross-sectional area of the pipe, the mass flow rate that precedes it will decrease.  So, the number of lanes on the road relates to the internal area of a pipe.  In water systems, pressure differences are what propel the water - the corresponding parameter for roadway systems would be the speed limit.  Red lights are like valves, and on it goes.

This might seem abstract and useless, but it is actually relevant in practice.  Urban planners that design roadways and public transit systems use softwares that call on equations of fluids.  It actually does not matter whether we are measuring fluid throughput or car throughput if the mathematics governing it all is the same.  The study of fluids preceded the implimentation of vehicular transit, and so, once transit systems were being developed, the laws of fluids were well understood, and could be applied to them.

Other such roadway/fluids analogies exist.  Sometimes you may find yourself in a traffic jam that suddenly opens up for no obvious reason, and then you're moving again.  This situation correlates to that of supersonic fluid flow - the border between car congestion and smooth flow is like a shock wave in a supersonic air duct.

The next time you are stuck in a traffic jam, you can choose to turn up the radio to distract yourself, or, envision your car as a water molecule squeezed up against other water molecules, and conduct a theoretical fluids analysis.  I'm guessing you'll choose the radio.

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