What is the physics of traffic speed limit?

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Solution 1

Curved roads are banked so that less transverse traction is required to prevent a vehicle from slipping. In some parts of the US (and perhaps elsewhere), there are "advisory" speed limit markings near tight turns, which are loosely related to safe speed allowed by the banking of the turn. Here's an example where the "ramp" limit is lower than the highway limit:

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The Wikipedia article for banked turns has a nice derivation for find the angle/speed/radius for a "frictionless turn"; if I were designing a road that's what I would put on the sign.

Beware there are lots of other (non-physics) issues that go into deciding speed limits.

Solution 2

Very roughly speaking, the danger to a living being riding in a crashing car is that of forces that arise within living tissues when that being is forced to change direction swiftly (being bumped in a sideswipe) or to stop suddenly. Those forces tear and injure living tissue. The physics and biophysics content of speedlimits is of setting states of motion on roads such that the probability of crashes are low enough to make the probability of these injuries acceptably low. The definition of "acceptably low" is one decided both by legal questions and also by the culture setting the speedlimits: what "appetite" that culture has for risk and so forth. Some of the physics is as follows.

I hope you have heard of Newton's second law. Basically, nothing changes its state of motion without being acted on by an outside agent - so you won't change direction or slow down unless something pushes you.

Now imagine you're in a crash. The frontend of the car slams into another, but the rest of the car keeps moving and only begins to slow when the frontend pushes back on it, with very high force. Thus the frontend of the car begins to crumple. As the force is transmitted backwards the rest of the car slows. But there's nothing holding you, at first, so you keep moving forward until you hit the end of the reach of your seatbelt, which, moving more slowly than you, begins pushing backwards on you. This force is what changes your state of motion.

But the seatbelt only pushes on certain parts of you. How does the rest of your body slow? Well, it's linked to the parts pushed on by the seatbelt through connective tissue. So this tissue must ultimately bear the forces that slow you. If you're not wearing a seatbelt, it will be your face hitting the windscreen or dashboard (it's not called a dash board for nothing - gruesome!) transmitting those forces.

So what you want to make sure of is that if a "reasonably foreseeable" crash happens, those connective tissue forces are small enough. The technology of the car helps. If you wear more webbing in your seatbely harness, the force needed to slow you is spread over more webbing and your connective tissue needs to do less to slow the unrestrained parts of your body. An alien technology that imparted a high gravitational field to every part of your body simultaneous could thus slow you arbitrarily fast - 1600gs as in Emperor Palpatine's personal craft, IIRC - without injury because every part of your body is slowed together and no stress would arise on your connective tissue. The car's crumple zones also set how slowly the car can slow. If your car is a good crumpler, it will crumple a long way and thus the deceleration is much lower. Crumple zones are like rubber around the car: the edges of the car can deform rather than forcing the whole, monolithic block to slow at the same acceleration.

The design of roads also helps: keeping speeds below limits ensures that turning cars will not slip and thus beget an accident, as in Rob's answer.

Lastly, the human minds driving the vehicles can only process information at a finite speed. Human drivers cannot take evasive action if events are happenning too fast. Speed limits therefore keep the likely flow of events slow enough that the information processing rate needed to control the vehicle is low enough for a human mind to cope with.

Then, given all these data from road engineering, vehicle engineering university departments, reaction time experiments from biophysics departments and likelihood of serious injury data from hospitals and morgues, the legal and cultural debate about acceptable speed limits begins ....

Solution 3

The lower speed limits are mainly determined by how fast a car needs to be able to stop. In town there are lots of children around, and you need to be able to stop quickly if a child runs into the road, so the speed limit in towns is low (30 mph in the UK). Where the risk is especially high, e.g. near schools, it's not uncommon for the speed limit to be reduced (to 20 mph in the UK). On the outskirts of town where pedestrian densities are lower the limit is often increased (to 40 mph in the UK).

The stopping distance is determined by two factors:

  1. how fast the driver can react

  2. how fast the brakes can bring the car to a stop

The UK highway code assumes the driver reaction time to be $t$ = 0.675 seconds, so the distance travelled before the driver reacts is:

$$ d_1 = vt = 0.675v $$

where $v$ is the car speed.

The distance the brakes take to stop a car is determined by its weight and the friction between the tyres and the road. This varies widely between different types of car, but the UK highways code takes an average deceleration (on a dry road) of about $a$ = 6.5 m/s$^2$. The distance taken to stop is related to the speed and the deceleration by:

$$ d_2 = \frac{v^2}{2a} = \frac{v^2}{13} $$

And the total distance taken to stop is then just the sum of these two:

$$ d_{stop} = d_1 + d_2 = 0.675v + \frac{v^2}{13} $$

For example at 30 mph (13.41 m/s) this equation gives $d_{stop} \approx 23$ m. So when the speed limit in town is set to 30 mph, what this really means is that the authorities have decided that cars need to be able to stop within 23 metres.

Note that I have only discussed the lower speed limits. The speed limits out of town are only loosely based on physics if at all. In the UK the limits have been variously 70 mph, 50 mph and 60 mph within my lifetime, and in other countries they vary widely. This shows there is no clearly defined principle for setting them.

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Updated on June 12, 2020

Comments

  • Shamisen Expert
    Shamisen Expert over 3 years

    In my highschool class the teacher said that the speed limit is designed to prevent drivers from crashing. Does anyone know how they are able to determine the speed limit?

    • HDE 226868
      HDE 226868 over 8 years
      This is a legal question, not one of physics. Different jurisdictions determine it in different places based on the surroundings.
    • John Alexiou
      John Alexiou over 8 years
      The department of transportation sets the limit as low as possible without people complaining too much. The design limits are much higher than the actual limits.
    • Mike
      Mike over 8 years
      I'm voting to close this question as off-topic because this is a legal/social question, not physics.
    • Shamisen Expert
      Shamisen Expert over 8 years
      But I am asking here not on Law.SE and my teacher is a physics teacher. I thought people used curvature $\kappa$, vectors and calculus to determine this number but apparently I overestimated them
    • Selene Routley
      Selene Routley over 8 years
      @HDE226868 I don't altogether agree. There is definite physics here, both traditional ``hard'' and also biophysics such effects as reaction times.
    • HDE 226868
      HDE 226868 over 8 years
      @WetSavannaAnimalakaRodVance That's something quite perceptive that I completely missed; thank you. The thing is, while curved roads influence speed - as rob pointed out - in many other places, such as straight roads, physics doesn't influence anything.
    • Kyle Kanos
      Kyle Kanos over 8 years
      Outside of banked roads, there really isn't any physics to it. Heck, there are different speed limits along the same stretch of road along I85 (60, 65, & 70 mph) with not much differentiating the stretches (population, road type, etc).
    • anna v
      anna v over 8 years
      @KyleKanos Surface traction needs physics and limits speed, depending on tires and construction, undulations due to terrain or changes since construction need physics for maximum speed too. One does not want a car to take off for example
    • Kyle Kanos
      Kyle Kanos over 8 years
      @annav: This is true, but the speed limits in that sense are well above posted speed limits--these limits are derived from non-physics reasons (notably safety concerns)
    • John Rennie
      John Rennie over 8 years
      I think we're missing the obvious here. The in town speed limits are there to protect pedestrians, cyclists, etc, and the link to physics is in the calculation of the stopping distance. The national speed limits are so variable that there is no obvious direct physical connection.