# Do real-life vehicles obey Snell's law?

It's fairly common in educational resources to draw an analogy between refraction - light bends as it moves to a medium in which it travels at a lower speed - and a car hitting a patch of mud, and turning as the tire on the road "goes past" the tire in the mud (examples: 1, 2, 3). In case you don't see the analogy, consult this slide from the second example:

If you actually work out the details, this analogy does seem to be accurate: a car really does obey Snell's law. Specifically, in order to get Snell's law, I believe we need the following assumptions about how cars behave:

1. A car can be modeled as two points (the front tires) which maintain a constant distance $$a$$ from one another.
2. The velocity vector of each tire is always normal to line passing through the two tires.
3. The speed of the first tire (the one which hits the mud first) is always $$k$$ times the speed of the second tire.

I haven't worked out the fully formalized details, but given these assumptions I think we have a Cauchy problem with a unique solution which basically obeys Snell's law. Assumption (1) is clearly true of a real car. But what about (2) and (3)? Is that really how traction works? How accurate is the "Snell model" for a real-world car?

(3) is only true during the interval while one tire is in the mud and the other is not. After both tires hit the mud, both tires move at the same speed and in the same directiion.

A real car can skid sideways. Moreover, we shouldn't overlook the effects of the rear tires in the case of a real car.

I've never liked the car-and-mud analogy. For a single tire hitting the mud, the model falls apart. In my opinion, Huygens' principle is far easier to use and understand.

Most real cars have four wheels. If the driver keeps the steering wheel straight and none of the wheels slip then you have to apply your assumption 2 to the rear tyres as well, which constrains the car to keep going in a straight line. It should be fairly easy to see that for the car to move in the way indicated in the image you posted, the rear wheels would have to swing sideways across the non-muddy ground as the car turns.

I would guess that if you start relaxing those assumptions, you'll likely just get further from the idealisation. So we can conclude that a real car will most likely not behave this way unless it's a front wheel drive, the ground is slippery, and the front tires have much better grip than the rear ones.

It might work for a two-wheel horse-drawn buggy, as long as the horse doesn't mind entering the mud and just keeps exerting a uniform forward force. (I don't know much about horses but I suspect this isn't a very good assumption either.) Maybe one of those two wheeled "hoverboard" things would behave this way.

I've always heard this analogy made with bulldozers, tanks, or other vehicles that operate on tracks instead of wheels --- not least because the actual mechanism for steering such vehicles is to drive the two tracks at different speeds.

As to whether such a vehicle obeys Snell's Law quantitatively, that involves making a bunch of assumptions. If the speeds of the two tracks are not very different, the vehicle can drive in a circle without either track slipping. You can figure out the radius of this circle if you know how far apart the tracks are and the difference in speeds. But as the track speeds get different, or as the tracks get longer, generally one or both of the treads will start to slip.

It might be fun to try and design a Snell's Law For Bulldozers, where a machine with a certain shape approaches a boundary between a fast region and a slow region at an angle, and compute how the angle changes across the boundary. But that model would have some bulldozer-related parameters that light doesn't have, and is expect that for most of those parameters the bulldozer wouldn't "refract" the same way as light for the same change in speed.