The Emperor has decreed that the speed limit for horses and horse-drawn vehicles within city limits is to be 5 leagues per hour. As usual he has left the implementation details to his Scientific Adviser.
Using only medieval-type measuring devices, how can the Adviser satisfy the Emperor's wishes?
You can assume that there are wooden odometers available
and medieval clocks
It presumably doesn't make any difference to the answer but a league can be assumed to be 3 miles.
No technology is needed at all:
From http://www.speedofanimals.com/animals/horse:
All horses move naturally with four basic gaits: the four-beat walk, which averages 6.4 kilometres per hour (4.0 mph); the two-beat trot or jog, which averages 13 to 19 kilometres per hour (8.1 to 12 mph) (faster for harness racing horses); and the leaping gaits known as the canter or lope (a three-beat gait that is 19 to 24 kilometres per hour (12 to 15 mph), and the gallop. The gallop averages 40 to 48 kilometres per hour (25 to 30 mph).
Only the gallop is over the limit: The speed limit can be enforced with a simple no-galloping rule. Any minimally-trained observer can tell the difference between the gaits.
With medieval roads? Don't even bother! Their axles will shatter if they try going too fast.
If you insist, put up speedbumps every hundred metres or so, or have twisted roads within your city limits. That will limit speed without needing the excess costs of constant monitoring. Anybody going too fast will either lose their wheels or make a sufficient racket to get bystanders' attention.
Frame Challenge
My knee-jerk reaction is that no one during medieval times would think in terms of distance-per-time. They're far more likely to think in terms of the behavior of the primary engine of motivation: the horse. In other words, the law would be that no one can run their horse above a canter. Everyone would understand that. Nobody would understand distance-per-time, even if you could invoke the tech to do it.
Add to this that historically politicians have never understood science and it's pretty much a guarantee that what you're seeking will never come to pass. The emperor would declare that no one can go faster than his horse (because a royal horse is the correct speed, after all).
I would just post an officer with an hourglass at one end of a street of known length. If he sees a horse or a vehicle passing a certain point at the beginning of the street, he turns the hourglass; if the road user passes the end of the street before the hourglass has finished, that means they're speeding, and eligible for a fine. If you make the fine high enough, e.g. seizure of the horse or carriage, people will make sure they'll never be speeding and keep a safe margin away from the maximum speed.
I'm not sure how precise medieval hourglasses were, but (sorry for using another system) your maximum speed amounts to 6.7 m/s; given a street of 100 meter, this is 14.9 seconds; you'll need 1.5 second precision (from both the hourglass and the observer) to obtain 0.5 league per hour precision.
If you want a reliable measuring equipment, you need to have better than medieval clocks and odometers.
A clock which can be transported on bumpy medieval roads and still measure the time in a good way is probably out of time for the middle age.
I think it is more practical to limit the horse gait. Step, trott and gallop come with peculiar velocities and can be easily recognized without any measuring device. Furthermore, it's harder to tamper with them.
These are medieval times. The Emperor's authority is absolute, and the Emperor's officers' authority can only be countermanded by the Emperor.
So you just need to show the traffic enforcement soldiers (because they will be soldiers) what a horse running at 5 leagues per hour looks like, and then say "anything faster than that, book them."
As with Judge Dredd, these soldiers are judge and jury. You have no right of appeal, no right of complaint, no right to do anything except pay them, which you do at the point of a sword. It doesn't matter whether they're right.
soldiers(professional standing army), most of the army was levies(drafted peasants and city-dwellers). So, probably, there were not a lot of soldiers to waste on watching some carts passing by.
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Commented
Dec 26, 2018 at 16:11
A technically simple - but reasonably expensive to implement - solution is to add grooves to all the roads (running crosswise, not lengthwise). If they are spaced evenly, then the frequency of the noise made by a cart traveling along it corresponds to the speed the cart is traveling at. The rule can be as simple as prohibiting the carts from making a noise higher than a particular note, and can be checked by anyone with minimal training.
It will be expensive to add grooves to all the roads, though, including maintaining them when they wear out, and it will be unpopular to make noisy streets even noisier.
Odometers/speedometers/ etc. will not work here because they can be mandated for carts but not for horses - you can make a speedometer today (and even 100 years ago) that could fit on a horse without weighing it down too much, but you could not have done that 500 years ago. So the only solutions are, as suggested by others, are to measure the speed externally by:
Limit based on horse gait (if it comes through the gate with the wrong gait...) By far the simplest with the catch that someone might try to train their horse to run differently, though I think that would be impractical at best. This works as long as the desired speed matches well with different gaits, and is the simplest yet also subjective. But then again, the word of the emperor's traffic enforcer is considered trustworthy by definition.
Hourglass. The good part about an hourglass in medieval times is that if you make one and it turns out to run too slow or too fast, due to the hole between the sections being hard to reproduce exactly the same between hourglasses, it really doesn't matter. The emperor & his scientific advisor produce one reference hourglass. When a new hourglass is manufactured, you add or remove sand until it runs for the same amount of time.
The emperor has another trick up his royal sleeve: If he wants to arrest someone, he can have his traffic enforcer release dogs into the street as his rival crosses the starting line. The horses react and gallop along uncontrollably for a block and the enforcer gets to write his ticket. Which can include, at his discretion, a visit to the dungeon.
Major roads (anywhere where it would be possible to exceed the speed limit) will have a... something... alongside them traveling at exactly 5 leagues per hour. As long as you are going slower than the something, you are fine. If you are passing it, you're in trouble...so that part is simple.
Now the more complicated part is what should be the "something". I think a marble run could do the trick. You'll want to shield it somehow (something translucent like waxed paper) to prevent wind from interfering, and you'll need to create some friction in the track to maintain a constant (non-accelerating) speed. Also you'll want something like a watermill to power the marble lift to get the marbles back to the top again. It might take a really smart person to come up with the idea in that era, but actually building it is should be within medieval technology (although building one on every major road will be a big undertaking).
An answer in what I think is the spirit of your question: Use the wooden odometer you have included in your question, and use it to haul a flag up a pole on the cart, via friction. Calibrate these things so that the flag slips down the driving element (say, twine) at a known rate such that if the cart goes faster than X speed, the twice drive the flag up the pole faster than it can slide down -- a lossy driven element. If the speed is below X, then the flag remains at the bottom of the pole, out of sight.
Then you local modder community can sell shifty apparati that decrease the friction, and your guards can inspect poles and twine and flags, all under the guidance of your science advisor.
I think this is what you're looking for based oion the way you have phrased your question.
Source: I did this in A.D. 1453, and the slow response time of shifting reinforcements around was part of why we lost Constantinople.
I like Glorfindel's proposal, but the hourglass for every traffic warden is not really necessary within city limits as long as there can be a bell tower.
The problem with that is that it only provides average speeds. If the average speed is above the limit then the top speed must have been above the limit as well. But an average speed below the limit does not prove that the cart never went above top speed.
Completely insane, of course.
Pressure plates a known distance apart in the road - one to open a water valve, and the other to shut it. The water thus pours into a graduated container, and the amount of water dispensed is inversely proportional to the speed of the animal or vehicle activating the plates. A technician is required to empty the container after each vehicle, and to intelligently resolve multiple-vehicle situations.
