Foggy advices

Classical Mechanics Level 3

Wattsia, the new capital of the land of Mechania, sometimes faces smog problems largely due to its large number of cars. The King of Mechania would like to tackle this problem and asks his 3 advisors what he should do.

Advisor 1: "My king, You should decrease the speed limit! The engines will need less force and will thus put less pollution in the air."

Advisor 2: "My king, my colleague speaks nonsense! The cars might emit less, but they will spend proportionally more time on the road. At the end of the day, they will have emitted as much gas as before. What is more, your citizens will be frustrated by the higher transit time!"

Advisor 3: "My king, my colleague misses an important point: you should instead increase the speed limit! Some part of the mechanism in our cars requires constant power; a shorter transit time means less pollution."

The King is confused by this divergence of opinions and reaches out to you for further advice.

Should he decrease or increase the speed limit, or it doesn't matter at all?

Details and Assumptions:

The contribution of smog for a single car is proportional to the mechanical energy delivered by the engine. The car engines in Mechania always work at full efficiency: they convert the chemical energy into mechanical energy with no loss.
In Mechania, the force required for a car to move forward is $a + b v,$ where $a$ and $b$ are positive real numbers which depend on the car and $v$ is the speed. You may assume this force is always in the direction of the movement.
The inhabitants of Mechania take the same trip (with the same car) independently of the speed limitations. Also, their waiting times at traffic lights, stops, etc. remain the same.
The inhabitants of Mechania respect the speed limits but generally try to go as fast as allowed.
Although surrounded by hills and mountains, the city of Wattsia itself is relatively flat so that going downhill without using the engine leaves you way below the speed limit. Also, neglect the wind (which is usually weak on smoggy days).

Increase Decrease Doesn't matter

1 solution

Antoine G
Jul 23, 2018

Relevant wiki: Work - Problem Solving

The energy the engine uses up during a trip is given by the work which equals force time displacement.

Speed limit change the force (see assumptions) but not the displacement. Lower speed lead to less force, which leads to less work, which leads to a smaller use of energy, which leads to less smog. So the speed limits should be $\fbox{decreased}$ .

Notes: On a real car, the force is usually proportional to $a + bv + cv_{rel}^2$ where $v_{rel}$ is the speed relative to the wind. This factor was here omitted as it could lead to confusion (an extremely strong backwind could mean reducing your speed is a bad idea). In practice, this factors makes going at high speed even more inefficient. The hypothesis on the flatness is also important for the same reason: being forced to brake while gravity could make you go forward faster leads to a slight increase in emissions (because of the constant energy the engine uses).

Also on a real car, the efficiency of the engine may vary according to the number of revolution per minute.

In real life, various cities use this measure, though they generally reduce the high speed limits (50 km/h remains 50km/h but 70 km/h might get downgraded to 60 or 50 km/h).

Useful tip: I don't own a car, but I was made aware of this phenomenon by two car users, who told me that reducing their speed on the highway reduced their fuel-consumption (so that they now drive consistently below the speed limit on highways and other fast lanes). Note the difference won't be necessarily visible on a single trip (because of wind, elevation, etc...). You have to do this a certain number of times and compare averages.

Foggy advices

1 solution

0 pending reports