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Highway 403 is one of the most frequently used highways in Ontario. During Rush Hour there are approximately 50,000 cars on the 100 km of Highway with only 3 lanes to work with (besides HOV lanes)! As I will discuss later in this investigation, when there is only 300km in total and 272.25 km are being taken up by cars whilst stationery, there will undoubtedly be traffic congestion at times. Highways in Ontario are important to the lives of many, including family and friends of mine. Whilst my family stayed over in Markham I had lost count of how many times we got stuck in traffic. Also, if there was less traffic my math teacher Mr Chun would be able to give Extra Help after school and maybe spend more time with his kids. Overall Highway congestion is a grand problem that affects many people. After talking to some other commuters of the 403 like Dr Rezunyk, I learnt the importance of appropriate distance between cars and how that concept, as well as the cars driven improperly, can cause unnecessary, highway long traffic jams. Highways are useful for commuting but dangerous when used improperly.
Although the formula for appropriate distance between cars is d(s)m = 0.004s2 + 0.06s + 2 (s being speed in kilometers per hour and d being distance in meters, measuring the distance from the front of one car to the front of another) the speed limit is 100km/h. More often than not drivers will aim to travel at the speed limit if not over. This brings the question, how many cars can fit on the highway whilst following the speed limit? Also with 50,000 cars on the highway during rush hour, what is the actual safest speed to be travelling at? There are many factors that come to play with keeping driving speeds efficient as possible and highway congestion as low. I remember when I took my own driving course in Alberta (where you can get a driver’s license at the age of 14) and drove on many different roads including highways. I realized how many things can influence safety of driving a vehicle and the congestion of a highway. To determine efficient speeds, and an effective number of cars on the highway, you must look at many aspects of driving such as: Car Length, Highway Usage, Highway Length, Car Speed, Distance between Cars.
The average vehicle length can be calculated by taking the average of all types of car lengths. Although, trucks are much less popular than cars, whereas the population of other car types are similar. The truck length will be added using its ratio to the number of cars. The average small car length is 4.1 meters, the average medium car length is 4.465 meters, the average family car length is 4.865 meters, the average estate car length is 5.234 meters, the average SUV length is 5.684 meters, the average multi-purpose vehicle is 4.386 meters. Through the accumulation of all this data it’s been determined that the average car length( )is 4.789m. Trucks are typically 16.15 meters long and according to inferred data from statistics Canada for every 69 vehicles 1 is a truck. (Assuming that the Ratio declines at a constant rate) We will use this equation to properly include the truck weight. Ratio(Truck) x Length(Truck) + Ratio(Other Vehicles) x Length (Other Vehicles) = Average Vehicle Length. Which with added data now reads (Truck) x 16.15m (Truck) + (Other Vehicles) x 4.789m (Other Vehicles) = 4.95m. 4.95m is our Average Vehicle Length.
This graph, d(s) = 0.004×2+0.06x+2, is the formula that determines the safe distance between cars from front bumper to front bumper at any speeds (Let x be speed. Let y be distance between cars needed (in meters). Whilst making this equation it was assumed that cars would be evenly spaced out and there will always be 50,000 cars on the highway. Also it was assumed that nothing would alter the highway length (accidents, repairs e.t.c) or the types of cars on the highway (advertisements, drastic sale changes e.t.c) If the average car is 4.95 meters long, 3 highway lanes (besides HOV lanes) and there are 55,000 cars on the highway, there is 27.75km to manage (9.25km per lane). Cars during rush hour take up 272.25km of the 300 km available.
Leaving 27.75 km, 9.25 km per lane, to work with. There are approximately 18,333 cars per lane. With 9.25km spacing them out, this means there should be approx. 0.505 meters between each car. After including Car length (by adding 4.95 to the 0.505 to fully accommodate the distance from front bumper to front bumper), according to the first graph you must be going at 22.832 km/h to achieve this distance between cars. Keeping at this speed would largely reduce congestion, but vehicles would no longer be slower because of having too many vehicles for the road to handle. Vehicles would be slower because of the drop in average speed. Instead of repeating these calculations a graph has been made to calculate the distance between cars on the highway more effectively. The ‘Amount of Cars On Highway’ is a graph with the equation ‘y = 300,000/x – 4.95’ (Let Y be Amount Of Space between Cars in Meters) (Let X be Amount Of Cars). This equation makes the same assumptions as the earlier one.
This equation works because 300,000 – 4.95x = y (Letting x be the Amount of Cars on the highway and y be Space distance available on the highway in Meters) will give you the amount of space left on the highway in meters. This equation works because 300,000 is the sum of the distance covered by the highway lanes and 4.95 is the average car length, both in meters. Instead of leaving the equation as 300,000 – 4.95x = y (y being the distance available on the highway) it’s been decided to let y be the distance between each car. To do this the whole equation was divided by x because to find the distance between each car you must divide ‘the distance available on the highway’ by the amount of cars. This results in the equation y = 300,000/x – 4.95. The speed limit on the 403 can either be 90km/h or 100km/h depending on your location. At 90km/h according to the equation you must be 39.8m behind the front of the car in front of you. At 100km/h according to the 1st equation you must be 48m behind the front of the car in front of you.
Now with the added information about average car length on average, you would be 34.85m behind the car in front of you at 90km/h and 43.05m behind the car in front of you at 100km of space (48 = 0.004(100)2 + 0.06(100) + 2, after including car length by subtracting 4.95 meters from the y value). This means that if you had all 50,000 cars on that it would not be possible to have all cars going at 100km/h, because 0.505 meters of space is available (according to earlier calculations) but 43.05 meters is needed. For there to constantly be 43.05 meters between each car there would only be 6,250 cars on the highway. ( 43.05 = 300,000/6,250 – 4.95 ) After countless calculations, graph construction, research and analysis it’s been concluded that 50,000 cars pass through the highway more effective through the current method of going fast as legally possible for as long as possible. Used by citizens currently. This has been the selected conclusion because both having the speed be at maximum (100 km/h) whilst keeping a safe distance and travelling as fast as possible while keeping a safe distance with 50,000 cars on the highway are less efficient than our current use of highways.
To further explain I will introduce a Google map of the expected time of travel on Highway 403 during Rush Hour. Using normal travelling techniques, the highway can be travelled in averagely 1 hr 48 minutes. By keeping appropriate distance between cars, (at 22.832 km/h) the highway can be travelled in 4 hr 38 minutes. Also at 100km/h only 6,250 cars could be on the highway safely. Although with normal travelling techniques, cars are averaging 55.56 km/h they are still going much faster than if they were put a safe distance from each other. There are some benefits to enforcing safe distances or limiting highway usage. For example; making more predictable travel times, preventing accidents (another cause of highway congestion). Although there are many more disadvantages of enforcing these methods. For example; the cost to enforce safe distances between cars, resources needed to limit the amount of cars allowed on the highways, dealing with people that don’t follow these new rules. Altogether, our current method of using highways is the most effective.
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