Pyramid Science

This is for researching science-based articles and the contents are for personal use although a wider potential interest is possible and so they are left here to view. No medical advice is given and a qualified medical practitioner should be consulted if any concerns are raised. Comments have been disabled, but any and all unsolicited or unauthorised links are absolutely disavowed.

Tuesday, October 26, 2010

Fast Driving And Increased Fuel Cost

Car Fuel Efficiency
Economical Driving
Economical Driving - The Physics
Fuel Consumption Ratings
Head Injury
Internal Combustion Engine

Economical Driving and the differences between 20, 30, 40, 50, 60, 70 and 80mph (v) have considerable consequences with respect to fuel consumption. Assuming the same car, the only real change is the actual speed and the resultant wind resistance (proportional to ½mv²) and since the mass is (relatively) the same, it can be ignored.

Increased speed is matched with increased frictional forces, though probably the effect is minimal compared to the increase in wind resistance created by speed.

Inertia to get the vehicle moving is ignored - the resistance to motion

The horizontal motion and the downward vertical pull by gravity combine to create an overall downward force. This will not disappear as the forces remain in operation  This is similar in principle to the forces on an orbiting satellite around a planet: the constant pull towards the planet as it moves away from the planet results in orbit around the planet. Frictional and gravitational forces are a constant. It is the stop/start and slowing/acceleration that causes the massive increase in consumption as the rate of change in the motion of the mass of the vehicle is constantly altering.

Speed/Time Taken (½mv²)

30mph: 100 miles = 200 mins (450)
40mph: 100 miles = 150 mins (800)
50mph: 100 miles = 120 mins (1250)
60mph: 100 miles = 100 mins (1800)
70mph: 100 miles   85 mins (2450)
80mph: 100 miles =   75 mins (3200)

  • Time to travel 100 miles at constant speed 
  • Resistance ≡ ½mv² (ignoring the mass and comparing the relative value, this is regarded as dimensionless)
Doubling speed results in a much greater energy requirement. That increase between 30mph and 60mph results in x4. Energy is proportional to v²/2 or (30 x 30)/2 = 450 and (60 x 60)/2 = 1800. The change between 40mph and 80mph also represents a doubling of speed with an increase of x4 resistance (800 and 3200). Both may halve the journey time, 200 mins -> 100 mins and 150 mins -> 75 mins, but to create this time saving for a speed increase x2.67, the fuel consumption difference between 30mph and 80mph is x7.11. In terms of energy (fuel used), even a x2 speed increase will result in consumption x4. Considering the higher speeds (60mph and 80mph), the fuel cost overhead is 7.11/4 = x1.78, an increase of 78% (4 + 3.11) for a speed increase of just 33% (60mph -> 80mph) and the actual journey time for the 100 miles is reduced by just 25 mins (100 mins at 60mph and 75 mins at 80mph = 25%), but the cost is raised considerably (+78%), more than x3 as much. Even the small 10 min time saving on the 100 mile journey (at constant speed) between 70mph and 80mph will have a large jump of 30% in the cost (3200/2450 = 1.3).
  • If there were to be a 30% hike in train fares to save 10 minutes on a 100 mile journey, there would be many complaints. When it comes to cars, a different mind-set can exist.
Travelling (legally) at 30mph or 70mph may increase speed by x2.33, but fuel consumption (increased wind resistance) jumps by x5.44 (2450/450) and a higher speed always costs more than slower speeds for the same vehicle. The only saving is time and can never be cost. A heavier vehicle costs considerably more at all speeds since a greater weight has to be propelled. The inertia involved with (stop/start) town driving makes the larger vehicle costlier. Large vehicles about town will always be more expensive, especially with heavy acceleration. A benefit of the long distance journey is that the overall inertia (getting the vehicle moving) is less. Another would be that the operating temperature for maximum efficiency is reached and maintained throughout the journey. Short journeys can never reach the optimum operating temperature. Once moving (from standstill), the inertia is effectively zero. But friction (contact of tyres with the road) is constant. Higher speeds result in more friction (tyres get hotter at speed) and without the constant application of power (fuel consumption), drag forces (friction, wind resistance and gravity pulling the mass downwards) would cause the vehicle to come to a standstill.

It is quite illustrative to compare short town driving journeys. Real journeys would involve frequent slow/stop, start/accelerate cycles and the inertia involved at each speed rate change would escalate all the following values significantly:
  • 30mph for 1 mile = 2.0 mins
  • 40mph for 1 mile = 1.5 mins
The time saving is just 30 seconds for the 33% speed increase. The fuel consumption at 30mph = (30 x 30)/2 = 450. The amount at 40mph = (40 x 40)/2 = 800. The difference is 350, so the extra fuel used in the 30 seconds faster driving is (800 - 450)/450 = 78%.
  • 30mph for 1 mile = 2.0 mins
  • 50mph for 1 mile = 1.2 mins
A time saving is increased to 50 seconds, but the fuel cost is much greater. 30mph = (30 x 30)/2 = 450, but at 50mph = (50 x 50)/2 = 1250. The extra fuel is then (1250 - 450)/450 = 178%.

Even a moderate (10mph) speed increase of 40mph to 50mph results in a significant fuel cost:
  • 40mph for 1 mile = 1.5 mins
  • 50mph for 1 mile = 1.2 mins
The time saving is just 30 seconds, but the fuel load increases by (40 x 40)/2 = 800 and (50 x 50)/2 = 1250 and (1250 - 800)/800 = 56%.

The raised stress level of the 'pushy' and confrontational driver is really unnecessary. Rapid approach to a red traffic light and 'tail-gating' when intimidation is the goal: the stupid and ignorant attempt at bullying and such people have a major psychological deficit. Intelligence is patently lacking. It is absurd to reach a stop at a red light by getting to the light quicker than a moderate approach by which time the light is more likely to have turned green. This would not require a stop/start cycle and the associated inertia and the likelihood is that the overall (short) journey time could be much the same. The energy difference between a few 100s metres at 40km/h or the same distance by natural deceleration (drag forces) to (or nearing) 0km/h can be significant. The time saving is an illusion.

Consider the relatively fast approach to a red traffic light over 400m at 40km/h. The duration is 36 seconds [40000m in 3600 seconds or 400m in 36 seconds] and then heavy braking to a standstill (0km/h).

A more moderate approach:
  • 200m at 40km/h (18 seconds), then 100m at 30km/h (12 seconds), 50m at 20km/h (9 seconds) and final 50m at 10km/h (18 seconds). The result is the same: final speed = 0km/h over 400m , but the time difference is then between 36 seconds (fast approach) and (18 + 12 + 9 + 18) = 57 seconds (moderate approach). This may achieve a reduction of 21 seconds for reaching the red light, but it's waiting time at the same red light. The (relative) energy used (≡ ½v²) is much more significant: 800 for 36 seconds = 28800 vs (800 x 18 = 14400) + (450 x 12 = 5400) + (200 x 9 = 1800) + (50 x 18 = 900)  or a total of 22500 = 22% less energy (fuel). The faster approach has the benefit of waiting at a red light for an extra 21 seconds. The natural deceleration by drag forces, that requires no fuel, also has the benefit of less wear on brakes.

If the driving technique involves a fast approach, then over time the additional fuel cost is very considerable. If the vehicle is heavier, the cost escalates significantly.

  • To 'benefit' from the higher-end of published performance data that 'sell' expensive cars, the cost overhead is enormous. Wealthy people probably don't notice (or care), but those buyers using credit and loan facilities, probably don't realise the cost trap being entered. 'Filling the tank' with ever-more expensive petrol will be an extraordinary wake-up call. The enormous mistake that could be made.

Egos are more expensive to run than
the extreme performance car

Travelling at 20mph near schools commonly meets with a lot of resistance. The 'inconvenience' of just a few seconds at the slower speed is enough to potentially kill or seriously injure a child (or any person).
  • 50m at 20mph takes (about 32km/h) takes less than 6 seconds. To travel the same distance driving at 30mph (48km/h) would take under 4 seconds. The 'inconvenience' equates to a staggering 2 seconds.
    Allegedly, reduced fuel economy is likely. No evidence or calculation is given to back up such a statement. That is absolutely not good enough. The arguments outlined on these pages:

    Car Fuel Efficiency
    Economical Driving
    Economical Driving - The Physics
    Fuel Consumption Ratings

    makes such a statement outrageous. To travel slower and an increased fuel load? Impossible. The description suggests a lot of stop/start cycles on possibly a busy road. Whether 10, 20 or 30 mph (average), the fuel load would inevitably be high. The actual driving conditions are rather vague: up hill in 2nd gear at 20mph or driving faster in (presumably) a lower gear defines an increased momentum. This cannot possibly make driving any 'safer' when the collision impact between a hard (vehicle) with a soft (human) body is being considered. The relative kinetic energy (difference) between the two speeds (20mph and 30mph) for the same (mass) vehicle is x2.25 (v²/2):

    (20 x 20)/2 vs (30 x 30)/2

    = 200 vs 450 = x2.25

    The momentum (mv) of an object (vehicle) travelling at the faster speed relative to a stationary object (human) is increased by a factor of x1.5 (20mph -> 30mph).

    • The easiest way to see this in action is to ride a 'bike. The muscular power required to make the machine move effectively translates directly to the fuel load an engine requires. The fatigue through exertion graphically illustrates how work varies with conditions.

    Thursday, October 07, 2010

    Pyramid Geometry

    The dimensions of the Great Pyramid of Giza possess some fascinating connections. Pi (π) is defined as the ratio of the diameter of a circle to its circumference (3.14159265...). The Fibonacci number (phi) is derived from the sequence 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610... where the sum of two preceding numbers gives the next number in the series. One number divided by the previous value moves towards phi and becomes ever closer to (approx) 1.6180339: or (5 + 1 )/2. But why √5 and not some other relationship? But... it works and is therefore empirical as an irrational number (π is an irrational number). These two constants and phi) are related and appear in the pyramid shape. This combines the dimensions of the circle with the triangle.

    One-half the base width (a =  ...144, 233, 377...) and the corresponding slope [hypotenuse] (h = ...233, 377, 610... ) provide the angle θ (144/233, 233/377, 377/610... a/h = a constant nearing 0.618034). In terms of the trigonometric ratio, arccos θ = 0.618034 gives θ = 51.83º. This is the angle of the pyramid slope. The height (o) for a base pyramid base width of 754ft yields a x tanθ or height = 377 x 1.2729 = 479.9ft. The value of π = 4a/height (1508/479.9) = 3.142 and this always applies: eg a = 144 and h = 233 (a/h = 0.618) gives height as 233 x 1.2729 = 296.586ft or π = (233 x 4)/296.586 = 3.14242749.... See Pi, Phi and Fibonacci Numbers.