Transcript Automobile Gasoline Mileage

Automobile Gasoline Mileage
Scenario we are interested in how fuel
economy varies with vehicular speed. We
suspect that driving at low speeds and in low
gear, automobiles convert power relatively
inefficiently, and at high speeds drag forces on
the vehicle increase rapidly. It seems reasonable
to expect that automobiles have one or more
speeds that yield optimum fuel mileage.
Question: What is the relationship between
the speed of a vehicle and its fuel mileage?
Simplify the problem
• Assume: For a particular driver, driving his
or her car on a given day on a level
highway at constant speeds near the
optimum speed for fuel economy.
• Fp: propulsion force
• Fr: resisting force
Model the propulsion force
•
•
•
•
K: energy contained in a gallon
C: the amount of fuel burned per unit time
V: velocity
Notice that
C
F p V C K  F p 
V
Model the resisting force
• V: velocity
• S: cross-sectional area perpendicular to
the direction of the moving car
• Assume Fr  S V 2  Fr V 2
C V 2  C V 3
• Since Fp=Fr, we have
V
• First conclusion: fuel consumption rate
increases as the cube of the velocity
• Gas mileage = distance / consumption
V
t
V
Gas Mileage 
 V 2
Ct C
• Second conclusion: gasoline mileage is
inversely proportional to the square of the
velocity