Automobile Gas Mileage
Claim: For every 5 mph over 50 mph, there is a loss in gas mileage
of 1 mile per gallon. How valid is this claim? Note that this
suggests a linear relationship:
Problem Identification – What is the relationship between the
speed of a vehicle and its fuel mileage
Assumptions:
Fuel mileage = f(propulsion forces, air temperature, tire condition,
road condition, engine condition, drag forces, driving habits,
vehicle shape, wind, weather, ….)
Restricted Problem Identification – For a particular driver, driving
his/her car on a given day on a level highway at a constant speed
near the optimal speed for fuel efficiency, how does fuel efficiency
vary with small increases in speed.
Note – We have assumed that many variables are constant. By
restricting the problem, we can develop a manageable model. We
use this model to see if the linear relationship hypothesized above
is qualitatively correct.
Developing the Model
Since the car’s velocity is constant, its acceleration is zero. So the
forces of propulsion and resistance must be equal:
Fp = Fr
Consider Fp: Let K be the amount of energy in a gallon of gas
and let Cr be the amount burned per unit time. Then the power
available to the car is CrK. Assume a constant rate of power
conversion. Then
Converted power ~ CrK
Power = Force x Velocity or Force = Power/Velocity
Cr K
So Fp ~ .
v
If we also assume a constant fuel rating K, this becomes:
Cr
Fp ~
v
Consider Fr: A common assumption is:
Fr ~ Sv2
where S is the cross sectional area of the car perpendicular to the
direction of motion.
Since S is a constant (why?), this becomes:
Fr ~ v 2
Since Fp = Fr,
Cr 2
~v
v
or
Cr ~ v3
This is not quite the relationship we need because Cr is fuel burned
per unit time. If we drive faster, we spend less time driving. We
need to look at:
gas mileage = distance/consumption
= vt / Crt = v / Cr
~ v / v3 = v-2
So the model predicts that gas mileage is inversely proportional to
the square of the velocity.
Homework – Read 2.4. Problem 1.