Look-at-Me Car

Copyright 2009 by Paul Niquette All rights reserved.

bservations in A Certain Bicyclist may be paraphrased as follows: Any vehicle moving throuh air must overcome wind resistance, which is a force that increases according as the square of velocity of the vehicle.  The power to overcome wind resistance increases according as the product of that force and velocity.  Thus, power to overcome aerodynamic drag increases according as the cube of velocity.

Let us conduct our analysis in the power domain using simplified parameters for a vehicle traveling at v mph...

Power to overcome aerodynamic drag = a v3
Power to overcome rolling and drive-train resistance = b v
Power supplied to accessories = c
where...
a hp/mph3 is a coefficient relating wind resistance to power
b hp/mph relates constant rolling resistance to power
c hp represents power for accessories
v mph denotes instantaneous vehicle velocity
Sophisticated solvers will note that a lumps aerodynamic drag coefficient with characteristic frontal area commonly used to calculate wind force; that b combines rolling coefficient along with drive-train resistance, which are linear multipliers of speed; that  c sums up internal power-plant loads as well as accessories which do not vary with vehicle speed.  We have that...
p[v] = a v3 + b v  + c
...where p[v] hp represents the requisite power to sustain vehicle motion at v mph.  Substituting information parsed from statements in the puzzle, one is able to write four equations...

{1}      p[150] = 1503 a + 150 b + c = 300
{2}      p[60] =  603 a + 60 + c = ?
{3}      p[0] = 03 a + 0 b + c = 5
{4}      603 a = (60%) p[60]

...and take note that there are four unknowns: a, b, c, and p[60].  It is not generally practical to make direct measurements of p[60] in a moving automobile.  Solvers have been given only this:  "At a 'highway speed' of, say, 60 mph, the Stingray's sleek design demanded a mere fraction of the engine's power, the rest being held in reserve for rapid acceleration."

Solving these four equations simultaneously gives the following values:

a = 0.0000823 hp/mph3
b = 0.114 hp/mph
c = 5 hp
p[60] = 29.6 hp
hus, for the 1966 Corvette C2, "mere fraction" apparently amounted to about 10% of the engine's power as being required for 'highway speed'.   The Look-at-Me Car is depicted below in a family of requisite horsepower curves for representative highway vehicles: Honda Civic, Ford Crown Victoria, Chevrolet Blazer, Hummer H3.

For calculating mileage, the puzzle invited solvers to assume an ideal Specific Horsepower of 2.0 hp/lb/hr.  A gallon of gasoline weighs 6 lb, which means that for every gallon-per-hour consumed gph, the engine will deliver 12 hp. By simply dividing delivered hp by 12, we obtain the fuel flow rate in gph.  Traveling at v mph, the vehicle consumes 1/12 gpm for each hp being demanded.  By dividing gpm into mph we get what we want, mpg...

...and our solution to the puzzle...
 

 [b] 23 mpg

ou are invited to adopt the expression "60-60 AME" (aerodynamic mileage estimate) to describe the solution.  Thus, our 60-60 AME for 1966 Corvette is 23 mpg. The main assumption in the 60-60 AME is that, at 60 mph (highway speed), 60% of engine power for a given vehicle is devoted to overcoming aerodynamic drag. The validity of that assumption holds up for a wide range of vehicle shapes and sizes.  Here's why. 

Whereas the solution to the Look-at-Me Car puzzle has so far been conducted entirely in the power domain (hp), both aerodynamic drag and rolling resistance are customarily treated as forces (in lb or any other convenient unit described in Erg and Ugh).  We might write...

f[v] = a' v2 + b' 
where f[v] lb represents the requisite tractive force to move the vehicle at v mph.
a' lb/mph2 is a parameter relating wind resistance (aerodynamic drag) to speed
b' lb is a parameter that represents rolling resistance, independent of speed.
The aerodynamic parameter a'  is a function of shape (the extent to which the vehicle is streamlined) and size (in particular the vehicle's frontal area presented to the windstream).  In Tin-Can Mystery, solvers confirmed that for objects of a given shape, various areas (surface area, cross-sectional area, wetted area) typically vary according as the two-thirds power of vehicle weight (w2/3) by virtue of the Cube-Square Law

The rolling resistance parameter b'  is not a function of shape but varies directly with the vehicle weight (in particular the 'normal' force against the roadway).  That force, of course, is supported by the wheels at their respective 'contact patches', which are four finite areas defined mostly by wheel diameter and tread-width.  Thus, b'  also varies according as the two-thirds power of vehicle weight (w2/3).  Exclamation point optional.

Using Ka and Kb for elementary constants of proportionality, the following relationships apply to a vehicle of a given shape, relevant to size

a' = Ka w2/3 and b' = Kb w2/3
a' / Ka  = b' / Kb
a' / b'  = K / Kb = 1/ K
b'  = K  a' 
...indicating that for any value of a' determined by shape, there exists a corresponding value of b' relevant to size.  We may then write...
f[v] = a' v2 + b' 
f[v] = a' v2 + K a' 
f[v] v = a'  v3 + K a' v , but 
p[v] = a' v3 +K a'  v  + c.
...and setting a = a' and b = b', then for the 60-60 AME to apply, we make substitutions in equations {2} and {4}...
p[60] =  603 a + 60 K a  + 5
603 a = (60%) p[60]
...in which sophisticated solvers will find only constant relationships, flexing from p[60] and taking note of the fact that p[60] is itself a constant, inasmuch as v = 60 mph in the analysis.
a = (40% p[60] - 5) / 60
b = K a
Using the formulation developed here, we are equipped to produce the 60-60 AMEs for various automobiles, including the 40 graphed below.
ome solvers may be surprised to see in the "Mileage vs Vehicle Speed" graphs above that each mileage curve has a maximum -- that driving slowly wastes gasoline.  Apparently, poking along at 5 mph burns about the same amount of fuel per mile as at highway speed of 60 mph. The 1966 Corvette apparently achieved its best fuel economy, 38 mpg, while moving at about 25 mph. The closed-form derivation requires a little differential calculus. 

To find the maximum for 12 v / p[v] = 12 v / (a v3 + b v  + c), set the first derivative to zero...

d{12 v / p[v]}/dv = d{12 v / (a v3 + b v  + c)}/dv = 0;
{12 (a v3 + b v  + c) - 12 v (3 a v2 + b)}/{a v3 + b v  + c}2  = 0;
(a v3 + b v  + c) - v (3 a v2 + b) = 0;
- 2 a v3 + c = 0.
...and solve for v and p[v].

Finally, as pointed out in To Brake or Not to Brake, mileage ratings for automobies... 

...are empirically determined by an EPA driver following a script on a dynamometer.  That these will be typically more generous than the steady-speed 60-60 AMEs is explainable by the low-speed segments of both the "Federal Urban Cycle" and the "Federal Highway Cycle." 
 


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