The Kinematics of Hypermiling or, what is your time REALLY worth?

Most aggressive drivers believe that their behavior and driving style saves
them lots of time getting from point A to B, and is worth the extra fuel
burned in the process.  This attempts to analyze just how much time that
works out to, and map a little theory into real-life situations that may help
anyone understand the pros and cons of spirited vs. conservative driving.

First, we need to make some assumptions.

We'll use two identical cars with everyday performance characteristics.  The
aggressive driver enjoys its acceleration, a middling-to-fair 0-60 in 10
seconds.  To keep those annoying inter-car gaps at a minimum while arriving at
the next stop, our aspiring street-racer is also willing to decelerate from
that 60 MPH in 15 seconds, just enough to keep the groceries from flying off
the back seat but still deriving relatively little benefit from coasting
and using the built-up kinetic energy.

Our hypermiler, on the other hand, knows the drivline's efficiency curves,
plays it for all it's worth, and likes taking a full 30 seconds to reach the
same 60 MPH, which would be considered grandmotherly-grade anemic by today's
standards.  As if that wasn't different enough, she's aware of the other
factors always at work against the car's forward progress -- tire rolling
resistance and air resistance, primarily, but perhaps she's also driving a
hybrid in which a fair portion of stopping energy can be captured by gentle
regenerative braking that doesn't stress the capacity of the hybrid system to
store it.  In a car like a Prius, for example, the benefits of longer, gentler
braking stack up even more due to efficient energy recapture, but simply trying
to use the brakes less in any car will leave more of the stopping work to the
other unavoidable forces already on the car.  A long, well-predicted glide down
to the stoplight takes this driver 45 seconds, minimizing or entirely ceasing
fuel consumption as early as feasible in the game -- no point in using any more
fuel to get there anyway, since momentum is a form of energy storage and all
the traffic has to ultimately stop.

To summarize -- I've given the aggressive driver a generous 3:1 acceleration
and deceleration factor:

			Aggressive	Conservative
	0-60 accel	   10		    30
	60-0 decel	   15		    45

Now, we're going to put these two drivers on the same road together and have
them run the same distance.  Maybe they're driving hybrids, maybe something
else -- doesn't really matter.  We're also making the assumption of linear
acceleration and deceleration and that both drivers have the leisure and space
to consistently perform as they're used to.  We're making a bit of a reach by
hard-limiting the top speed of both cars to 60 MPH, but also requiring that
both *attain* that speed, even if just for an instant, to make calculations
easier and because the emphasis of this article is on the trip-time effects of
acceleration and deceleration rather than steady-state conditions.  The whole
debate about speed limits is for another time.

The minimum trip length, therefore, is what the hypermiler manages to cover
during acceleration to just 60 MPH and then deceleration right back down to
0.  This chart shows her velocity profile relative to time in seconds:

taking 75 seconds to complete the run.

Remember your high school physics?  The distance covered by this speed profile
is calculated from the integral of her speed, or the area under the velocity
curves.  For a right triangle, area is 0.5 (b * h) or half of what the base
and height would define as a rectangle.  The general formula for distance,
(1/2)a(t^2) also reduces to this using the given assumptions.  Our units are
kept simple -- seconds on the time axis and miles per hour on the speed axis,
and when multiplied together to determine area we get what I'm calling
"distance units" or DU -- which is simply 1/3600 of a mile.

So our hypermiler covers

	(0.5)(60 * 30) + (0.5)(60 * 45)  =  2250 DU

or 0.625 mile over 75 seconds from stop to stop.  This is used as the bounding
case because the aggressive driver will do the same leg in less time, by
rapidly outdistancing the hypermiler at first while coming up to 60 MPH,
traveling for a while, and then cramming on the brakes at the end.  We're not
yet sure how long the fast car needs to stay at speed, so the length of
steady-state travel is left indefinite:

But the problem now is making the AREA under the aggressive driver's curve the
same as the hypermiler, since they must travel the same length of road.

Over the first 30 seconds, the aggressive driver's profile can be broken into
the acceleration phase, and the remainder steady-state at 60 MPH.  We can also
tack the deceleration component on right afterward, but the combined area
under these still isn't quite enough to complete the run:

	(0.5)(60 * 10) + (60 * 30) + (0.5)(60 * 15)  = 1950 DU

falling just shy of the stopping point by 300 distance units.  From that,
it's easy enough to calculate that the aggressive driver needs to travel an
additional five seconds at 60 MPH before starting to decelerate, bringing his
traveled distance to the same 2250 DU.  So now we can draw both of these
profiles and be confident that the same area exists under each:

The aggressive driver lurches to a halt at 50 seconds, while the hypermiler
casually drifts in at 75 seconds.  The aggressive driver's 25 second lead
clocks him a time advantage of 33% over the hypermiler.  In fact, this time
delta will *always* be 25 seconds as more length is added to the course, since
when both cars are doing 60 MPH there's no additional difference in travel
time other than that the zoomy one stays 10 seconds, or 1/6 of a mile, ahead
of the laid-back one.  (Sounds like a nice safe following distance, in fact...)

We can now superimpose the speed profiles, placing the "insert point" for both
cars' steady-state period of travel at the arrow.  The aggressor is red, the
hypermiler is (obviously) green:

and now we can begin inserting additional distance, in blue.  Since we've
mapped and equalized both cars' acceleration and deceleration phases as
immutable sections, any added distance increases the area under both curves
by the same amount -- in other words, we're just moving the stoplight farther
and farther away.  Here's what one full mile looks like, in which both cars
remain at 60 MPH for 22.5 seconds:

which shrinks the street-racer's time advantage to 26%.  As distance continues
to expand, the time delta begins to have less and less significance.

Here's how it works out numerically:

And when quick-n-dirty graphed as % gain over distance traveled:

The time benefits are beginning to look fairly miniscule by comparison to
the overall trip, provided that both drivers adhere to the speed limit.  

But wait, haven't we been reading just about everywhere that changing driving
habits away from full-bore acceleration and sudden stops can easily save anyone
20% in overall fuel usage?  For a trip distance greater than a mile, our
supposed "time advantage" rapidly yields to the increased fuel consumption
needed to attain it.  Now, this is an idealized scenario for just one
start/stop cycle, and real life on the roads is different -- but different
in ways that over the long haul, favors the hypermiler's goals.

Think of it this way: if you're looking for a challenge on the road, it is
requires much more thought and finesse to calculate the energy *envelope*
needed to get from here to there, as opposed to mashing a pedal to beat the
other guy out of a fresh green.

Now, one may immediately come up with the counterargument that the 25 seconds
will make all the difference with regard to getting through the NEXT traffic
light or not.  Sure, an aggressive start might get the lead car to the next
light before it goes yellow, but on the other hand with random light cycles,
that lead car might also have to spend some time stopped at a red one --
meanwhile, the hypermiler in that long, well-predicted glide may very well
reach it just as it goes green and now has some speed advantage accumulated
too, and used no fuel or brake lining in the process.  The aggressive driver
then has to burn to *catch up* and keep going!  The math starts to get gnarly
from there, but suffice to say that in everyday traffic, the signals are the
"great equalizer" -- without illegally running reds or being in an emergency
vehicle on call, the aggressive driver has to wait just like everyone else and
the perceived time savings quickly becomes just that -- a perception.  There
is absolutely nothing wrong with spending a large part of the "red time" in
coasting TO the light, as opposed to just sitting there looking up at it.  The
waste heat difference is significant.

Now, where do we get to go 60 MPH between lights? -- on those fairly major
secondary highways, where we nonetheless have signals but the distances between
them are often quite long, extending the "blue" part of the trip even longer
and shrinking the relative difference.  For a run on an interstate, it truly
doesn't matter how fast anyone starts or stops, which every trucker who's
climbed out of a mountainside rest stop knows quite well -- even if he has to
crawl up that next hill at 40 MPH with the 4-ways on, the load gets where it's
going on time.

About the only scenario that gives the puncher-of-pedals a continuously
increasing time advantage is a sequence of stop signs a couple of city blocks
apart, with no cross traffic.  Hops can be immediately chained together, and
he'll come out of it with glowing brake rotors and an empty tank, but way
ahead of the hypermiler.  Fortunately, such environments only occur in certain
regions of towns and are thus small and/or rare.

So make your own tradeoff.  20% more fuel burned for a 4 or less percent time
factor most of the time?  Doesn't seem worth it, does it?  And we haven't even
touched on the increased safety and decreased frustration factors that the less
aggressive driver routinely enjoys.