Speaking of threadjacking, that reminds me, and this is for everyone, not just the OP.
You are on the xway in your 300+ HP BMW and just stopped to pay a toll. You are cruising at a legal 65 mph. What is the most economical (in terms of fuel only!) manner to accelerate to "cruising speed". Easy on the throttle or floor it?
If the overall distance matters, say the fuel used from the toll plaza to 20 miles down the road at speed.
I get the same number each way using simple calculations.
First- get the time for the work being done
v(f) = v(i) + a*t where v(f) = 30 m/sec, near 60 mph, v(i) = o = initial speed.
30 = 3m/sec**2 * 10sec
30 = 5m/sec**2 * 6sec
Now get the distance for the work:
d = v(i)t + (a*t**2)/2
for 3 m/sec**2 acceleration, distance = 150 m
and 90 m for the faster 5 m/sec**2 acceleration
Get a force for each acceleration:
F=m*a, let m=1kg since it's the same car (kg=kilograms)
for the slow 3 m/sec**2, F=3gm/sec**2
Faster acceleration, F= 5gm/sec**2
Work= force * distance
w= 3 kg/sec**2 * 150m = 450 kgm/sec**2
w=5 kg/sec**2 * 90m = 450 kgm/sec**2
The work done is the same in each case (unless i worked myself into a circular argument). You will need more power to accelerate faster as the faster acceleration needs more energy per unit time.
In a more practical sense, it is more complicated.
This study uses fuel consumption simulators for 15 late-model automobiles to determine how one ought to drive to maximize fuel economy. The simulation is based on extensive on-road and dynamometer testing of the 15 cars. Dynamic programming is used to determine the optimal way to accelerate from rest to cruising speed, to drive a block between stop signs, and to cruise on hilly terrain while maintaining a given average speed. The dependence of fuel economy on cruising speed is also characterized for various road grades. Findings include that optimal speeds are generally higher for larger cars and higher on downgrades than on upgrades, and that the relative fuel penalty for exceeding the speed limit is no worse for small cars than large cars. Optimal control for accelerate-and-cruise and for driving between stop signs varies considerably from car to car; in the latter case fuel economy is much improved by achieving a rather low peak speed. Optimal control on hills is consistent from car to car and can achieve fuel economy 7% to 30% better than that of constant-speed driving on 3% to 6% grades. Results that appear generalizable to other cars are reduced to advice for the fuel-conscious driver.
Ref:
http://www.sciencedirect.com/science/article/pii/0191260788900362