The Tappet Challenge

It seems my brother Nate is becoming a major source of blog material.  The other day, he pointed me to NPR's podcast of last Saturday's Car Talk.  Near the end of the show (44:38, if you'd like to hear for yourself) the hosts take a call from a woman looking to prove her husband wrong about his driving style.  The husband claims that you can get better gas mileage by accelerating up to some top speed, then allowing the car to coast until it slows to a minimum speed, before accelerating once again to the top speed.  Click and Clack maintain that, whether or not it really is better for your mileage, any money you save will be outweighed by the damage you do to your car.  They invite (or perhaps dread) any physicists listening to send them evidence one way or the other, so here I am.

We'll assume there are only two significant forces on the car: the wind resistance slowing it down, and the car's own acceleration speeding it up.  Given this, we can write a differential equation for the car's position:

where a is the acceleration applied by the car, and c is a constant associated with the wind resistance.  Solving the equation gives

which is admittedly pretty awful.  However, we're not really interested in t, so we can differentiate this to get the velocity, and solve the two equations simultaneously to get

We'd like to know the distance traveled to get from an initial speed vi to a final speed vf.  We can find this by taking the difference between the two points:

Now that we have this equation, we can find the distance the car travels during one cycle of accelerating and decelerating.  If we call the car's minimum and maximum speeds vlo and vhi, then the distance spent accelerating is

and the distance decelerating is

Using these, we can define an efficiency, ε, analogous to a measure like miles per gallon:

where E is the energy expended in traveling the distance.  It will be given by

where m is the mass of the car.  Plugging everything in, we have

where

We'd like to compare this to the efficiency of maintaining a constant velocity.  First, we need to know the average speed of the accelerating car.  Going back to our original differential equation, we can get the time in terms of the change in velocity:

We can use this to find the average velocity with

but the result is pretty awful, so I won't write it out just yet.  In this case, our efficiency is given by

so that makes our nasty equation even worse.  To spare you the horror, I decided to toss in some dummy values and see what sort of results we get.  To get a comparison of the two efficiencies, I divided the constant speed efficiency by the accelerating efficiency, so any values greater than 1 indicate that constant speed is the way to go.

Using the 65/55 speed range they discuss in the show, we get

Acceleration	Ratio
1	1.00697
3	1.00697
5	1.00696
7	1.00696
9	1.00696
11	1.00695
13	1.00694
15	1.00693
17	1.00692
19	1.00691
21	1.00688
23	1.00685
25	1.00679

so it seems Click and Clack were correct in their theory that the husband's method does not save gas.  I tried fiddling around with the speeds a bit, and I couldn't find a case where accelerating and decelerating is a good idea.  However, these equations I'm using are a real mess, so I can't be sure I've got everything exactly right.

Thanks for another great tip, Nate! I considered closing with, "Don't drive like my brother," but I don't even have a license...