Constant speed vs. repeated acceleration

[tauristar]

Assuming only air friction, do you think it is equally as energy efficient to reach a destination in a vehicle traveling a constant 60 MPH as it is to reach that destination through repeated accelerating to 60 MPH and allowing the vehicle to coast to a stop? That is, you accelerate to 60, then turn off the car. Once it comes to a stop you accelerated to 60 again. Assume no drivetrain losses as well.

 

[russ_watters]

Welcome to PF!
Is this homework? Do you have a guess?

 

[tauristar]

No it is not homework. My guess is if the vehicle is switched off, those friction losses will eventually bring the vehicle to a stop, however, during the stopping process NO fuel is used. During acceleration back to the same speed, you are again battling friction losses, but at the same time you are supplying enough additional force to get the vehicle up to speed. If that additional force supplied equals the energy saved during the coasting period, then the energy consumption should be the same regardless of whether you are at a constant speed or repeatedly accelerating and coasting. Unless I am missing something here . . . which I feel like I am.

 

[berkeman]

Assuming only air friction, do you think it is equally as energy efficient to reach a destination in a vehicle traveling a constant 60 MPH as it is to reach that destination through repeated accelerating to 60 MPH and allowing the vehicle to coast to a stop? That is, you accelerate to 60, then turn off the car. Once it comes to a stop you accelerated to 60 again. Assume no drivetrain losses as well.

No it is not homework. My guess is if the vehicle is switched off, those friction losses will eventually bring the vehicle to a stop, however, during the stopping process NO fuel is used. During acceleration back to the same speed, you are again battling friction losses, but at the same time you are supplying enough additional force to get the vehicle up to speed. If that additional force supplied equals the energy saved during the coasting period, then the energy consumption should be the same regardless of whether you are at a constant speed or repeatedly accelerating and coasting. Unless I am missing something here . . . which I feel like I am.

Welcome to the PF.

One thing that you are not taking into account is the average speed of the two methods. Why is that an important consideration in your efficiency calculation?

 

[tauristar]

I believe it is important since fuel efficiency tends to decline at the higher speeds, so if the average of one method is 60mph and the other is 30mph then the lower average speed will save more fuel. Is this correct?

 

[olivermsun]

The real world problem is quite complicated for a whole list of reasons, among them:
a) the rolling friction is not quite constant with speed, although it might be reasonable to take an average value for this
b) the air resistance is, as you say, very nonconstant with speed (~ v^2) and so you cannot just average the speed and plug that into the drag equation
c) the power efficiency of the car is not at all constant (e.g., different for different accelerations).

In practice I have read results showing that some patterns of speeding up and slowing down can (in some cases, and usually not to zero speed) result in better overall economy for a certain car. I wouldn't really recommend it for other reasons though (e.g., safety). ;)

 

[berkeman]

I believe it is important since fuel efficiency tends to decline at the higher speeds, so if the average of one method is 60mph and the other is 30mph then the lower average speed will save more fuel. Is this correct?

Well, I was thinking about it from the other direction. For the comparison to be valid, you would need to have the same average speed for the two methods of travel, in order to say anything meaningful about their relative fuel efficiencies.

 

[Dmytry]

the accelerating and coasting would require less energy for same distance. It is very easy really. Assuming only air friction - the drag force equals energy transferred to air per distance travelled. Accelerating and coasting, except for brief times at 60mph, always has lower drag force, i.e. transfers less energy to the air per distance travelled.
There is only air friction, so the energy transferred to air is equal to the energy lost for travel.

Now the much more interesting question would be - is it more efficient to accelerate and coast, than to move with same average speed at constant speed?
There again moving at constant speed is most efficient but it is a little bit more difficult to show, especially without calculus. Basically, boils down to average of cubes vs cube of averages. (the energy lost per time, i.e. the power put into moving the air, is proportional to velocity cubed . We have average of velocities at all times cubed, for average power transferred to air, versus a cube of average velocity).

ahh, and also, you need some infinitesimal regular friction at least, for car to ever stop. If it is only air friction, it will never stop. Also without any non-air friction, it will get from point A to point B eventually by Brownian motion, but it will be a looooong while. Or a minimum speed at which you declare car stopped and start accelerating again.