Mathematical Musings

Good calculations! Can help explain the cubed thing: Drag from air resistance does square with speed, so (as power = force x velocity) the power required to overcome the air resistance cubes with speed.

But rolling resistance isn't such a clear cut thing, it's almost linear at low speeds, then increases at a lower power (not cubed) when you get faster. A typical rule of thumb used by car designers is that rolling drag = 1/3 total drag at freeway speeds.

[dstrout]

<div class='quotetop'>QUOTE(rwlade\";p=\"90794)</div>

Great post, Dave. Intuitively, the velocity cubed relationship makes sense. Where did that equation come from?
[/b]

It's a pretty standard rule of thumb ( I studied aeronautical engineering many years ago -- don't use it now). I dug it up from here:

http://www.bobnorwood.com/300%20mph%C2%A0%...and%20Power.htm

since I don't carry exact constants in my head. I thought it was 9x10 ^- something x Cd x A x V^3. I couldn't remember the fudge factor tho', and details make all the difference.

It's an interesting page to read in general. Turns out we only need 428 h.p. to push our little prius to 200 m.p.h. Problem is, I think a 350 V8 will put us over our max weight limit Sometimes you just can't win...

dave.

[DanMan32]

<div class='quotetop'>QUOTE(clett\";p=\"90795)</div>

Good calculations! Can help explain the cubed thing: Drag from air resistance does square with speed, so (as power = force x velocity) the power required to overcome the air resistance cubes with speed.

But rolling resistance isn't such a clear cut thing, it's almost linear at low speeds, then increases at a lower power (not cubed) when you get faster. A typical rule of thumb used by car designers is that rolling drag = 1/3 total drag at freeway speeds.
[/b]

Actually the square thing is besides the air and rolling resistance.
I tried to figure this out, as it really didn't make sense to me.
But energy is a function of force and distance, and force is a function of mass and acceleration.
E=Fd=MAd

If you bring acceleration and distance to a common variable for velocity using differentiation and integration, A=deltav/t and d=deltaVt/2, you get M(v^2)/2 but I think that only applies for linear acceleration.
As my calculus instructor would always shout out, there's supposed to be a +C somewhere there for the integration.

[jayman]

<div class='quotetop'>QUOTE(DanMan32\";p=\"90819)</div>

If you bring acceleration and distance to a common variable for velocity using differentiation and integration, As my calculus instructor would always shout out[/b]

Dan:

So your Calculus instructor shouted at you too?? I wonder if we had the same prof?? By the time year four rolled around in college, that prof was probably ready to shove me into an industrial wood chipper.

Even more unfortunate - for him anyway - was he also taught that college Physics class I took way back then. I thought it was really cool how if you knew the acceleration of an object, you could play around with Integration to find instantaneous velocity and then total distance.

This prof was also a consulting engineer to Litton and apparently he did a lot of hush-hush work for weapon systems. So naturally when he discussed objects it was always a missile/rocket in a ballistic trajectory: boost, cruise, reentry, kaboom, etc.

To ease understanding, it's appropriate to make generalizations like stating V^2 for KE and V^3 for drag. We can further clarify by using Integration to a common variable, but to be precise we would need to know, among other things: frontal area, tire rolling resistance, road texture, drag from lubricants, humidity, downforce on object, velocity, etc etc.

I don't mind exercising my brain, but that just might give it a strain or a hernia! I haven't had to perform that sort of math for a long time now.

Anyhoo, based on studies already completed by DOT, NHTSA, etc, 55 MPH is the mythical "sweet spot" for convenience of velocity, energy efficiency, occupant safety, and environmental awareness. A speed of 75-85 might seem faster, but the energy penalty is such that you're really not getting there too much faster.

With large SUV type vehicles, you take a fuel economy hit of 40% or more just by driving at 75 MPH instead of 55 MPH. Those things don't get stellar fuel economy even at 55 MPH, so when you figure out how more often you have to stop to gas up, what time are you "saving?"

By the same token, you can equip a road transport tractor with a powerful motor, speed-optimized gearing, disable the electronic governor, put on some wind fairings, and high speed tires. It would easily cruise at 120 MPH.

However, the fuel economy at that speed would be so horrendous that nobody even bothers. Most reputable and cost-conscious road transport companies set the governor no faster than 65 MPH.

[DanMan32]

I started calculus my senior year in high school. I forgot the instructor's name. He left the following year. He was a cool guy. If we forgot to add a the unknown constant, he was being cute by saying "+C!"

He also called interesting formulas "a Raquel", as his favorite actress was Raquel Welsh. He thought both were a 'beaut'.

[dstrout]

Anyhoo, based on studies already completed by DOT, NHTSA, etc, 55 MPH is the mythical \"sweet spot\" for convenience of velocity, energy efficiency, occupant safety, and environmental awareness. A speed of 75-85 might seem faster, but the energy penalty is such that you're really not getting there too much faster.

I've always wondered if that still holds. 55 mph came about after the 1973 oil embargo, and cars were very different back then. My '03 mustang, for example, gets 18-20 mpg around town / commuting, but 24-25 on a steady 75 mph highway cruise. I think that the gearing just happens to put it at a very good point in the engine & aerodynamic curves.

It's too bad the car makers can't publish each cars best highway speed. But I'm sure they'd get sued the first time some idiot crashes doing 56 mph because the car maker told them to. :cussing:

dave.

[jayman]

Dave:

Good point. Most of the studies I've mentioned from the regular acronyms have had to do with heavy road transport, especially with regards to safety eg stopping distance.

Intuitively, one might expect a "sweet spot" of anywhere from 45-65 MPH, but only empirical results need apply. So far the data appears to be scarce, indicating a "sweet spot" of 50-55 MPH for optimum fuel economy and crash survivability.

There are some interesting documents out there:

http://www.sourcewatch.org/wiki.phtml?titl...laws_of_physics

Jay

[DanMan32]

Problem with town/commuting avgs is that theres a lot of idling there. For the Prius, that is negligable as it hardly idles, but not a conventional car.

For non-stop, level speed driving, you'll probably find the sweet spot at the speed where you just come into last gear with torque lockup, which is usually 45MPH or so.

[200Volts]

I made up a new accronym:
M.E.S. or Maximum Efficiency Speed
I calculate this as the speed where MPH=MPG. For the Prius this is about 54.
I just traveled in a new Accord Hybrid (very plush leather, "burled" plastic, smooth and quiet-active noise canceling in the doors) for 2 days that got 36 mpg average with a MES about 42.
Our old 1993 Lincoln MK8 (a slippery V8 ideal for highway drives, it lowered itself 1" above 45mph-air shocks) had a MES about 28 with and average MPG of 26(yes , even with 300 HP)