Remember R, the Ideal Gas Constant? It's used in:
PV = nRT
It has about eleventy-seven values depending on the system of units you're using - SI (kg, m, s, K) or cgs (g, m, s, K) or US ("lbs", ft, s, F) or ...
Similarly, the power needed to overcome the friction due to air is proportional to the cube of the speed. If you measure power in HP and speed in mph, then constant factors in the equation will be different than if you measure power in KW and speed in m/s. But the functional relationship is the same no matter what units you choose - power required varies as the cube of the speed.
If it's not clear, then think about finance. The amount of "money" you need to buy a bottle of whiskey at a particular duty-free store doesn't depen on what currency you use. It might be $100 or 10,728.57 Yen. It's the same amount of "money" - just the units change. (Assuming no transaction costs, of course.)
Finally, you write:
What we're interested in here is the cube [or square, etc] of the change in speed (or, IOW, the cube of the ratio of speeds) -- but NOT the "cube of the speed".
No. Starting at 0 and going to 50 mph takes a lot less power than starting from 500 and going to 550 mph. The change is the same, but the power required is very different. The incremental change is much smaller in the latter case too (55/50 vs "50/0"), but the power required is much larger.
It really just falls out of the [link|http://www.getfaster.com/Techtips/Physics6.html|equation]:
[image|http://www.getfaster.com/BIS/Techtipdata/_8387_tex2html_wrap153.gif|0|Equation for HP to overcome air resistance|37|394]
If v is the vehicle speed in miles per hour, then P is in horsepower. The constants in the equation will be different if the units of speed and power are different.
Units can be very tricky. E=mc^2 and similar relationships in physics implies all sorts of interconnections between units that can be hard to get ones head around...
HTH a bit.
Cheers,
Scott.