This thread is pretty much dead and cold but I think it's relevant so I'm gonna try a little CPR.
I think that Gib65 is making an interesting point here that and I'm not sure if the rest of the forum is quite on the same page.
Why is it that we can so easily conceptualize the idea of division in units but not multiplication? We can understand the idea of "feet per second" (ft/sec) - it's a certain number of feet for every second elapsed. We can understand the idea of "miles per gallon" (mil/g)- it's a number of miles traversed for every gallon used.
But multiplication is much harder to understand. What is a "foot-pound", a "meter-second", a "kilogram-watt"? I don't mean "what are they?" literally. I understand what they are and how to use them mathematically. BUT, if 1 meter/sec is a meter for every second elapsed then what is 1 meter x sec? What does that mean? Why can't we conceptualize it like we can conceptualize division based units?
I think the answer has to do with the innate limitations that we (as humans) have in understanding the concept of time. Meters/sec makes the best example.
As self-proclamed geeks we all understand the idea that time can mathematically be manipulated like any other component of an equation. Mathematically time can go forward, backwards, breakdance, whatever. Of course in reality time only flows in one direction, at least for us. By doing this it's very nature is to distribute events across itself, essentially, to divide. If you traverse 30 meters in 2 seconds, you have moved 30m at 15 m/s. We can understand this easily because we are used to manipulating time in this way. It is, in fact, the only way that we are used to manipulating it.
Now at this point you may say, "Yes but time doesn't have to be an element of the unit for it to be conceptualized. Take for instance miles/gal. 'A certain amount of miles traversed for every gallon used', no time units at all!"
Yes, this is true BUT miles/gal INFERS that time is moving in a fashion that we are used to dealing with, where as meters x sec does not.
When you really think about it, it's rather bizarre that we can conceptualize any mathematically manipulated unit. Miles/hr is a totally intuitive concept to us but why? All it really means is the distance traveled in miles, divided by the time it took to traverse it in hours. It is entirely a mathematical concept and it makes no sense that it should be more or less intuitive than a "foot-pound".
At this point you may think that this is just one of those situations in which you have to throw up your hands and say, "whatever, it just IS". But I find that there are SOME multiplied units that are easier to conceptualize than others. Take, for instance, a Newton, or, for the purposes of our discussion let's call it a "kilogram-meter per second squared" (kg x m/s^2). This multiplied (and divided) unit is, of course, derived from the formula F=ma, just as m/s is derived for the formula V=d/t.
Think about it, you CAN conceptualize this unit. If someone punches you in the face it is easy to understand that the amount of pain you experience is a product of the mass of their fist and the amount that said fist is accelerating. Small fist x big accel.- not so bad. Big fist x small accel.- tolerable. Small fist x small accel.- barely noticable. Big fist x big accel.- expensive dental bill. The differences in how your jaw feels afterwards IS a conceptualization of a "kilogram-meter per second squared".
Remember, things like Force and Kinetic Energy are fundamentally different than things like Velocity and Frequency, so conceptualizing their units requires a different sort of thinking. If you expect them to feel the same, you will be disappointed.
I hope I have contributed in some positive way here, and I hope that SOMEONE that posted here still checks their account, considering that this thread has been left alone for 5 years. This is just something I have been thinking a lot about recently so I felt the need to share. Happy to be a part of the forum