Physicists love units. The best way to wind up a physicist is to tell him you were driving at 100 down the road. One hundred what? Just hope you don’t get pulled over by a traffic cop with a physics degree or he’ll ticket you for leaving your unit off, even if you were within the speed limit.
A unit is usually a very meaningful thing. One hundred kilometres an hour means, at that constant speed (and I mean speed not velocity, direction is irrelevant) you will travel one hundred kilometres in one hour. An freefall acceleration of ten metres per second, per second (or 10 metres per second squared) means every second something is in free fall it gains ten metres per second of velocity. Easy.
Not quite. In the area of stochastic physics, which I have worked with for several years, we have units that are metres per square-root second. What on earth is a square-root second? It still confuses me no end. Sure, I can work with it, and write computer programmes to use equations that involve it, but quite grasping it in my mind is something I haven’t succeeded in doing.
I know I haven’t succeeded yet because I find it so hard to describe what’s happening to my summer students.
(N.B. It’s related to the random walk (‘drunkard’s walk) problem – where the root mean square distance you travel is proportional to the square root of time.)
However, not understanding a concept does not exclude you from using it. I remember years ago working with Lagrange’s method of undetermined multipliers, (N.B. don’t click on this link unless you have a solid grasp of calculus) and being able to use it to work out problems, but not really having a clue as to WHY it worked. No text book seemed to help me. Then I remember one winter in Bedford walking to a bus stop and having a sudden flash of inspiration. At last I got it. It didn’t help solving problems, but it made the process that much less mysterious.
Maybe, just maybe, one day the same thing will happen with square-root units.