Phenomenon that have $\sqrt{x}$ functions? Grading curve, inverse-square law, etc? 
I am primarily interested in the NATURE of square root functions. ie: What kinds of behavior, situations, or models are best described with a square root function ?
I have seen a grading curve scale as $y=10\sqrt{x}$
This makes lower grades increase much more than higher grades.
This is illustrated on the graph above where it is steeper for lower x values.

x   10*sqrt(x)
--------------
0    0
1    10
25   50
49   70
64   80
81   90
100  100

What are some other situations where the main theme is that lower numbers are magnified and higher numbers, not so much?    
Does the inverse-square law of gravity apply here?  The lower the distance, the higher the gravitational pull (and not in a linear manner, but a squared relationship) <-- what is the right phrasing of that idea?
 A: The charachteristic of $\sqrt x$ that interests you appear to be that its derivative (ie how much is it steep) if $\displaystyle \frac1{2\sqrt x}$. This means that for $x$ very small, $\displaystyle \frac1{2\sqrt x}$ is very big (so the curve is very steep); for $x$ very large, $\displaystyle \frac1{2\sqrt x}$ is very small.
You can find any number of function that behave this way. For example $\log(x)$, $\sqrt[n]x$ etc.
In general think of any function $f(x)$ such that $f(x) \to \infty$ if $x$ is small, and $f(x) \to 0$ if $x$ is large; then $F(x) = \int_0^x f(t)dt$ will behave in a similar fashion
A: The time it takes an object beginning at rest to fall a distance $h$ (constant gravity over the distance, air or other resistence neglected) is proportional to $\sqrt {h}.$
This reminds me of something I once wondered about (early to mid 1990s, and I used my reasoning on a teaching statement in 1996), and which I posted a description of in this 18 January 2001 k12.ed.math post archived at Math Forum. An excerpt of that post follows:

Related to this is something I wondered about a few years ago, and the analysis of it surprised me. If you watch little bugs crawl about, it seems that their reaction times are much better than ours. Also, if something falls from a height of $1$ mm or $2$ mm, it takes too little time for us to see it falling. So I wondered how things falling from $1$ mm or so are perceived by tiny bugs. Do they perceive objects to fall at a faster rate, the same rate, or a slower rate than we perceive objects to fall, when the objects fall a distance roughly equal to their size? That is, a $1$ mm bug watches something fall from a $1$ mm height, while a (tall) $2$ m person watches something fall from a $2$ m height. I believe the answer is slower.
Let $h$ denote the length of the organism. The time it takes an object to fall from a height $h$ is proportional to $\sqrt h$ (high school physics), while an organism's reaction time is proportional to $h$ (nerve impulses travel roughly at a constant rate). As $h$ becomes smaller and smaller, the values of $\sqrt h$ become larger and larger relative to the values of $h.$ Thus, if we were as small as ants and dropped something, it would appear to slowly float down to the ground.

