$x \propto y^2$ Vs $x \propto y$ $$x \propto y^2$$
How is it different from saying:
$$x \propto y$$
That is; when we say that Two variables are proportional then it means that two variables are related such that when one is zero other is too. And change in one variable is accompanied by change in other. This is a general definition for proportionality. Then if we write $x \propto y^2$, by definition, we implicitly mean that $x \propto y$. So, why write $x \propto y^2$ instead of a simple one $x \propto y$?
Is it due to the calculation of constant of variation? Viz., the constant of proportionality onliy lies between $x$ and $y^2$ relation and not between $x$ and $y$ relation? Is it for that purpose that we specify them?
 A: If $x$ is proportional to $y^2$, then when $y$ doubles, $x$ is four times as much.
If $x$ is just proportional to $y$, then $x$ doubles when $y$ doubles.
For example, the position of a ball falling towards the ground is proportional to $t^2$ ($t$ is time). So if the length of time doubles, the ball has fallen four times as far. This is very different from the ball's velocity, which is proportional to $t$. If the amount of time doubles, the final velocity of the ball also doubles.
EDIT: It may be tempting to think this way: $x\propto y$ means that there is some constant such that $x=cy$. But if $x=y^2$, then $x=y\cdot y$, so we could just write $x$ is proportional to $y$ since $x$ is something times $y$. This is not correct. We only call things proportional when one relates to the other via a constant.
A: The proportionality symbol carries the implication that if
$x \propto y \implies x=ky$ for some nonzero real k.
That being said, it's clear that $ x \propto y \ne x \propto y^2 $ since $ ky \ne ky^2 $
A: While dealing with Functions we can do that.
For example. Whether $f(x)=x$, $f(x)=x^2$ or $f(x)=x^3$ we can say that $f(x)=y$ , in general.
While dealing with "$\propto$", don't forget the implication.
For example. $x \propto y \implies x=ky, k \neq 0$
I am pretty sure that you are thinking like we think while dealing with functions. $k$ is not a function there but a simple multiple. We can easily relate (with "equals to" sign) any two quantities in mathematics with prefixing just a $f$ (or any symbol denoting function). But we can not express two quantities in equation arbitrarily with placing a constant. So , we have to describe enough when we place a constant in contrast to the situation, when we place $f$ symbol. So, it is obligatory there to write $y^2$ in stead of $y$.
