I have a question about qudratic expressions and correlation coefficients. I know how to prove that if $y=mx+b$ then the correlation coefficent between $x$ and $y$ is 1. If $y$ is a quadratic function of $x$, I don't get CC of $1$ even though all the change in $y$ is due to change in $x$ or $t$.
Ignoring air resistance, the height of a dropped object is given by $H=h_0-(1/2)gt^2$. Where $H$ is height above the ground $h_0$ is initial height above ground, $g=9.8 m/s^2$ and $t$ is time in seconds.
Crunch the numbers and you find that $CC(t,H)=\frac{<Ht>-<H><t>}{\sigma_H\sigma_t}<1$.
Since that term qudratic in $t$ accounts for all the variation in $H$, why isn't $CC(t,H)=1$?