For a given parametric parabola of the form

which could well represent a blade of grass growing in direction b and bending down to gravity in direction c, which of course has derivative

the length between its base defined at t=0 and its tip at t=T, is, as we know (after applying the binomial expansion),

where

which happens to have the not-so-simple solution

Probably T=1, but still, seems that when defined through a velocity and an acceleration, the length of a parabolic shape of a blade of grass is indeed surprisingly complex – it involves a logarithm and a couple of square roots, the point being here that i can’t find an intuitive interpretation to it really. However, we should be grateful there actually exists an analytic solution for it.

Again, we know so little beyond second degree polynomials!