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),
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!