Could there be a property related to the velocity cubed, like there is momentum and kinetic energy? There is a property represented by $mv$.
There is a property that is represented by $\dfrac{mv^2}{2}$.
Is there a property (or could there possibly theoretically be one) that is represented by $\dfrac{mv^3}{6}$, $\dfrac{mv^4}{24}$, etc.?
 A: The power of wind varies with velocity cubed. As a formula, it is $$P=\frac12\rho Av^3$$ $v$ is the average velocity of wind. This is described in detail here.
A: In special relativity, the total energy of a particle in a given frame is given by $E=\gamma mc^2$ where $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ and $c$ is the speed of light. In the low velocity limit the gamma function is well approximated by a taylor series expansion in $\frac{v}{c}\ll1$.
It's expansion is $$\gamma = 1 + \frac{v^2}{2c^2} + \frac{3v^4}{8c^4}+...$$ and so the energy has the expansion
$$E=mc^2 + \frac{1}{2}mv^2 + ...$$
where ... represents terms higher order in $\frac{v}{c}$, which is very small in the limit that $v\ll c$.
We see the classic $mc^2$ term that tells us the particle's energy in terms of it's mass. The next term is the Newtonian kinetic energy term.
We could do higher order, the next term looks like $\frac{3mv^4}{8c^2}$ which is quartic in velocity. However, if we are going to regimes where higher-order terms become non-negligible it's better to work with special relativity directly.
That doesn't mean this explicit expansion to third-order isn't important sometimes. For example, this term is incorporated in the Schrodinger equation of the hydrogen atom. It gives rise to the fine splitting of the hydrogen atom's emission spectra.
A: Besides the obvious:
$$ E = mc^2\times \sum_{k=0}^{\infty}\frac{\prod_{i=1}^k(2k-1)}{2^kk!}\big(\frac v c\big)^{2k}$$
you have power lost to drag force:
$$P=vF_D=v\Big(\frac{\rho v^2A}2c_D\Big)=\frac 1 2 (Ac_D)\rho v^3$$
