$m a=-\alpha x^{2}$

$a=\frac{-\alpha x^{2}}{m}$

$\frac{v d v}{d x}=-\frac{\alpha}{m} x^{2}$

$\int_{v_{0}}^{0} v d v=\int_{0}^{x}-\frac{\alpha}{m} x^{2} d x$

$\left(\frac{v^{2}}{2}\right)^{0}=-\frac{\alpha}{m}\left(\frac{x^{3}}{3}\right)_{0}^{x}$

$\frac{-v_{0}^{2}}{2}=-\frac{\alpha}{m} \frac{x^{3}}{3}$

$x=\left(\frac{3 m v_{0}^{2}}{2 \alpha}\right)^{\frac{1}{3}}$