A particle is projected with velocity v _ 0 along x -axis. A...

$F = - \alpha{x}^{2}$

$ma = - \alpha{x}^{2}$

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

$\dfrac{vdv}{dx} = - \dfrac{\alpha}{m}{x}^{2}$

${∫}_{{v}_{0}}^{0}vdv = {∫}_{0}^{x} - \dfrac{\alpha}{m}{x}^{2}dx$

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

$\dfrac{- {v}_{0}^{2}}{2} = - \dfrac{\alpha}{m}\dfrac{{x}^{3}}{3}$

$x = \left(\dfrac{3m{v}_{0}^{2}}{2\alpha}\right)\dfrac{1}{3}$