$v=\alpha \sqrt{x}$

Squaring both sides $v^2=\alpha^2 x$

Comparing above equation with $3^{\text {rd }}$ equation of kinematics.

$v^2=u^2+2 a x$

$\alpha^2 x =2 a x$

$\Rightarrow a=\frac{\alpha^2}{2}$

$\,\downarrow$

$\text { Constant } \rightarrow \text { not a function of time }$

so,