$\Rightarrow u =\sqrt{2 gh }$

$S =u t+\frac{1}{2} a t^{2}$

$\frac{ h }{3}=\sqrt{2 gh } t +\frac{1}{2}(- g ) t ^{2}$

$\frac{ gt ^{2}}{2}-\sqrt{2 ght }+\frac{ h }{3}=0 \quad$ (Roots are $t _{1} \& t _{2}$ )

$\frac{t_{2}}{t_{1}}=\frac{\sqrt{2 g h}+\sqrt{2 g h-4 \times \frac{g}{2} \times \frac{h}{3}}}{\sqrt{2 g h}-\sqrt{2 g h-4 \times \frac{g}{2} \times \frac{h}{3}}}=\frac{\sqrt{2 g h}+\sqrt{\frac{4 g h}{3}}}{\sqrt{2 g h}-\sqrt{\frac{4 g h}{3}}}=\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$