$h=5\, m \text { (given) }$

$t_2-t_1=10 \,s$

$T \rightarrow$ Time taken to reach the highest point.

$t_1=T-\sqrt{T^2-\frac{2 h}{g}}, t_2=T+\sqrt{T^2-\frac{2 h}{g}}$

$t_2-t_1=T+\sqrt{T^2-\frac{2 h}{g}}-T+\sqrt{T^2-\frac{2 h}{g}}$

$\Rightarrow 10=2 \sqrt{T^2-\frac{2 \times 5}{10}}$

$\Rightarrow 5=\sqrt{T^2-1} \Rightarrow 25=T^2-1$

$T^2=26$

$\Rightarrow T=\sqrt{26}$

$\text { Total time of flight } \Rightarrow 2 T=2 \sqrt{26}=\sqrt{4 \times 26}=\sqrt{104} \,s$