c
(c)

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

\(t_2-t_1=10 \,s\)

\(T \rightarrow\) Time taken to reach the highest point.

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

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

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

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

\(T^2=26\)

\(\Rightarrow T=\sqrt{26}\)

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