Let the total height of tower \(=H\)

Total time of journey \(=t\)

Time taken to cover the \(\frac{5 h}{9}\) is = last second

So, \(s_t-s_{t-1}=\frac{5 h}{9}\)

\(\Rightarrow \frac{1}{2} g t^2-\frac{1}{2} g(t-1)^2=\frac{5}{9} \times \frac{1}{2} g t^2\) \(\left[\because h=\frac{1}{2} g t^2\right]\)

\(\Rightarrow \frac{1}{2} g\left(t^2-t^2-1+2 t\right)=\frac{1}{2} g t^2 \times \frac{5}{9}\)

\(\Rightarrow (2 t-1)=\frac{5}{9} t^2\)

\(\Rightarrow 18 t-9=5 t^2\)

\(\Rightarrow 5 t^2-18 t+9=0\)

\(\Rightarrow 5 t^2-15 t-3 t+9=0\)

\(\Rightarrow 5 t(t-3)-3(t-3)=0\)

\(\Rightarrow(5 t-3)(t-3)=0\)

\(t=\frac{3}{5}, t=3 s \quad\left(t=\frac{3}{5}\right.\), doesn't satisfy the given criterion, so we neglect it)