$u =\sqrt{ ng \ell}$

$v =\sqrt{4 g \ell}$

$L =\sqrt{5 g \ell}$

Angle of slack is $\theta$ with vertical

Using energy conservation

$\Rightarrow \frac{1}{2} m u^2=\frac{1}{2} m v^2+m g \ell(1+\cos \theta) \ldots \text { (i) }$

and $m g \cos \theta+7=\frac{m v^2}{\ell} \quad \ldots$ $(ii)$

By $(i)$ and $(ii)$ we get

$\left[\cos \theta=\frac{n-2}{3}\right]=\frac{2}{3}$

$\Rightarrow \frac{2}{3} m g=\frac{ mv ^2}{\ell}$ using $\ldots(ii)$

$v =\sqrt{\frac{2}{3} g \ell}$

At $H _{\max } v ^{\prime}= v \cos \theta=\sqrt{\frac{2}{3} g \ell} \times \frac{2}{3}= v ^{\prime}=\sqrt{\frac{8}{27} g \ell}$