Let a mud blob is detached from the circumference of wheel with initial speed $u$ at angle $\theta$ as shown below.

Height upto which mud blob can be thrown is

$h=$ Maximum height of projectile$+$Height at which mud blob is thrown

$h=\frac{u^2{\sin }^2\theta }{2g}+R\cos \theta ....\left(\mathrm{i}\right)$

$h$ is maximum when $\frac{dh}{d\theta }=0$

$\Rightarrow \frac{d}{d\theta }\left(\frac{u^2{\sin }^2\theta }{2g}+R\cos \theta \right)$

$\Rightarrow \frac{u^2}{2g}.2\sin \theta \cos \theta -R\sin \theta =0$

$\Rightarrow \frac{u^2}{g}·\cos \theta =R$ or $\cos \theta =\frac{Rg}{u^2}$

$\Rightarrow {\sin }^2\theta =1-\frac{R^2{g}^2}{u^2}$

Substituting these values in Eq (i), we get

${\lambda }_{\max }=\frac{u^2}{2g}\left(1-\frac{R^2{g}^2}{u^4}\right)+R\left(\frac{Rg}{u^2}\right)$

$=\frac{u^2}{2g}-\frac{R^{2}g}{2u^2}+\frac{R^{2}g}{u^2}=\frac{u^2}{2g}+\frac{R^{2}g}{2u^2}$