Height upto which mud blob can be thrown is
$h=$ Maximum height of projectile $+$ Height at which mud blob is thrown
$\therefore \quad h=\frac{u^{2} \sin ^{2} \theta}{2 g}+R \cos \theta \quad \dots(i)$
$h$ is maximum when $\frac{d h}{d \theta}=0$
$\Rightarrow \frac{d}{d \theta}\left(\frac{u^{2} \sin ^{2} \theta}{2 g}+R \cos \theta\right)$
$\Rightarrow \frac{u^{2}}{2 g} \cdot 2 \sin \theta \cos \theta-R \sin \theta=0$
$\Rightarrow \quad \frac{u^{2}}{g} \cdot \cos \theta=R \text { or } \cos \theta=\frac{R g}{u^{2}}$
$\Rightarrow \quad \sin ^{2} \theta=1-\frac{R^{2} g^{2}}{u^{2}}$
Substituting these values in Eq $(i)$, we get
$\lambda_{\max } =\frac{u^{2}}{2 g}\left(1-\frac{R^{2} g^{2}}{u^{4}}\right)+R\left(\frac{R g}{u^{2}}\right)$
$=\frac{u^{2}}{2 g}-\frac{R^{2} g}{2 u^{2}}+\frac{R^{2} g}{u^{2}}=\frac{u^{2}}{2 g}+\frac{R^{2} g}{2 u^{2}}$
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