and $x=\frac{1}{2} g t^{2}$
$\therefore R=\sqrt{2 g(3 H-x)} \sqrt{\frac{2 x}{g}}$
$=2 \sqrt{\left(3 H x-x^{2}\right)}$
For $R=\max , \frac{d R}{d x}=0$
$=\frac{d}{d x} \sqrt{3 H x-x^{2}}$
$=\frac{1}{2 \sqrt{3 H x-x^{2}}}(3 H-2 x)$
$\Rightarrow x=\frac{3}{2} H=1.5 H$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$(a)$ $\left(x^2-v t\right)^2$
$(b)$ $\log \left[\frac{(x+v t)}{x_0}\right]$
$(c)$ $e^{\left\{-\frac{(x+v t)}{x_0}\right\}^2}$
$(d)$ $\frac{1}{x+v t}$
