A particle of mass m moving with velocity ${\mathrm{V}}_0$ strikes a simple pendulum of mass m and sticks to it. The maximum height attained by the pendulum will be

Initial momentum of particle = ${\mathrm{mV}}_0$
Final momentum of system (particle + pendulum) = 2mv
By the law of conservation of momentum
$\Rightarrow {\mathrm{mV}}_0 = 2 \mathrm{mv} \Rightarrow \mathrm{Initial} \mathrm{velocity} \mathrm{of} \mathrm{system} \mathrm{v} = \frac{{\mathrm{V}}_0}{2}$

$\therefore$ Initial K.E. of the system $= \frac{1}{2}\left(2\mathrm{m}\right){\mathrm{v}}^2 = \frac{1}{2}\left(2\mathrm{m}\right){\left(\frac{{\mathrm{V}}_0}{2}\right)}^2$
If the system rises up to height h then P.E. = 2mgh
By the law of conservation of energy
$\frac{1}{2}\left(2\mathrm{m}\right){\left(\frac{{\mathrm{V}}_0}{2}\right)}^2 = 2\mathrm{mgh} \Rightarrow \mathrm{h} = \frac{{{\mathrm{V}}^2}_0}{8\mathrm{g}}$