Two bodies of the same mass are connected with an inelastic string of length l are placed together on a…

The velocity of $\mathrm{B}$ just before the string is taut

${\mathrm{v}}_1=\sqrt{{\mathrm{u}}^2-2\mathrm{gl}}=\sqrt{6\mathrm{gl}-2\mathrm{gl}}=2\sqrt{\mathrm{gl}}$

After the string is taut, both will move up with same speed, ${\mathrm{v}}_2=\frac{{\mathrm{mv}}_1}{\mathrm{m}+\mathrm{m}}=\frac{{\mathrm{v}}_1}{2}=\sqrt{\mathrm{gl}}$

By this time centre of mass in at $\mathrm{h}=\frac{\mathrm{l}}{2}$

A further rise in the centre of mass $=\frac{{{\mathrm{v}}_2}^2}{2\mathrm{g}}=\frac{\mathrm{gl}}{2\mathrm{g}}=\frac{\mathrm{l}}{2}$

Hence maximum height $=\frac{\mathrm{l}}{2}+\frac{\mathrm{l}}{2}=\mathrm{l}$