A shell of mass 200 g is ejected from a gun of mass 4 kg by an explosion that generates 1.05 kJ of energy. The initial velocity of the shell is

In the given problem, conservation of linear momentum and energy hold good.
Conservation of momentum yields

${\mathrm{m}}_1{\mathrm{v}}_1+{\mathrm{m}}_2{\mathrm{v}}_2 = 0$
or $4{\mathrm{v}}_1+0.2 {\mathrm{v}}_2 = 0------\left(\mathrm{i}\right)$
Conservation of energy yields
$\frac{1}{2}{\mathrm{m}}_1{{\mathrm{v}}^2}_1+\frac{1}{2}{\mathrm{m}}_2{{\mathrm{v}}^2}_2 = 1050$

or $\frac{1}{2}\times 4{{\mathrm{v}}^2}_1+\frac{1}{2}\times 0.2\times {{\mathrm{v}}^2}_2 = 1050$

or $2{{\mathrm{v}}^2}_1+0.1 {{\mathrm{v}}^2}_2 = 1050-----\left(\mathrm{ii}\right)$

Solving Eqs. (i) and (ii), we have
${\mathrm{v}}_1 = 100 \mathrm{m}/\mathrm{s}$