An artillery piece of mass M _ 1 fires a shell of mass M _ 2...

As linear momentum will be conserved, $\left|\overset{⇀}{p}1\right| = \left|\overset{⇀}{p}2\right|$

Now, kinetic energy can be written as $KE = \dfrac{{p}^{2}}{2M}$.

Also $p$ is the same for both masses. We can write,

$KE ∝ \dfrac{1}{m}$

$⇒ \dfrac{K{E}_{1}}{K{E}_{2}} = \dfrac{\dfrac{{p}^{2}}{2{M}_{1}}}{\dfrac{{p}^{2}}{2{M}_{2}}} = \dfrac{{M}_{2}}{{M}_{1}}$