In a collinear collision, a particle with an initial speed v 0 strikes a stationary particle of the same…

$K{E}_2=\frac{3}{2}m{v}_0^2$. Conservation of momentum, $v_1+v_2=v_0$. Conservation of energy, $\frac{1}{2}m{v}_1^2+\frac{1}{2}m{v}_2^2=\frac{1}{2}\times \frac{3}{2}m{v}_0^2$
$\Rightarrow v_1^2+v_2^2=\frac{3}{2}{v}_0^2$$\Rightarrow v_1^2+v_2^2+2v_1{v}_2=v_0^2$$\Rightarrow \frac{3}{2}{v}_0^2+2v_1{v}_2=v_0^2$
$\Rightarrow 2v_1{v}_2=-\frac{1}{2}{v}_0^2$${\Rightarrow (v_1-v_2)}^2=\frac{3}{2}{v}_0^2+\frac{1}{2}{v}_0^2$$=2v_0^2$
$\Rightarrow v_1-v_2=\sqrt{2}{v}_0$.