A body of mass m moving along the positive x-axis with speed v...

The initial momentum of the system is the vector sum of the momenta of the two bodies.

Initial momentum vector, $\vec{p}_i = m v \hat{i} + m v \hat{j}$

The magnitude of the initial momentum is:
$|\vec{p}_i| = \sqrt{(mv)^2 + (mv)^2} = \sqrt{2} m v$

In a completely inelastic collision, the bodies stick together, so the total mass becomes $m + m = 2m$. Let the final velocity be $\vec{v}_f$.

Final momentum vector, $\vec{p}_f = (2m) \vec{v}_f$

By conservation of linear momentum, $\vec{p}_i = \vec{p}_f$. Therefore, their magnitudes must also be equal:
$|\vec{p}_f| = |\vec{p}_i|$

$(2m) |\vec{v}_f| = \sqrt{2} m v$

$|\vec{v}_f| = \dfrac{\sqrt{2} m v}{2m} = \dfrac{v}{\sqrt{2}}$

Answer: $\dfrac{v}{\sqrt{2}}$