An undeformed spring of spring constant k is connected to a bead of mass m which can move along a…

Conserving the linear momentum of the system along the rod

${\text{mv cos 45}}^{\circ }=2{\text{mv}}_1$

${\text{v}}_1=\frac{\text{v}}{2\sqrt{2}}$

Applying work-energy theorem

$-\frac{1}{2}{\text{kx}}^2=-\frac{1}{2}\left(2\text{m}\right){\left(\frac{\text{v}}{2\sqrt{2}}\right)}^2$

$\Rightarrow \text{x}=\frac{\mathrm{v}}{2}\sqrt{\frac{\text{m}}{\text{k}}}$