Kinetic energy of block is converted into potential energy of stretched wire.

$\Rightarrow \quad \frac{1}{2} m v^2=\frac{1}{2} \text { Stress } \times \text { Strain } \times \text { Volume }$

$\Rightarrow m v^2=\frac{\text { Stress }}{\text { Strain }} \times(\text { Strain })^2 \times \text { Volume }$

$\Rightarrow m v^2=Y \times \frac{\Delta l^2}{L^2} \times A \times L \Rightarrow \Delta l^2=\frac{m v^2 L}{A Y}$

$\therefore \quad \Delta l=v \sqrt{\frac{m L}{A Y}}$

So, distance travelled by block in stretching the wire is $\Delta l=v \sqrt{\frac{m L}{A Y}}$.