\[L_0=\frac{x_2-x_1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\frac{L}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\: \: ,\: \: \Rightarrow \: \: L=L_0\sqrt{1-\frac{v^{2}}{c^{2}}}\]