\[ct'=\frac{ct-\frac{v}{c}x}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\frac{ct-ct_v}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\frac{2ct_2}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\]