Two blocks of masses $m_1=2m$ and $m_2=m$  are connected by a light string passing over a friction less pulley as shown in figure. When they are released,

The acceleration of each mass is the same and is given by $\text{a=}\frac{\left(m_1-m_2\right)g}{\left(m_1+m_2\right)}$

$\text{a=}\frac{\left(m_1-m_2\right)g}{\left(m_1+m_2\right)}\text{=}\frac{g}{3}$

The tension in the string is T= $\frac{{\text{2m}}_{\text{1}}{\text{m}}_{\text{2}}\text{g}}{\left({\text{m}}_{\text{1}}{\text{+m}}_{\text{2}}\right)}$

$\frac{{\text{2m}}_{\text{1}}{\text{m}}_{\text{2}}\text{g}}{\left({\text{m}}_{\text{1}}{\text{+m}}_{\text{2}}\right)}=\frac{4}{3}mg$

Since the masses start from rest, the distance moved by ${\text{m}}_1$ in time t is

$\text{s= ut+}\frac{\text{1}}{\text{2}}{\text{at}}^{\text{2}}\text{=0+}\frac{\text{1}}{\text{2}}{\text{at}}^{\text{2}}\text{=}\frac{\text{1}}{\text{2}}{\text{at}}^{\text{2}}$

$\text{s=}\frac{g{t}^2}{6}$