Using energy conservation for Atwood machine:

Initial state: both at rest. After mass $m_1$ descends by h: Loss in PE = Gain in KE $m_1$gh - $m_2$gh = $\frac{1}{2}$(m_{1+m}_2)*$v^2$ v = sqrt(2(m_{1-m}_2)$\frac{gh}{m_1+m_2}$)

Advantage of energy method:

• No need to find tension or acceleration
• Directly gives velocity after a certain displacement
• Works for complex pulley arrangements

With friction on the table: $m_1$gh - $mu_k$$m_2$gh = $\frac{1}{2}$(m_{1+m}_2)$v^2$ (for hanging $m_1$ and table $m_2$)