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_1gh - m_2gh = (1/2)(m_1+m_2)*v^2 v = sqrt(2(m_1-m_2)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_1gh - mu_km_2g*h = (1/2)(m_1+m_2)*v^2 (for hanging m_1 and table m_2)