1. Collision

Block A ($m_1$) hits Block B ($m_2$) and sticks. Initial velocity of A is $u$.

Final velocity of blocks: $v_f = \frac{m_1 u}{m_1 + m_2}$.

2. Relative Motion

The ball momentarily retains velocity $u$. Velocity relative to blocks: $v_{rel} = u – v_f$.

$$v_{rel} = u – \frac{m_1 u}{m_1 + m_2} = u \left( \frac{m_2}{m_1 + m_2} \right)$$

3. Circular Motion Condition

For a full circle, $v_{rel} \ge \sqrt{5gl}$.

$$u \left( \frac{m_2}{m_1 + m_2} \right) \ge \sqrt{5gl} \implies u \ge \frac{m_1 + m_2}{m_2} \sqrt{5gl}$$