1.A block of mass $m=2\text{ kg}$ compresses a spring of constant $k=400\text{ N/m}$ by $0.1\text{ m}$ on a smooth floor and released. Find speed of block as it leaves spring.
Sol: $\frac{1}{2} k x^2 = \frac{1}{2} m v^2 \implies v = x \sqrt{\frac{k}{m}} = 0.1 \sqrt{\frac{400}{2}} = 0.1 \sqrt{200} = \sqrt{2} \approx 1.414\text{ m/s}$.
2.Find the minimum initial speed required for a particle attached to a light inextensible string of length $L$ to complete a full vertical circle.
Sol: At top: $v_{top} = \sqrt{gL}$. By energy conservation: $\frac{1}{2} m u^2 = \frac{1}{2} m v_{top}^2 + m g (2L) \implies u = \sqrt{5gL}$.
3.A head-on elastic collision occurs between mass $m_1$ moving with speed $v_1$ and stationary mass $m_2$. Find final velocities $v_1'$ and $v_2'$.
Sol: $v_1' = \left(\frac{m_1 - m_2}{m_1 + m_2}\right) v_1$, $v_2' = \left(\frac{2m_1}{m_1 + m_2}\right) v_1$.
