An 8.0 g bullet is fired into a 1.6 kg block hanging on strings, and stays inside…

In a collision with no outside force, momentum is conserved: $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$. In an elastic collision kinetic energy is conserved too. Head-on and elastic, with the target at rest: $v_1 = \dfrac{m_1 - m_2}{m_1 + m_2}u_1$, $v_2 = \dfrac{2m_1}{m_1 + m_2}u_1$.

When only conservative forces (gravity, springs) do work, mechanical energy is conserved: $K_1 + U_1 = K_2 + U_2$. Forces that do no work, like a smooth surface's push or a string at right angles to the motion, do not spoil it.