Conservation of Linear Momentum: For a system with no external forces, the total linear momentum remains constant. In a collision, the total momentum before equals the total momentum after: $m_1 \mathbf{v}_1 + m_2 \mathbf{v}_2 + ... + m_n \mathbf{v}_n = m_1 \mathbf{u}_1 + m_2 \mathbf{u}_2 + ... + m_n \mathbf{u}_n$
Conservation of Angular Momentum: For a rotating system with no external torque, the angular momentum is conserved. For a constant mass system: $\omega_1 I_1 = \omega_2 I_2$ or equivalently: $v_1 r_1 = v_2 r_2$

Application Examples

Perfectly Inelastic Collision: The example calculates the common velocity ( $u$ ) of two bodies moving in the same direction after colliding and sticking together. $u = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}$
Recoil Velocity: The recoil velocity ( $u_r$ ) of a rifle is calculated by applying conservation of momentum, assuming the system starts from rest. $u_r = -\frac{m_b v_b}{m_r}$ The recoil kinetic energy ( $E$ ) is then $E = \frac{1}{2} m_r u_r^2$ .
Angular Momentum with Changing Radius: The example shows how the rotational speed ( $n$ ) of a point mass changes when the radius of its circular path changes, due to conservation of angular momentum. $n_2 = n_1 \frac{r_1}{r_2}$

References