Conservation of Momentum Calculator

This calculator relies on the fundamental law of physics stating that the total momentum of a closed system remains constant if no external forces act upon it. Mathematically, this is expressed as $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$, where $m$ represents mass and $v$ represents velocity. For elastic collisions, kinetic energy is also conserved. The calculator uses specialized equations depending on the collision type: elastic impacts utilize mass ratios to determine distinct final velocities, while completely inelastic impacts assume both objects stick together, sharing a unified final velocity $v_{1f} = v_{2f} = \frac{m_1v_{1i} + m_2v_{2i}}{m_1 + m_2}$. Kinetic energy loss is subsequently evaluated by comparing initial total kinetic energy against post-collision levels.

Consider a head-on elastic collision where Object 1 has a mass of $2\text{ kg}$ traveling at $4\text{ m/s}$ ($v_{1i} = 4$), and Object 2 has a mass of $3\text{ kg}$ traveling in the opposite direction at $-2\text{ m/s}$ ($v_{2i} = -2$). First, compute initial kinetic energy: $\text{KE}_1 = 0.5(2)(4)^2 + 0.5(3)(-2)^2 = 16 + 6 = 22\text{ J}$. Next, apply the elastic final velocity formulas: $v_{1f} = \frac{2 - 3}{2 + 3}(4) + \frac{2(3)}{2 + 3}(-2) = (-0.2)(4) + (1.2)(-2) = -0.8 - 2.4 = -3.2\text{ m/s}$. For Object 2: $v_{2f} = \frac{2(2)}{2 + 3}(4) + \frac{3 - 2}{2 + 3}(-2) = (0.8)(4) + (0.2)(-2) = 3.2 - 0.4 = 2.8\text{ m/s}$. Because it is an elastic collision, the total kinetic energy after the impact remains exactly $22\text{ J}$, resulting in zero kinetic energy loss.

When setting up your problem, pay close attention to the directional signs of your velocities; moving to the right is typically positive while moving to the left is negative. Ensure your mass units are consistent, utilizing kilograms uniformly across all inputs. Finally, remember that real-world collisions are rarely perfectly elastic due to sound and thermal energy dissipation, meaning theoretical elastic models serve primarily as an idealized upper baseline.