Conservation of Momentum Worked Examples Cheat Sheet

Key Facts

• Linear momentum is $p = mv$, where $p$ is momentum, $m$ is mass, and $v$ is velocity.
• The total momentum of an isolated system is conserved, so $\sum p_{\text{initial}} = \sum p_{\text{final}}$.
• For a two-object collision in one dimension, $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$.
• In a perfectly inelastic collision, objects stick together and the final velocity is $v_f = \frac{m_1v_{1i} + m_2v_{2i}}{m_1 + m_2}$.
• In an elastic collision, both momentum and kinetic energy are conserved, so $\sum p_i = \sum p_f$ and $\sum KE_i = \sum KE_f$.
• Impulse equals change in momentum, so $J = F\Delta t = \Delta p = m\Delta v$ for a constant net force.
• Momentum is a vector, so velocities in opposite directions must have opposite signs in equations.
• For an explosion or recoil starting from rest, the total final momentum is $0$, so $m_1v_{1f} + m_2v_{2f} = 0$.

Vocabulary

Momentum

Momentum is the quantity of motion of an object, calculated by $p = mv$.

Isolated system

An isolated system is a group of objects with no net external force acting on it during the interaction.

Impulse

Impulse is the change in momentum caused by a force acting over a time interval, given by $J = F\Delta t$.

Elastic collision

An elastic collision is a collision in which total momentum and total kinetic energy are both conserved.

Inelastic collision

An inelastic collision is a collision in which momentum is conserved but kinetic energy is not conserved.

Perfectly inelastic collision

A perfectly inelastic collision is a collision in which the objects stick together and move with one shared final velocity.