Momentum and Collisions

Momentum and Impulse

Linear momentum is $p = mv$, a vector pointing with velocity. The impulse-momentum theorem connects force and time to momentum change: $J = F\Delta t = \Delta p$ This explains safety design: airbags, crumple zones, and bending your knees all lengthen $\Delta t$ for the same $\Delta p$, lowering peak force.

Conservation of Momentum

In any collision or explosion with no external force, total momentum is conserved: $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$ Set up the equation with consistent signs — momentum is a vector, and one direction must be negative.

Collision Types

Elastic collisions conserve both momentum and kinetic energy (idealized billiard balls, gas molecules). Inelastic collisions conserve momentum but lose kinetic energy to deformation and heat. Perfectly inelastic collisions are the extreme case: objects stick together and move with common final velocity $v_f = (m_1v_{1i} + m_2v_{2i})/(m_1 + m_2)$, maximizing kinetic energy loss. Recoil problems are explosions run in reverse: total momentum starts and stays zero.

Key Takeaways

• Momentum is conserved in every collision; kinetic energy only in elastic ones.
• Longer impact time means smaller force for the same momentum change.
• Objects sticking together signals perfectly inelastic — one shared final velocity.
• Watch signs: opposite directions carry opposite momentum.