Collisions are key events in physics, where objects interact and exchange energy and momentum. Understanding elastic and inelastic collisions helps us analyze everything from billiard ball impacts to car crashes, revealing how energy and momentum are conserved or transformed.

One-dimensional collisions are simpler to solve, using conservation laws and equations. Two-dimensional collisions require vector analysis, breaking motion into components. These principles apply to real-world scenarios, from particle physics experiments to astronomical events like comet impacts.

Types of Collisions

Elastic vs inelastic collisions

• Elastic collisions
  • Total kinetic energy and momentum conserved during impact
  • Objects bounce off each other with no deformation (billiard balls, atomic collisions)
• Inelastic collisions
  • Momentum conserved but kinetic energy partially lost
  • Energy converted to heat, sound, or deformation (car crashes, clay balls sticking together)
• Perfectly inelastic collisions
  • Objects combine into single mass after impact
  • Maximum kinetic energy loss while conserving momentum (bullet embedding in wood)

One-dimensional collision problem-solving

• Conservation of momentum: $p_1 + p_2 = p_1' + p_2'$ applies to all collisions
• Elastic collisions
  • Kinetic energy conserved: $\frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 = \frac{1}{2}m_1v_1'^2 + \frac{1}{2}m_2v_2'^2$
  • Relative velocity reversal: $v_1 - v_2 = -(v_1' - v_2')$ helps solve for final velocities
• Inelastic collisions
  • Find final velocity using momentum conservation
  • Energy loss: $\Delta E = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 - \frac{1}{2}(m_1 + m_2)v_f^2$
• Coefficient of restitution
  • Elasticity measure: $e = \frac{v_2' - v_1'}{v_1 - v_2}$ ranges 0 to 1
  • 0 for perfectly inelastic, 1 for perfectly elastic collisions

Two-Dimensional Collisions

Vector analysis of two-dimensional collisions

• Decompose vectors into x and y components
• Apply momentum conservation separately in each direction
  • $p_{1x} + p_{2x} = p_{1x}' + p_{2x}'$ and $p_{1y} + p_{2y} = p_{1y}' + p_{2y}'$
• Use vector addition to find final momenta
• Calculate deflection angles with trigonometry (glancing collisions)

Conservation principles in collision problems

• Vector momentum conservation: $\vec{p_1} + \vec{p_2} = \vec{p_1'} + \vec{p_2'}$ for all collisions
• Apply kinetic energy conservation for elastic collisions
• Solve simultaneous equations for unknown variables
• Center of mass frame
  • Simplifies calculations, total momentum zero
  • Useful for analyzing particle collisions
• 2D collision types include glancing and head-on impacts
• Real-world applications in particle physics experiments and astronomical events (comet impacts)