Elastic Collisions in 2D or 3D

We now want to study the elastic collision of two bojects in two or three spatial dimensions. For the one-dimensional case of an elastic collision, we have worked our a general solution. This is possible, because there are two unknowns (v₃ and v₄) and two equations (momentum conservation and kinetic energy conservation) to determine them. In general one needs at least as many equations as there are unknowns to solve a system of equations.

• In 2 dimensions, we have 4 unknowns (2 components of ₃ and ₄ each), but only 3 equations for the conserved quantities
  (E_kin, P_x, and P_y).



• In 3 dimensions, we have 6 unknowns (3 components of ₃ and ₄ each), but only 4 equations for the conserved quantities
  (E_kin, P_x, P_y, P_z).



• There is therefore no general unique solution without imposing additional constraints. (Makes billiards interesting!)



• Interesting case: m₁= m₂= m, and ₂= 0
  • Conserved kinetic energy:
  • mv₁²= mv₃² + mv₄²
  
  
  
  • Dividing out m/2 results in:
  • v₁²= v₃² + v₄²
  • This means (Pythagoras) that there will be a 90° angle between ₃ and ₄.

© MultiMedia Physics, 1999