If two dimensionless particles collide, then the vectors of their momenta before the collision must intersect. These two vectors thus define a plane, and the sum of their momenta before the collision lies in this plane.
The usual assumption for a collision is that, during the short duration of the collision, the impulse provided by external forces is negligible in comparison with the total momentum, so momentum is conserved:
p_{1i} + p_{2i} = p_{1f} +
p_{2f}
p_{1f} and p_{2f} also intersect and the sum of these vectors must, by conservation of momentum, also lie in this plane too. So twoparticle collisions can be considered as collisions in two dimensions.
Two further simplifications are possible: one can choose a frame of reference in which one particle is stationary and the other travels initially in the x direction. These simplifications are made in the case shown at right, but using billiard balls, not particles, so an offset displacement between their centres is possible.
