Collisions in two dimensions

Collisions between two particles be analysed in two dimensions, as we do here. (This is a support page to the multimedia tutorial Momentum.)

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:

    p1i + p2i   =  p1f +  p2f

p1f and p2f also intersect and the sum of these vectors must, by conservation of momentum, also lie in this plane too. So two-particle 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.


In these two film clips, the offset has been changed.

In both cases,

    p1i + p2i   =  p1f +  p2f

may be written as two scalar equations, representing x and y components:

    p1 + 0  =  p1f cos θ1 +  p2f cos θ2 and
    0 + 0   =  p1f sin θ1 -  p2f sin θ2

This leaves four scalar unknowns, p1f, p2f, θ1and θ2 and only two scalar equations. An energy equation could give another: in an elastic equation, we have

    p12/2m1  =  p1f2/2m1 + p2f2/2m2

The fourth equation would come from the collision geometry. In this case, we could assume that the force between the two particles, and thefore the impulse, is in the line between their centres, which passes through the point of contact. So θ l lies along the line between their centres at contact.

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