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M.E. Irizarry-Gelpí

Physics impostor. Mathematics interloper. Husband. Father.

Two-Dimensional Elastic Kinematics


Recently I came across a neat result for a non-relativistic 2-to-2 elastic collision in one time plus one space dimensions. You have two kinds of particles. Each incoming particle has a momentum p, a velocity u, and kinetic energy. Each outgoing particle has a momentum q, a velocity v, and kinetic energy. Non-relativistically, momentum and velocity are related via

p=mu,q=mv.

There is conservation of total linear momentum,

p1+p2=q1+q2m1u1+m2u2=m1v1+m2v2.

Since this collision is elastic, the total kinetic energy is also conserved:

m1u21+m2u22=m1v21+m2v22

Using these two constraints, you can solve for the outgoing velocities in terms of the incoming velocities. There are two sets of solutions. You can have the forward scattering case:

v1=u1,v2=u2;

or the more complicated case:

v1=(m1m2)u1+2m2u2m1+m2,v2=(m2m1)u2+2m1u1m1+m2.

When the masses are equal (m1=m2=m), the more complicated case reduces to backward scattering:

v1=u2,v2=u1.

Note that you always have

u1u2=v2v1

This particular equation is invariant under a constant shift in velocity, which corresponds to a Galileo transformation. This result is general and does not assume working in the center-of-mass frame.

In a future post I will look at the relativistic version of this system.