- Fri 28 October 2016
- Physics
- #kinematics, #two-dimensional
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
There is conservation of total linear momentum,
Since this collision is elastic, the total kinetic energy is also conserved:
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:
or the more complicated case:
When the masses are equal (m1=m2=m), the more complicated case reduces to backward scattering:
Note that you always have
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.