Two-Dimensional Elastic Kinematics | M.E. Irizarry-Gelpí

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

$$ p = m u, \qquad q = m v. $$

There is conservation of total linear momentum,

$$ p_{1} + p_{2} = q_{1} + q_{2} \quad \Longrightarrow \quad m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}. $$

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

$$ m_{1} u_{1}^{2} + m_{2} u_{2}^{2} = m_{1} v_{1}^{2} + m_{2} v_{2}^{2} $$

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:

$$ v_{1} = u_{1}, \qquad v_{2} = u_{2}; $$

or the more complicated case:

$$ v_{1} = \frac{(m_{1} - m_{2}) u_{1} + 2 m_{2} u_{2}}{m_{1} + m_{2}} , \qquad v_{2} = \frac{(m_{2} - m_{1}) u_{2} + 2 m_{1} u_{1}}{m_{1} + m_{2}}. $$

When the masses are equal ($m_{1} = m_{2} = m$), the more complicated case reduces to backward scattering:

$$ v_{1} = u_{2}, \qquad v_{2} = u_{1}. $$

Note that you always have

$$ u_{1} - u_{2} = v_{2} - v_{1} $$

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.