# Elastic collision

An elastic collision is a collision in which the total kinetic energy of the colliding bodies after collision is equal to their total kinetic energy before collision. Elastic collisions occur only if there is no conversion of kinetic energy into other forms, as in the collision of atoms (Rutherford backscattering is one example).

In the case of macroscopic bodies this will not be the case as some of the energy will become heat. In a collision between polyatomic molecules, some kinetic energy may be converted into vibrational and rotational energy of the molecules, but otherwise molecular collisions appear to be elastic.

Collisions that are not elastic are known as inelastic collisions.

## Equations and calculation in the one-dimensional case

Total kinetic energy remains constant throughout, hence: ${\displaystyle m_{1}v_{1}^{2}+m_{2}v_{2}^{2}=m_{1}v_{1}'^{2}+m_{2}v_{2}'^{2}}$

Total momentum remains constant as well: ${\displaystyle \,\!m_{1}v_{1}+m_{2}v_{2}=m_{1}v_{1}'+m_{2}v_{2}'}$

Solving for ${\displaystyle v_{1}'}$ and ${\displaystyle v_{2}'}$ results in the following:

${\displaystyle v_{1}'={\frac {v_{1}(m_{1}-m_{2})+2m_{2}v_{2}}{m_{1}+m_{2}}}}$

and

${\displaystyle v_{2}'={\frac {v_{2}(m_{2}-m_{1})+2m_{1}v_{1}}{m_{1}+m_{2}}}}$

Property:

${\displaystyle v_{1}'-v_{2}'=v_{2}-v_{1}}$

i.e.:

• the relative velocity of one particle with respect to the other is reversed by the collision
• the average of the momenta before and after the collision is the same for both particles

As can be expected, the solution is invariant under adding a constant to all velocities, which is like using a frame of reference with constant translational velocity.

The velocity of the center of mass does not change by the collision. With respect to the center of mass both velocities are reversed by the collision: in the case of particles of different mass, a heavy particle moves slowly toward the center of mass, and bounces back with the same low speed, and a light particle moves fast toward the center of mass, and bounces back with the same high speed.

From the solution above we see that in the case of a large ${\displaystyle v_{1}}$, the value of ${\displaystyle v_{1}'}$ is small if the masses are approximately the same: hitting a much lighter particle does not change the velocity much, hitting a much heavier particle causes the fast particle to bounce back with high speed.

Therefore a neutron moderator (a medium which slows down fast neutrons, thereby turning them into thermal neutrons capable of sustaining a chain reaction) is a material full of atoms with light nuclei (with the additional property that they do not easily absorb neutrons): the lightest nuclei have about the same mass as a neutron.