Elastic collision
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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: <math>(1/2)m_{1}v_{1}^2+(1/2)m_{2}v_{2}^2=(1/2)m_{1}v_{1}'^2+(1/2)m_{2}v_{2}'^2</math>
Total momentum remains constant as well: <math>\,\! m_{1}v_{1}+m_{2}v_{2}=m_{1}v_{1}'+m_{2}v_{2}'</math>
Solving for <math>v_{1}'</math> and <math>v_{2}'</math> results in the following:
- <math>v_{1}' = \frac{v_{1}(m_{1}-m_{2})+2m_{2}v_{2}}{m_{1}+m_{2}}</math>
and
- <math>v_{2}' = \frac{v_{2}(m_{2}-m_{1})+2m_{1}v_{1}}{m_{1}+m_{2}}</math>
For example:
ball 1:
mass = 3 kg v = 4 m/s
ball 2:
mass = 5 kg v = −6 m/s
After collision:
Ball 1: v = −17.7 m/s
Ball 2: v = 5.4 m/s
Property:
- <math>v_{1}'-v_{2}' = v_{2}-v_{1}</math>
- 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 <math>v_{1}</math>, the value of <math>v_{1}'</math> 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.
Two-dimensional collisions
Newton's Rule (i.e. the conservation of momentum) applies to the components of velocity resolved along the common normal surfaces of the colliding bodies at the point of contact. In the case of the two spheres the velocity components involved are the components resolved along the line of centers during the contact. Consequently, the components of velocity perpendicular to the line of centers will be unchanged during the impact.
To solve an equation involving two colliding bodies in two-dimensions the overall velocity of each body must be split into two perpendicular velocities: one tangent to the point of collision, the other along the line of collision. The velocities that are tangent to the point of collision do not change, while the velocities along the line of collision are used in the same equations as a one-dimensional collision. The final velocity can then be calculated from the two new component velocities.
See also
es:Colisión elástica ru:Абсолютно упругий удар sl:Prožni trk zh:彈性碰撞