Editing Momentum (section)

===Application to collisions===
If two particles, each of known momentum, collide and coalesce, the law of conservation of momentum can be used to determine the momentum of the coalesced body. If the outcome of the collision is that the two particles separate, the law is not sufficient to determine the momentum of each particle. If the momentum of one particle after the collision is known, the law can be used to determine the momentum of the other particle. Alternatively if the combined [[kinetic energy]] after the collision is known, the law can be used to determine the momentum of each particle after the collision.<ref>Resnick and Halliday (1966), ''Physics'', Section 10-3. Wiley Toppan, Library of Congress 66-11527</ref> Kinetic energy is usually not conserved. If it is conserved, the collision is called an ''[[elastic collision]]''; if not, it is an ''[[inelastic collision]]''.

====Elastic collisions====
{{Main|Elastic collision}}
[[File:Elastischer stoß.gif|thumb|right|Elastic collision of equal masses]]
[[File:Elastischer stoß3.gif|thumb|right|Elastic collision of unequal masses]]
An elastic collision is one in which no [[kinetic energy]] is transformed into heat or some other form of energy. Perfectly elastic collisions can occur when the objects do not touch each other, as for example in atomic or nuclear scattering where electric repulsion keeps the objects apart. A [[gravity assist|slingshot maneuver]] of a satellite around a planet can also be viewed as a perfectly elastic collision. A collision between two [[Pool billiards|pool]] balls is a good example of an ''almost'' totally elastic collision, due to their high [[stiffness|rigidity]], but when bodies come in contact there is always some [[dissipation]].<ref>{{cite web |url=http://hyperphysics.phy-astr.gsu.edu/hbase/elacol.html |title=Elastic and inelastic collisions |work=Hyperphysics |first=Carl |last=Nave |date=2010 |access-date=2 August 2012  |archive-url=https://web.archive.org/web/20120818114930/http://hyperphysics.phy-astr.gsu.edu/hbase/elacol.html |archive-date=18 August 2012 }}</ref>

A head-on elastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. If the velocities are {{math|{{var|v}}{{sub|A1}}}} and {{math|{{var|v}}{{sub|B1}}}} before the collision and {{math|{{var|v}}{{sub|A2}}}} and {{math|{{var|v}}{{sub|B2}}}} after, the equations expressing conservation of momentum and kinetic energy are:

<math display="block">\begin{align} m_{A} v_{A1} + m_{B} v_{B1} &= m_{A} v_{A2} + m_{B} v_{B2}\\
\tfrac{1}{2} m_{A} v_{A1}^2 + \tfrac{1}{2} m_{B} v_{B1}^2 &= \tfrac{1}{2} m_{A} v_{A2}^2 + \tfrac{1}{2} m_{B} v_{B2}^2\,.\end{align}</math>

A change of reference frame can simplify analysis of a collision. For example, suppose there are two bodies of equal mass {{mvar|m}}, one stationary and one approaching the other at a speed {{mvar|v}} (as in the figure). The center of mass is moving at speed {{math|{{sfrac|{{var|v}}|2}}}} and both bodies are moving towards it at speed {{math|{{sfrac|{{var|v}}|2}}}}. Because of the symmetry, after the collision both must be moving away from the center of mass at the same speed. Adding the speed of the center of mass to both, we find that the body that was moving is now stopped and the other is moving away at speed {{mvar|v}}. The bodies have exchanged their velocities. Regardless of the velocities of the bodies, a switch to the center of mass frame leads us to the same conclusion. Therefore, the final velocities are given by<ref name=FeynmanCh10/>

<math display="block">\begin{align}  v_{A2} &= v_{B1}\\
v_{B2} &= v_{A1}\,. \end{align}</math>

In general, when the initial velocities are known, the final velocities are given by<ref>{{cite book|last1=Serway|first1=Raymond A.|first2=John W. Jr. |last2=Jewett |title=Principles of physics: a calculus-based text|date=2012|publisher=Brooks/Cole, Cengage Learning|location=Boston, Massachusetts|isbn=978-1-133-10426-1|page=245|edition=5th}}</ref>

<math display="block">\begin{align}
 v_{A2} &= \left( \frac{m_{A} - m_{B}}{m_{A} + m_{B}} \right) v_{A1} + \left( \frac{2 m_{B}}{m_{A} + m_{B}} \right) v_{B1} \\
 v_{B2} &= \left( \frac{m_{B} - m_{A}}{m_{A} + m_{B}} \right) v_{B1} + \left( \frac{2 m_{A}}{m_{A} + m_{B}} \right) v_{A1}\,.
\end{align}</math>

If one body has much greater mass than the other, its velocity will be little affected by a collision while the other body will experience a large change.

====Inelastic collisions====
{{Main|Inelastic collision}}

[[File:Inelastischer stoß.gif|thumb|right|a perfectly inelastic collision between equal masses]]
In an inelastic collision, some of the kinetic energy of the colliding bodies is converted into other forms of energy (such as [[heat]] or [[sound]]). Examples include [[traffic collisions]],<ref>{{cite web |url=http://hyperphysics.phy-astr.gsu.edu/hbase/carcr.html#cc1 |title=Forces in car crashes |work=Hyperphysics |first=Carl |last=Nave |date=2010 |access-date=2 August 2012 |url-status=live |archive-url=https://web.archive.org/web/20120822034313/http://hyperphysics.phy-astr.gsu.edu/hbase/carcr.html#cc1 |archive-date=22 August 2012 }}</ref> in which the effect of loss of kinetic energy can be seen in the damage to the vehicles; electrons losing some of their energy to atoms (as in the [[Franck–Hertz experiment]]);<ref>{{cite web |url=http://hyperphysics.phy-astr.gsu.edu/hbase/FrHz.html |title=The Franck-Hertz Experiment |work=Hyperphysics |first=Carl |last=Nave |date=2010 |access-date=2 August 2012 |url-status=live |archive-url=https://web.archive.org/web/20120716180316/http://hyperphysics.phy-astr.gsu.edu/hbase/FrHz.html |archive-date=16 July 2012 }}</ref> and [[particle accelerator]]s in which the kinetic energy is converted into mass in the form of new particles.

In a perfectly inelastic collision (such as a bug hitting a windshield), both bodies have the same motion afterwards. A head-on inelastic collision between two bodies can be represented by velocities in one dimension, along a line passing through the bodies. If the velocities are {{math|{{var|v}}{{sub|A1}}}} and {{math|{{var|v}}{{sub|B1}}}} before the collision then in a perfectly inelastic collision both bodies will be travelling with velocity {{mvar|v}}{{sub|2}} after the collision. The equation expressing conservation of momentum is:

<math display="block">\begin{align} m_A v_{A1} + m_B v_{B1} &= \left( m_A + m_B \right) v_2\,.\end{align}</math>

If one body is motionless to begin with (e.g. <math> u_2 = 0 </math>), the equation for conservation of momentum is

<math display="block">m_A v_{A1} = \left( m_A + m_B \right) v_2\,,</math>

so

<math display="block"> v_2 = \frac{m_{A}}{m_{A}+m_{B}} v_{A1}\,.</math>

In a different situation, if the frame of reference is moving at the final velocity such that <math> v_2 = 0 </math>, the objects would be brought to rest by a perfectly inelastic collision and 100% of the kinetic energy is converted to other forms of energy. In this instance the initial velocities of the bodies would be non-zero, or the bodies would have to be massless.

One measure of the inelasticity of the collision is the [[coefficient of restitution]] {{math|{{var|C}}{{sub|R}}}}, defined as the ratio of relative velocity of separation to relative velocity of approach. In applying this measure to a ball bouncing from a solid surface, this can be easily measured using the following formula:<ref>{{cite book|last=McGinnis|first=Peter M.|title=Biomechanics of sport and exercise|date=2005|publisher=Human Kinetics|location=Champaign, Illinois |isbn=978-0-7360-5101-9|page=85|edition=2nd|url=https://books.google.com/books?id=PrOKEcZXJ58C&q=coefficient+of+restitution+bounciness&pg=PA85|url-status=live|archive-url=https://web.archive.org/web/20160819020542/https://books.google.com/books?id=PrOKEcZXJ58C&pg=PA85&lpg=PA85&dq=coefficient+of+restitution+bounciness|archive-date=2016-08-19}}</ref>

<math display="block">C_\text{R} = \sqrt{\frac{\text{bounce height}}{\text{drop height}}}\,.</math>

The momentum and energy equations also apply to the motions of objects that begin together and then move apart. For example, an [[explosion]] is the result of a chain reaction that transforms potential energy stored in chemical, mechanical, or nuclear form into kinetic energy, acoustic energy, and electromagnetic radiation. [[Rocket]]s also make use of conservation of momentum: propellant is thrust outward, gaining momentum, and an equal and opposite momentum is imparted to the rocket.<ref>{{cite book | last = Sutton | first = George | title = Rocket Propulsion Elements |edition=7th |chapter-url=https://books.google.com/books?id=LQbDOxg3XZcC | publisher = John Wiley & Sons | location = Chichester | date = 2001 | isbn = 978-0-471-32642-7 |chapter=Chapter 1: Classification}}</ref>