Editing Inelastic collision

{{Short description|Collision in which energy is lost to heat}}
{{More citations needed|date=February 2023}}
[[Image:Bouncing ball strobe edit.jpg|thumb|right|350px|A [[bouncing ball]] captured with a stroboscopic flash at 25 images per second. Each impact of the ball is inelastic, meaning that energy dissipates at each bounce. Ignoring [[air resistance]], the square root of the ratio of the height of one bounce to that of the preceding bounce gives the [[coefficient of restitution]] for the ball/surface impact.]]
An '''inelastic collision''', in contrast to an [[elastic collision]], is a [[collision]] in which kinetic energy is not conserved due to the action of [[Friction#Internal friction|internal friction]].

In collisions of macroscopic bodies, some [[kinetic energy]] is turned into vibrational energy of the [[atom]]s, causing a [[heat]]ing effect, and the bodies are deformed.

The [[molecule]]s of a [[gas]] or [[liquid]] rarely experience perfectly [[elastic collision]]s because kinetic energy is exchanged between the molecules' translational motion and their internal [[Degrees of freedom (physics and chemistry)|degrees of freedom]] with each collision. At any one instant, half the collisions are &ndash; to a varying extent &ndash; inelastic (the pair possesses less kinetic energy after the collision than before), and half could be described as “super-elastic” (possessing ''more'' kinetic energy after the collision than before). Averaged across an entire sample, molecular collisions are elastic. <ref>{{Cite journal |last=Hernandez |first=Hugo |date=2023 |title=Confusion and Illusions in Collision Theory |url=https://rgdoi.net/10.13140/RG.2.2.24913.10088 |journal=ForsChem Research Reports |language=en |volume=8 |doi=10.13140/RG.2.2.24913.10088 |access-date=25 August 2024 |via=ResearchGate}}</ref>

Although inelastic collisions do not conserve kinetic energy, they do obey [[conservation of momentum]].<ref>{{cite book| title = Vector equations for engineers: Dynamics | author = Ferdinand Beer Jr. and E. Russell Johnston | edition = Sixth | publisher = McGraw Hill | year = 1996 | pages = 794–797 | isbn = 978-0070053663 | quote = If the sum of the external forces is zero ... ''the total momentum of the particles is conserved''. ''In the general case of impact'', i.e., when ''e'' is not equal to 1, ''the total energy of the particles is not conserved''.}}</ref> Simple [[ballistic pendulum]] problems obey the conservation of kinetic energy ''only'' when the block swings to its largest angle.

In [[nuclear physics]], an inelastic collision is one in which the incoming [[subatomic particle|particle]] causes the [[atomic nucleus|nucleus]] it strikes to become [[excited state|excited]] or to break up. [[Deep inelastic scattering]] is a method of probing the structure of subatomic particles in much the same way as Rutherford probed the inside of the atom (see [[Rutherford scattering]]). Such experiments were performed on [[proton]]s in the late 1960s using high-energy [[electron]]s at the [[Stanford Linear Accelerator]] (SLAC). As in Rutherford scattering, deep inelastic scattering of electrons by proton targets revealed that most of the incident electrons interact very little and pass straight through, with only a small number bouncing back. This indicates that the charge in the proton is concentrated in small lumps, reminiscent of Rutherford's discovery that the [[electric charge|positive charge]] in an atom is concentrated at the nucleus. However, in the case of the proton, the evidence suggested three distinct concentrations of charge ([[quark]]s) and not one.

==Formula==

The formula for the velocities after a one-dimensional collision is:
<math display=block>
\begin{align}
v_a &= \frac{C_R m_b (u_b - u_a) + m_a u_a + m_b u_b} {m_a+m_b} \\
v_b &= \frac{C_R m_a (u_a - u_b) + m_a u_a + m_b u_b} {m_a+m_b}
\end{align}
</math>

where

*''v''<sub>a</sub> is the final velocity of the first object after impact
*''v''<sub>b</sub> is the final velocity of the second object after impact
*''u''<sub>a</sub> is the initial velocity of the first object before impact
*''u''<sub>b</sub> is the initial velocity of the second object before impact
*''m''<sub>a</sub> is the mass of the first object
*''m''<sub>b</sub> is the mass of the second object
*''C''<sub>R</sub> is the [[coefficient of restitution]]; if it is 1 we have an [[elastic collision]]; if it is 0 we have a perfectly inelastic collision, see below.

In a [[center of momentum frame]] the formulas reduce to:

<math display=block>
\begin{align}
v_a &= -C_R u_a \\
v_b &= -C_R u_b
\end{align}
</math>

For two- and three-dimensional collisions the velocities in these formulas are the components perpendicular to the tangent line/plane at the point of contact.

If assuming the objects are not rotating before or after the collision, the [[Normal (geometry)|normal]] [[Impulse (physics)|impulse]] is:

<math display=block>J_{n} = \frac{m_{a} m_{b}}{m_{a} + m_{b}} (1 + C_R) (\vec{u_{b}} - \vec{u_{a}}) \cdot \vec{n}</math>

where <math>\vec{n}</math> is the normal vector.

Assuming no friction, this gives the velocity updates:

<math display=block>
\begin{align}
\Delta \vec{v_{a}} &= \frac{J_{n}}{m_{a}} \vec{n} \\
\Delta \vec{v_{b}} &= -\frac{J_{n}}{m_{b}} \vec{n}
\end{align}
</math>

==Perfectly inelastic collision==
[[Image:Inelastischer stoß.gif|A completely inelastic collision between equal masses|frame|center]]

A '''perfectly inelastic collision''' occurs when the maximum amount of kinetic energy
of a system is lost. In a perfectly inelastic collision, i.e., a zero [[coefficient of restitution]],  the colliding particles stick together. In such a collision, kinetic energy is lost by bonding the two bodies together. This bonding energy usually results in a maximum kinetic energy loss of the system. It is necessary to consider conservation of momentum: (Note: In the sliding block example above, momentum of the two body system is only conserved if the surface has zero friction. With friction, momentum of the two bodies is transferred to the surface that the two bodies are sliding upon. Similarly, if there is air resistance, the momentum of the bodies can be transferred to the air.) The equation below holds true for the two-body (Body A, Body B) system collision in the example above. In this example, momentum of the system is conserved because there is no friction between the sliding bodies and the surface.
<math display=block>m_a u_a + m_b u_b = \left( m_a + m_b \right)  v </math>
where ''v'' is the final velocity, which is hence given by
<math display=block> v=\frac{m_a u_a + m_b  u_b}{m_a + m_b}</math>The reduction of total kinetic energy is equal to the total kinetic energy before the collision in a [[center of momentum frame]] with respect to the system of two particles, because in such a frame the kinetic energy after the collision is zero. In this frame most of the kinetic energy before the collision is that of the particle with the smaller mass. In another frame, in addition to the reduction of kinetic energy there may be a transfer of kinetic energy from one particle to the other; the fact that this depends on the frame shows how relative this is. The change in kinetic energy is hence:
<math display="block"> \Delta KE = {1\over 2}\mu u^2_{\rm rel} = \frac{1}{2}\frac{m_a m_b}{m_a + m_b}|u_a - u_b|^2 </math>

where μ is the [[reduced mass]] and u<sub>rel</sub> is the [[relative velocity]] of the bodies before collision. With time reversed we have the situation of two objects pushed away from each other, e.g. shooting a [[projectile]], or a [[rocket]] applying [[thrust]] (compare the [[Tsiolkovsky rocket equation#Derivation|derivation of the Tsiolkovsky rocket equation]]).

==Partially inelastic collisions==
Partially inelastic collisions are the most common form of collisions in the real world. In this type of collision, the objects involved in the collisions do not stick, but some kinetic energy is still lost. Friction, sound and heat are some ways the kinetic energy can be lost through partial inelastic collisions.

== See also ==
* [[Collision]]
* [[Elastic collision]]
* [[Coefficient of restitution]]


==References==
{{Reflist}}

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[[Category:Classical mechanics]]
[[Category:Collision]]
[[Category:Particle physics]]
[[Category:Scattering]]

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