Editing Elastic collision (section)

==Two-dimensional==

For the case of two non-spinning colliding bodies in two dimensions, the motion of the bodies is determined by the three conservation laws of momentum, kinetic energy and [[angular momentum]]. The overall velocity of each body must be split into two perpendicular velocities: one tangent to the common normal surfaces of the colliding bodies at the point of contact, the other along the line of collision. Since the collision only imparts force along the line of collision, the velocities that are tangent to the point of collision do not change.  The velocities along the line of collision can then be used in the same equations as a one-dimensional collision. The final velocities can then be calculated from the two new component velocities and will depend on the point of collision.  Studies of two-dimensional collisions are conducted for many bodies in the framework of a [[two-dimensional gas]].
[[Image:Elastischer stoß 2D.gif|frame|center|Two-dimensional elastic collision]]

In a [[center of momentum frame]] at any time the velocities of the two bodies are in opposite directions, with magnitudes inversely proportional to the masses. In an elastic collision these magnitudes do not change. The directions may change depending on the shapes of the bodies and the point of impact. For example, in the case of spheres the angle depends on the distance between the (parallel) paths of the centers of the two bodies. Any non-zero change of direction is possible: if this distance is zero the velocities are reversed in the collision; if it is close to the sum of the radii of the spheres the two bodies are only slightly deflected.

Assuming that the second particle is at rest before the collision, the angles of deflection of the two particles, <math>\theta_1</math> and <math>\theta_2</math>, are related to the angle of deflection <math>\theta</math> in the system of the center of mass by<ref>{{harvnb|Landau|Lifshitz|1976|p=[https://archive.org/details/mechanics00land/page/46 46]}}</ref>
<math display="block">\tan \theta_1=\frac{m_2 \sin \theta}{m_1+m_2 \cos \theta},\qquad
\theta_2=\frac{{\pi}-{\theta}}{2}.</math>
The magnitudes of the velocities of the particles after the collision are:
<math display="block">\begin{align}
v'_1 &= v_1\frac{\sqrt{m_1^2+m_2^2+2m_1m_2\cos \theta}}{m_1+m_2} \\
v'_2 &= v_1\frac{2m_1}{m_1+m_2}\sin \frac{\theta}{2}.
\end{align}</math>

===Two-dimensional collision with two moving objects===
The final x and y velocities components of the first ball can be calculated as:<ref>{{cite web |last= Craver |first= William E. |date= 13 August 2013 |title= Elastic Collisions |url= https://williamecraver.wixsite.com/elastic-equations |access-date= 4 March 2023}}{{Self-published source|date=March 2023}}</ref>
<math display="block">\begin{align}
v'_{1x} &= \frac{v_{1}\cos(\theta_1-\varphi)(m_1-m_2)+2m_2v_{2}\cos(\theta_2-\varphi)}{m_1+m_2}\cos(\varphi)+v_{1}\sin(\theta_1-\varphi)\cos(\varphi + \tfrac{\pi}{2})
\\[0.8em]
v'_{1y} &= \frac{v_{1}\cos(\theta_1-\varphi)(m_1-m_2)+2m_2v_{2}\cos(\theta_2-\varphi)}{m_1+m_2}\sin(\varphi)+v_{1}\sin(\theta_1-\varphi)\sin(\varphi + \tfrac{\pi}{2}),
\end{align}</math>
where {{math|''v''<sub>1</sub>}} and {{math|''v''<sub>2</sub>}} are the scalar sizes of the two original speeds of the objects, {{math|''m''<sub>1</sub>}} and {{math|''m''<sub>2</sub>}} are their masses, {{math|''θ''<sub>1</sub>}} and {{math|''θ''<sub>2</sub>}} are their movement angles, that is, <math>v_{1x} = v_1\cos\theta_1,\; v_{1y}=v_1\sin\theta_1</math> (meaning moving directly down to the right is either a −45° angle, or a 315° angle), and lowercase phi ({{mvar|φ}}) is the [[contact angle]]. (To get the {{mvar|x}} and {{mvar|y}} velocities of the second ball, one needs to swap all the '1' subscripts with '2' subscripts.)

This equation is derived from the fact that the interaction between the two bodies is easily calculated along the contact angle, meaning the velocities of the objects can be calculated in one dimension by rotating the x and y axis to be parallel with the contact angle of the objects, and then rotated back to the original orientation to get the true x and y components of the velocities.<ref>Parkinson, Stephen (1869) "An Elementary Treatise on Mechanics" (4th ed.) p. 197. London. MacMillan</ref><ref>Love, A. E. H. (1897) "Principles of Dynamics" p. 262.  Cambridge. Cambridge University Press</ref><ref>Routh, Edward J. (1898) "A Treatise on Dynamics of a Particle" p. 39. Cambridge. Cambridge University Press</ref><ref>Glazebrook, Richard T. (1911) "Dynamics" (2nd ed.) p. 217. Cambridge. Cambridge University Press</ref><ref>Osgood, William F. (1949) "Mechanics" p. 272. London. MacMillan</ref><ref>Stephenson, Reginald J. (1952) "Mechanics and Properties of Matter" p. 40. New York. Wiley</ref>

In an angle-free representation, the changed velocities are computed using the centers {{math|'''x'''<sub>1</sub>}} and {{math|'''x'''<sub>2</sub>}} at the time of contact as
{{NumBlk|:|<math display="block">\begin{align}
\mathbf{v}'_1 &= \mathbf{v}_1-\frac{2 m_2}{m_1+m_2} \ \frac{\langle \mathbf{v}_1-\mathbf{v}_2,\,\mathbf{x}_1-\mathbf{x}_2\rangle}{\|\mathbf{x}_1-\mathbf{x}_2\|^2} \ (\mathbf{x}_1-\mathbf{x}_2),
\\
\mathbf{v}'_2 &= \mathbf{v}_2-\frac{2 m_1}{m_1+m_2} \ \frac{\langle \mathbf{v}_2-\mathbf{v}_1,\,\mathbf{x}_2-\mathbf{x}_1\rangle}{\|\mathbf{x}_2-\mathbf{x}_1\|^2} \ (\mathbf{x}_2-\mathbf{x}_1)
\end{align}</math>|{{EquationRef|1}}}}
where the angle brackets indicate the [[inner product]] (or [[dot product]]) of two vectors.

===Other conserved quantities===
In the particular case of particles having equal masses, it can be verified by direct computation from the result above that the scalar product of the velocities before and after the collision are the same, that is <math>\langle \mathbf{v}'_1,\mathbf{v}'_2 \rangle = \langle \mathbf{v}_1,\mathbf{v}_2 \rangle.</math> Although this product is not an additive invariant in the same way that momentum and kinetic energy are for elastic collisions, it seems that preservation of this quantity can nonetheless be used to derive higher-order conservation laws.<ref>{{Cite journal | last1 = Chliamovitch | first1 = G. | last2 = Malaspinas | first2 = O. | last3 = Chopard | first3 = B. | doi = 10.3390/e19080381 | title = Kinetic theory beyond the Stosszahlansatz | journal = Entropy | volume = 19 | issue = 8 | year = 2017| page = 381 | bibcode = 2017Entrp..19..381C | doi-access = free }}</ref>

===Derivation of two dimensional solution===
The [[Impulse_(physics)|impulse <math>\mathbf J</math>]] during the collision for each particle is:
{{NumBlk|:|<math display="block">\begin{align}
\mathbf{p'_1}-\mathbf{p_1} &= \mathbf{J_1},
\\
\mathbf{p'_2}-\mathbf{p_2} &= \mathbf{J_2}
\end{align}</math>|{{EquationRef|2}}}}
Conservation of Momentum implies <math>\mathbf{J}\equiv\mathbf{J_1}=-\mathbf{J_2} </math>
<!--:<math>\mathbf{p'_1}+\mathbf{p'_2}=\mathbf{p_1}+\mathbf{p}_2</math>-->.

Since the force during collision is perpendicular to both particles' surfaces at the contact point, the impulse is along the line parallel to <math>\mathbf{x}_1-\mathbf{x}_2 \equiv\Delta \mathbf x </math>, the relative vector between the particles' center at collision time:
: <math>\mathbf J =\lambda\, \mathbf \hat n,</math> for some <math>\lambda</math> to be determined and <math>\mathbf \hat n \equiv \frac{\Delta \mathbf x}{\|\Delta \mathbf x\|}</math>
Then from ({{EquationRef|2}}):
{{NumBlk|:|<math display="block">\begin{align}
\mathbf{v'_1} &= \mathbf {v_1} + \frac{\lambda}{m_1} \mathbf \hat n,
\\
\mathbf{v'_2} &= \mathbf {v_2} - \frac{\lambda}{m_2} \mathbf \hat n
\end{align}</math>|{{EquationRef|3}}}}
From above equations, conservation of kinetic energy now requires:
:<math>\lambda^2\frac{m_1+m_2}{m_1 m_2}+2\lambda\,\langle \mathbf \hat n, \Delta \mathbf v\rangle = 0 ,\quad</math>with <math>\quad\Delta \mathbf v\equiv \mathbf{v}_1-\mathbf{v}_2.</math>
The both solutions of this equation are <math>\lambda = 0 </math> and <math>\lambda = -2 \frac{m_1 m_2}{m_1+m_2}\langle \mathbf \hat n, \Delta \mathbf v\rangle</math>, where <math>\lambda = 0 </math> corresponds to the trivial case of no collision. Substituting the non trivial value of <math>\lambda</math> in ({{EquationRef|3}}) we get the desired result ({{EquationRef|1}}).

Since all equations are in vectorial form, this derivation is valid also for three dimensions with spheres.