Editing Elastic collision (section)

===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.