= Elastic collision
{wiki}
An <elastic collision> preserves total <kinetic energy> between the incoming and outgoing states. For isolated colliding bodies, <momentum conservation> also holds. The <kinetic energy> condition supplies information beyond <momentum conservation> alone. https://openstax.org/books/university-physics-volume-1/pages/9-4-types-of-collisions[OpenStax University Physics Volume 1, section 9.4: Types of Collisions]{ref}
Consider a stipulated one-dimensional example with rightward positive: $m_1=2\,\mathrm{kg}$, $m_2=1\,\mathrm{kg}$, incoming <velocities> $u_1=3\,\mathrm{m/s}$ and $u_2=0$. The total <momentum> is $P=6\,\mathrm{kg\,m/s}$ and total <kinetic energy> is $K=9\,\mathrm J$. In <SI> numerical values, outgoing <velocities> $v_1,v_2$ satisfy
$$
2v_1+v_2=6,\qquad v_1^2+\frac12v_2^2=9.
$$
Substituting $v_2=6-2v_1$ gives $3(v_1-1)(v_1-3)=0$. The root $v_1=3$, $v_2=0$ repeats the incoming state; the outgoing separating state is $v_1=1\,\mathrm{m/s}$, $v_2=4\,\mathrm{m/s}$. Its <momenta> add to $2+4=6\,\mathrm{kg\,m/s}$ and its <kinetic energies> add to $1+8=9\,\mathrm J$. The incoming and outgoing <relative velocities> both have magnitude $3\,\mathrm{m/s}$, consistent with <coefficient of restitution> $e=1$.
\Image[/momentum-collision-before-after.png]
{title=The same incoming momentum gives different elastic and sticking collision outcomes}
The <inelastic collision> example below uses exactly these incoming data. Both outcomes preserve total <momentum>, while only the <elastic collision> preserves total <kinetic energy>. These are ideal numerical examples, not experimental measurements.
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