Expressions for Final Velocities after a Head-on Elastic Collision

Combining the elastic-collision relation $v_2-v_1=u_1-u_2$ (from e=1) with conservation of momentum, $m_1u_1+m_2u_2=m_1v_1+m_2v_2$, and eliminating $v_2$, gives the general formulas for the final velocities after a head-on ELASTIC collision:

$v_1=\frac{m_1-m_2}{m_1+m_2}u_1+\frac{2m_2}{m_1+m_2}u_2$

$v_2=\frac{2m_1}{m_1+m_2}u_1+\frac{m_2-m_1}{m_1+m_2}u_2$

(the second obtained simply by interchanging the subscripts 1 and 2 in the first, since the labelling was arbitrary).

Three important PARTICULAR CASES follow directly:

1. EQUAL (or identical) masses, $m_1=m_2$: the formulas reduce to $v_1=u_2$ and $v_2=u_1$ -- the two bodies simply EXCHANGE velocities.
2. A much HEAVIER striking body colliding with a much lighter, initially-at-rest struck body ($m_1\gg m_2$, $u_2=0$): approximating $\frac{m_1-m_2}{m_1+m_2}\approx1$ and $\frac{2m_1}{m_1+m_2}\approx2$ gives $v_1\approx u_1$ and $v_2\approx 2u_1$ -- the massive striking body is practically unaffected, while the tiny struck body flies off at roughly DOUBLE the striking body's speed.
3. A much LIGHTER striking body colliding with a much heavier, initially-at-rest struck body ($m_1\ll m_2$, $u_2=0$): the formulas give $v_1\approx-u_1$ and $v_2\approx0$ -- the light object REBOUNDS with essentially the same speed, while the massive struck object is practically unaffected (as good as an elastic ball bouncing off the Earth's hard surface). …