Q.Discuss the following as special cases of elastic collisions and obtain their exact or approximate final velocities in terms of their initial velocities.

[!TLDR] (i) identical masses: velocities exchange, $v_1=u_2,\,v_2=u_1$. (ii) heavy striker on light rest body: striker barely slows, $v_1\approx u_1$, struck body flies off at $v_2\approx 2u_1$. (iii) light striker on heavy rest body: striker rebounds, $v_1\approx -u_1$, heavy body barely moves, $v …

Starting from the general head-on elastic-collision formulas (4.8.4), $v_1=\frac{m_1-m_2}{m_1+m_2}u_1+\frac{2m_2}{m_1+m_2}u_2$ and $v_2=\frac{2m_1}{m_1+m_2}u_1+\frac{m_2-m_1}{m_1+m_2}u_2$:

1. IDENTICAL colliding bodies, $m_1=m_2$: the first fraction in each formula vanishes ($\frac{m_1-m_2}{m_1+m_2}=0$) and the second becomes 1, giving $v_1=u_2$ and $v_2=u_1$ -- the two bodies simply EXCHANGE their velocities exactly.
2. A VERY HEAVY object ($m_1$) striking a much LIGHTER, initially-at-rest object ($m_2\ll m_1$, $u_2=0$): with $m_1\gg m_2$, $\frac{m_1-m_2}{m_1+m_2}\approx 1$ and $\frac{2m_1}{m_1+m_2}\approx 2$, giving $v_1\approx u_1$ (the heavy striker is essentially UNAFFECTED, continuing at almost its original speed) and $v_2\approx 2u_1$ (the light struck object flies off at roughly DOUBLE the striker's speed). …

Take the general elastic head-on collision formulas and apply the appropriate limiting approximations (equal masses; one mass much larger; one mass mu …

Applying the mass-ratio approximations without also setting $u_2=0$ where specified, or reversing which body is the 's …