Conservation of momentum with kinetic energy ratio

Steps:

• Initial momentum: $mu$; post-collision, let velocity of $m$ be $-v$ (rebounds), of $4m$ be $w$, so $-mv + 4mw = mu$.
• Initial KE of $m$: $\frac{1}{2}mu^2$; post-collision KE of $m$: , so .
• Substitute into momentum: $-m\frac{u}{2} + 4mw = mu$; simplify to $4w = \frac{3u}{2}$, so .

Why D is correct:

• Matches $w = \frac{3u}{8}$ from momentum conservation equation.

Why the others are wrong:

• A: $\frac{u}{8}$ ignores rebound velocity magnitude.
• B: $\frac{3u}{16}$ underestimates from incorrect KE-to-speed conversion.

Final answer: D