Elastic collisions conserve momentum and kinetic energy Steps:

• Compute mass ratio μ = M/m from momentum: μ = (v_{m,f} - v_{m,i}) / (v_{M,i} - v_{M,f}).
• Require μ > 0 for positive masses.
• Verify KE conservation: μ v_{M,i}^2 + v_{m,i}^2 = μ v_{M,f}^2 + v_{m,f}^2.
• Options B and D yield μ = 2 > 0 with equal KE; A and C fail momentum.

Why C is correct:

• μ = (8 - 0)/(2 - 5) = -8/3 < 0, violating momentum conservation for positive masses (stationary m cannot accelerate M forward).

Why the others are wrong:

• A: Denominator zero requires v_{m,f} = v_{m,i} for momentum conservation, but limit μ → ∞ approximates elastic case for M ≫ m despite v_{m,f} = 5 ≠ 4.
• B: μ = 2 > 0 and KE = 81 both sides, conserved.
• D: μ = 2 > 0 and KE = 216 both sides, conserved.

Final answer: C