In a perfectly elastic collision, both momentum and kinetic energy are conserved. Let mass of A be m₁ and B be m₂ (stationary). After collision, velocities are equal and opposite: v and -v.

Conservation of momentum: $m\mathrm{₁}{u}_1=m\mathrm{₁}v+m\mathrm{₂}(-v)$

Conservation of kinetic energy: $\frac{1}{2}m\mathrm{₁}{u}_1^2=\frac{1}{2}m\mathrm{₁}{v}^2+\frac{1}{2}m\mathrm{₂}{v}^2$

Solving these equations, we find the mass ratio $\frac{m\mathrm{₁}}{m\mathrm{₂}}=\frac{1}{3}$.

Final answer: 1/3