A object of mass m₁ collides with another object of mass m₂ , which is at rest. After the collision the objects move with equal speeds in opposite direction. The ratio of the masses m₂ :m₁ is :

After the collision the objects move in opposite direction let with velocity v₁ then from law of conservation of momentum

 $$\begin{gathered} {\mathrm{P}}_{\mathrm{i}}={\mathrm{P}}_{\mathrm{f}} \\ \Rightarrow {\mathrm{m}}_1\mathrm{v}=\left({\mathrm{m}}_2-{\mathrm{m}}_1\right){\mathrm{v}}_1 \end{gathered}$$
 

$\Rightarrow {\mathrm{v}}_1=\frac{{\mathrm{m}}_1\mathrm{v}}{\left({\mathrm{m}}_2-{\mathrm{m}}_1\right)}\dots \dots \dots \left(\mathrm{i}\right)$
If collision is elastic, then, $\mathrm{e}=1=\frac{{\mathrm{v}}_1-\left(-{\mathrm{v}}_1\right)}{\mathrm{v}-0}\mathrm{A}\text{lso, }2{\mathrm{v}}_1=\mathrm{v}.....\left(i\mathrm{i}\right)$

From equations (i) & (ii) $\frac{1}{2}=\frac{{\mathrm{m}}_1}{{\mathrm{m}}_2-{\mathrm{m}}_1}\therefore \frac{{\mathrm{m}}_2}{{\mathrm{m}}_1}=\frac{3}{1}$