A particle of mass 0.1 kg moving with an initial speed v collides with another particle of same mass kept at rest. If after collision total energy becomes 0.2 J. Then

${\text{u}}_1={\text{v, u}}_2=0$

$\frac{1}{2}{\text{m}}_{\text{1}}{\text{v}}_{\text{1}}^{\text{2}}+\frac{1}{2}{\text{m}}_{\text{2}}{\text{v}}_{\text{2}}^{\text{2}}=0.2$

${\text{v}}_1^2+{\text{v}}_2^2=4\to \left(1\right)$

${\text{v}}_1=\frac{{\text{m}}_{\text{1}}-{\text{em}}_{\text{2}}}{{\text{m}}_{\text{1}}+{\text{m}}_{\text{2}}}{\text{u}}_{\text{1}}+\frac{\left(1+\text{e}\right){\text{m}}_{\text{2}}{\text{u}}_{\text{2}}}{{\text{m}}_{\text{1}}+{\text{m}}_{\text{2}}}$

${\text{v}}_1=\frac{\left(1-\text{e}\right)}{2}{\text{u}}_{\text{1}}\Rightarrow \left(\frac{1-\text{e}}{2}\right)\text{v}$

${\text{v}}_2=\frac{\left(1-\text{e}\right)}{2}{\text{u}}_{\text{2}}\Rightarrow \left(\frac{1-\text{e}}{2}\right)\text{v}$

From (1)

${\left(\frac{\text{1}-\text{e}}{\text{2}}\right)}^{\text{2}}{\text{v}}^{\text{2}}+{\left(\frac{\text{1}+\text{e}}{\text{2}}\right)}^{\text{2}}{\text{v}}^{\text{2}}=\left(4\right)$

$\frac{{\text{v}}^{\text{2}}}{\text{4}}\left[{\left(1-\text{e}\right)}^2+{\left(1+\text{e}\right)}^2\right]=4$

${\text{v}}^{\text{2}}\left[2\left(1+{\left(\text{e}\right)}^{\text{2}}\right)\right]=16$

${\text{v}}^{\text{2}}\left(1+{\left(\text{e}\right)}^{\text{2}}\right)=8$

${\text{v}}^{\text{2}}=\frac{8}{1+{\text{e}}^{\text{2}}}$

Elies between 0 & 1

$\Rightarrow {\text{e}}_{\max }=1$

${\text{e}}_{\min }=0$

If e is maximum v is minimum

If e is minimum v is maximum

$\text{e}=1\Rightarrow {\text{v}}_{\text{min}}=2$

$\text{e}=0\Rightarrow {\text{v}}_{\text{max}}=2\sqrt{2}$