A body of mass m₁ collides elastically with another body of mass m₂ at rest. The maximum  transfer of energy takes place when

Given that

A body of mass  m₁ collides elastically with another body of mass m₂ at rest.

$\text{ First body has mass }=m_1$

$\text{ Initial velocity }={\mathrm{u}}_1$

$\text{ Final velocity }={\mathrm{v}}_1$

$\text{ Second body has mass }={\mathrm{m}}_2$

$\text{ Initial velocity }={\mathrm{u}}_2$

$\text{ Final velocity }={\mathrm{v}}_2$

We can have

$\begin{array}{r}{\mathrm{v}}_1=\left(\frac{{\mathrm{m}}_1-{\mathrm{m}}_2}{{\mathrm{m}}_1+{\mathrm{m}}_2}\right){\mathrm{u}}_1\\ {\mathrm{v}}_2=\left(\frac{2{\mathrm{m}}_1}{{\mathrm{m}}_1+{\mathrm{m}}_2}\right){\mathrm{u}}_1\end{array}$

$\frac{{\mathrm{v}}_2}{{\mathrm{v}}_1}=\frac{2{\mathrm{m}}_1}{{\mathrm{m}}_1-{\mathrm{m}}_2}$

To get maximum transfer of energy we need  v₁ to become zero

$\Rightarrow \left(\frac{{\mathrm{m}}_1-{\mathrm{m}}_2}{{\mathrm{m}}_1+{\mathrm{m}}_2}\right){\mathrm{u}}_1=0$

we get

$\begin{array}{r}\therefore {\mathrm{m}}_1-{\mathrm{m}}_2=0\\ \Rightarrow {\mathrm{m}}_1={\mathrm{m}}_2\end{array}$

Then we can have maximum transfer of energy taking place.