As $\alpha$-particle is free to move, initial kinetic energy of system will be

$k_i=\frac{1}{2} \mu v^2$

where, $\mu=$ reduced mass of system

$=\frac{m \cdot 4 m}{m+4 m} \text {. }$

Now, by energy conservation, we have Initial kinetic energy = Potential energy at minimum separation $r$

$\frac{1}{2} (\frac{m .4 m}{m+4 m})v^2=\frac{1}{4 \pi \varepsilon_0} \cdot \frac{2 e^2}{r}$

$\Rightarrow r=\frac{5 e^2}{4 \pi \varepsilon_0 m v^2}$